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(TOPS-3E:NU) Test of Problem Solving - Elementary, Third Edition Normative Update

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The TOPS-3E: NU focuses on a student’s linguistic ability to think and reason. Language competence is the verbal indicator of how a student’s language skills affect his or her ability to think, reason, problem solve, infer, classify, associate, predict, determine causes, sequence, and understand directions. The test focuses on a broad range of language-based thinking skills, including clarifying, analyzing, generating solutions, evaluating, and showing affective thinking.

While other tests may assess students’ thinking skills by tapping mathematical, spatial, or nonverbal potential, the TOPS-3E: NU measures discreet skills that form the foundation of language-based thinking, reasoning, and problem-solving ability.

Although the skills tested by the TOPS-3E: NU are necessary for developing social competence, it is not primarily a test of pragmatic or social language skills. Rather, it should be part of a battery of tests and observations used to assess pragmatic competence.

New features:

• Characteristics of the normative sample were stratified by age relative to region, gender, ethnicity, and socioeconomic factors.
• The Total Score was renamed the Problem Solving Index and calculated as a standard score with a mean of 100 and a standard deviation of 15.
• Each item on the test was evaluated using both conventional item analysis to choose “good” items and differential item analysis to find and eliminate potentially biased items.
• The index score was thoroughly examined for floor and ceiling effects.
• The test was subjected to diagnostic accuracy analyses, particularly rigorous techniques involving the computation of the receiver operating characteristic/area under the curve (ROC/AUC) statistic.
• The Manual was reorganized and rewritten to provide more detailed information on the administration, interpretation, and statistical characteristics of the test.
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The upshot | a quick puzzle to test your problem solving.

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A Quick Puzzle to Test Your Problem Solving

By DAVID LEONHARDT and YOU JULY 2, 2015

A short game sheds light on government policy, corporate America and why no one likes to be wrong. Related Article

Here’s how it works:

We’ve chosen a rule that some sequences of three numbers obey — and some do not. Your job is to guess what the rule is.

We’ll start by telling you that the sequence 2, 4, 8 obeys the rule:

Obeys the rule

Now it’s your turn. Enter a number sequence in the boxes below, and we’ll tell you whether it satisfies the rule or not. You can test as many sequences as you want.

Enter your first sequence here:

I don't want to play; just tell me the answer.

Guess wrong

The answer was extremely basic. The rule was simply: Each number must be larger than the one before it. 5, 10, 20 satisfies the rule, as does 1, 2, 3 and -17, 14.6, 845. Children in kindergarten can understand this rule.

But most people start off with the incorrect assumption that if we’re asking them to solve a problem, it must be a somewhat tricky problem. They come up with a theory for what the answer is, like: Each number is double the previous number. And then they make a classic psychological mistake.

They don’t want to hear the answer “no.” In fact, it may not occur to them to ask a question that may yield a no.

Remarkably, 80 percent of people who have played this game so far have guessed the answer without first hearing a single no. A mere 7 percent heard at least three nos — even though there is no penalty or cost for being told no, save the small disappointment that every human being feels when hearing “no.”

It’s a lot more pleasant to hear “yes.” That, in a nutshell, is why so many people struggle with this problem.

Confirmation Bias

This disappointment is a version of what psychologists and economists call confirmation bias. Not only are people more likely to believe information that fits their pre-existing beliefs, but they’re also more likely to go looking for such information. This experiment is a version of one that the English psychologist Peter Cathcart Wason used in a seminal 1960 paper on confirmation bias. (He used the even simpler 2, 4 and 6, rather than our 2, 4 and 8.)

Most of us can quickly come up with other forms of confirmation bias — and yet the examples we prefer tend to be, themselves, examples of confirmation bias. If you’re politically liberal, maybe you’re thinking of the way that many conservatives ignore strong evidence of global warming and its consequences and instead glom onto weaker contrary evidence. Liberals are less likely to recall the many incorrect predictions over the decades, often strident and often from the left, that population growth would create widespread food shortages. It hasn’t.

This puzzle exposes a particular kind of confirmation bias that bedevils companies, governments and people every day: the internal yes-man (and yes-woman) tendency. We’re much more likely to think about positive situations than negative ones, about why something might go right than wrong and about questions to which the answer is yes, not no.

Sometimes, the reluctance to think negatively has nothing to do with political views or with a conscious fear of being told no. Often, people never even think about asking questions that would produce a negative answer when trying to solve a problem — like this one. They instead restrict the universe of possible questions to those that might potentially yield a “yes.”

Government Policy

In this exercise, the overwhelming majority of readers gravitated toward confirming their theory rather than trying to disprove it. A version of this same problem compromised the Obama administration’s and Federal Reserve’s (mostly successful) response to the financial crisis. They were too eager to find “green shoots” of economic recovery that would suggest that the answer to the big question in their minds was, just as they hoped and believed: “Yes, the crisis response is aggressive enough, and it’s working.” More damaging was the approach that President George W. Bush’s administration, and others, took toward trying to determine whether Iraq had weapons of mass destruction a decade ago — and how the Iraqi people would react to an invasion. Vice President Dick Cheney predicted in 2003, “We will, in fact, be greeted as liberators.”

Corporate America

Corporate America is full of more examples. Executives of Detroit’s Big Three didn’t spend enough time brainstorming in the 1970s and 1980s about how their theory of the car market might be wrong. Wall Street and the Fed made the same mistake during the dot-com and housing bubbles. To pick an example close to home, newspapers didn’t spend enough time challenging the assumption that classified advertisements would remain plentiful for decades.

One of the best-selling business books in history — about negotiation strategy — is “Getting to Yes.” But the more important advice for us may instead be to go out of our way to get to no. When you want to test a theory, don’t just look for examples that prove it. When you’re considering a plan, think in detail about how it might go wrong.

Some businesses have made this approach a formal part of their decision-making: Imagine our strategy has failed; what are the most likely reasons it did? As Jason Zweig has written in The Wall Street Journal, “Gary Klein, a psychologist at Applied Research Associates, of Albuquerque, N.M., recommends imagining that you have looked into a crystal ball and have seen that your investment has gone bust.”

When you seek to disprove your idea, you sometimes end up proving it — and other times you can save yourself from making a big mistake. But you need to start by being willing to hear no. And even if you think that you are right, you need to make sure you’re asking questions that might actually produce an answer of no. If you still need to work on this trait, don’t worry: You’re only human.

Guess right

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Unit 1: Place value

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Self-Assessment • 20 min read

How Good Is Your Problem Solving?

Use a systematic approach..

By the Mind Tools Content Team

Good problem solving skills are fundamentally important if you're going to be successful in your career.

But problems are something that we don't particularly like.

They're time-consuming.

They muscle their way into already packed schedules.

They force us to think about an uncertain future.

And they never seem to go away!

That's why, when faced with problems, most of us try to eliminate them as quickly as possible. But have you ever chosen the easiest or most obvious solution – and then realized that you have entirely missed a much better solution? Or have you found yourself fixing just the symptoms of a problem, only for the situation to get much worse?

To be an effective problem-solver, you need to be systematic and logical in your approach. This quiz helps you assess your current approach to problem solving. By improving this, you'll make better overall decisions. And as you increase your confidence with solving problems, you'll be less likely to rush to the first solution – which may not necessarily be the best one.

Once you've completed the quiz, we'll direct you to tools and resources that can help you make the most of your problem-solving skills.

How Good Are You at Solving Problems?

Instructions.

For each statement, click the button in the column that best describes you. Please answer questions as you actually are (rather than how you think you should be), and don't worry if some questions seem to score in the 'wrong direction'. When you are finished, please click the 'Calculate My Total' button at the bottom of the test.

Answering these questions should have helped you recognize the key steps associated with effective problem solving.

This quiz is based on Dr Min Basadur's Simplexity Thinking problem-solving model. This eight-step process follows the circular pattern shown below, within which current problems are solved and new problems are identified on an ongoing basis. This assessment has not been validated and is intended for illustrative purposes only.

Below, we outline the tools and strategies you can use for each stage of the problem-solving process. Enjoy exploring these stages!

Step 1: Find the Problem (Questions 7, 12)

Some problems are very obvious, however others are not so easily identified. As part of an effective problem-solving process, you need to look actively for problems – even when things seem to be running fine. Proactive problem solving helps you avoid emergencies and allows you to be calm and in control when issues arise.

These techniques can help you do this:

PEST Analysis helps you pick up changes to your environment that you should be paying attention to. Make sure too that you're watching changes in customer needs and market dynamics, and that you're monitoring trends that are relevant to your industry.

Risk Analysis helps you identify significant business risks.

Failure Modes and Effects Analysis helps you identify possible points of failure in your business process, so that you can fix these before problems arise.

After Action Reviews help you scan recent performance to identify things that can be done better in the future.

Where you have several problems to solve, our articles on Prioritization and Pareto Analysis help you think about which ones you should focus on first.

Step 2: Find the Facts (Questions 10, 14)

After identifying a potential problem, you need information. What factors contribute to the problem? Who is involved with it? What solutions have been tried before? What do others think about the problem?

If you move forward to find a solution too quickly, you risk relying on imperfect information that's based on assumptions and limited perspectives, so make sure that you research the problem thoroughly.

Step 3: Define the Problem (Questions 3, 9)

Now that you understand the problem, define it clearly and completely. Writing a clear problem definition forces you to establish specific boundaries for the problem. This keeps the scope from growing too large, and it helps you stay focused on the main issues.

A great tool to use at this stage is CATWOE . With this process, you analyze potential problems by looking at them from six perspectives, those of its Customers; Actors (people within the organization); the Transformation, or business process; the World-view, or top-down view of what's going on; the Owner; and the wider organizational Environment. By looking at a situation from these perspectives, you can open your mind and come to a much sharper and more comprehensive definition of the problem.

Cause and Effect Analysis is another good tool to use here, as it helps you think about the many different factors that can contribute to a problem. This helps you separate the symptoms of a problem from its fundamental causes.

Step 4: Find Ideas (Questions 4, 13)

With a clear problem definition, start generating ideas for a solution. The key here is to be flexible in the way you approach a problem. You want to be able to see it from as many perspectives as possible. Looking for patterns or common elements in different parts of the problem can sometimes help. You can also use metaphors and analogies to help analyze the problem, discover similarities to other issues, and think of solutions based on those similarities.

Traditional brainstorming and reverse brainstorming are very useful here. By taking the time to generate a range of creative solutions to the problem, you'll significantly increase the likelihood that you'll find the best possible solution, not just a semi-adequate one. Where appropriate, involve people with different viewpoints to expand the volume of ideas generated.

Tip: Don't evaluate your ideas until step 5. If you do, this will limit your creativity at too early a stage.

Step 5: Select and Evaluate (Questions 6, 15)

After finding ideas, you'll have many options that must be evaluated. It's tempting at this stage to charge in and start discarding ideas immediately. However, if you do this without first determining the criteria for a good solution, you risk rejecting an alternative that has real potential.

Decide what elements are needed for a realistic and practical solution, and think about the criteria you'll use to choose between potential solutions.

Paired Comparison Analysis , Decision Matrix Analysis and Risk Analysis are useful techniques here, as are many of the specialist resources available within our Decision-Making section . Enjoy exploring these!

Step 6: Plan (Questions 1, 16)

You might think that choosing a solution is the end of a problem-solving process. In fact, it's simply the start of the next phase in problem solving: implementation. This involves lots of planning and preparation. If you haven't already developed a full Risk Analysis in the evaluation phase, do so now. It's important to know what to be prepared for as you begin to roll out your proposed solution.

The type of planning that you need to do depends on the size of the implementation project that you need to set up. For small projects, all you'll often need are Action Plans that outline who will do what, when, and how. Larger projects need more sophisticated approaches – you'll find out more about these in the article What is Project Management? And for projects that affect many other people, you'll need to think about Change Management as well.

Here, it can be useful to conduct an Impact Analysis to help you identify potential resistance as well as alert you to problems you may not have anticipated. Force Field Analysis will also help you uncover the various pressures for and against your proposed solution. Once you've done the detailed planning, it can also be useful at this stage to make a final Go/No-Go Decision , making sure that it's actually worth going ahead with the selected option.

Step 7: Sell the Idea (Questions 5, 8)

As part of the planning process, you must convince other stakeholders that your solution is the best one. You'll likely meet with resistance, so before you try to “sell” your idea, make sure you've considered all the consequences.

As you begin communicating your plan, listen to what people say, and make changes as necessary. The better the overall solution meets everyone's needs, the greater its positive impact will be! For more tips on selling your idea, read our article on Creating a Value Proposition and use our Sell Your Idea Skillbook.

Step 8: Act (Questions 2, 11)

Finally, once you've convinced your key stakeholders that your proposed solution is worth running with, you can move on to the implementation stage. This is the exciting and rewarding part of problem solving, which makes the whole process seem worthwhile.

This action stage is an end, but it's also a beginning: once you've completed your implementation, it's time to move into the next cycle of problem solving by returning to the scanning stage. By doing this, you'll continue improving your organization as you move into the future.

Problem solving is an exceptionally important workplace skill.

Being a competent and confident problem solver will create many opportunities for you. By using a well-developed model like Simplexity Thinking for solving problems, you can approach the process systematically, and be comfortable that the decisions you make are solid.

Given the unpredictable nature of problems, it's very reassuring to know that, by following a structured plan, you've done everything you can to resolve the problem to the best of your ability.

This assessment has not been validated and is intended for illustrative purposes only. It is just one of many Mind Tool quizzes that can help you to evaluate your abilities in a wide range of important career skills.

If you want to reproduce this quiz, you can purchase downloadable copies in our Store .

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How Good Is Your Problem Solving?

© iStockphoto Entienou

Use a systematic approach.

Good problem solving skills are fundamentally important if you're going to be successful in your career.

But problems are something that we don't particularly like.

They're time-consuming.

They muscle their way into already packed schedules.

They force us to think about an uncertain future.

And they never seem to go away!

That's why, when faced with problems, most of us try to eliminate them as quickly as possible. But have you ever chosen the easiest or most obvious solution – and then realized that you have entirely missed a much better solution? Or have you found yourself fixing just the symptoms of a problem, only for the situation to get much worse?

To be an effective problem-solver, you need to be systematic and logical in your approach. This quiz helps you assess your current approach to problem solving. By improving this, you'll make better overall decisions. And as you increase your confidence with solving problems, you'll be less likely to rush to the first solution – which may not necessarily be the best one.

Once you've completed the quiz, we'll direct you to tools and resources that can help you make the most of your problem-solving skills.

How Good Are You at Solving Problems?

Instructions.

For each statement, click the button in the column that best describes you. Please answer questions as you actually are (rather than how you think you should be), and don't worry if some questions seem to score in the 'wrong direction'. When you are finished, please click the 'Calculate My Total' button at the bottom of the test.

Your last quiz results are shown.

You last completed this quiz on , at .

Score Interpretation

Answering these questions should have helped you recognize the key steps associated with effective problem solving.

This quiz is based on Dr Min Basadur's Simplexity Thinking    problem-solving model. This eight-step process follows the circular pattern shown below, within which current problems are solved and new problems are identified on an ongoing basis. This assessment has not been validated and is intended for illustrative purposes only. 

Figure 1 – The Simplexity Thinking Process

Reproduced with permission from Dr Min Basadur from "The Power of Innovation: How to Make Innovation a Part of Life & How to Put Creative Solutions to Work" Copyright ©1995

Below, we outline the tools and strategies you can use for each stage of the problem-solving process. Enjoy exploring these stages!

Step 1: Find the Problem

(Questions 7, 12)

Some problems are very obvious, however others are not so easily identified. As part of an effective problem-solving process, you need to look actively for problems – even when things seem to be running fine. Proactive problem solving helps you avoid emergencies and allows you to be calm and in control when issues arise.

These techniques can help you do this:

• PEST Analysis   helps you pick up changes to your environment that you should be paying attention to. Make sure too that you're watching changes in customer needs and market dynamics, and that you're monitoring trends that are relevant to your industry.
• Risk Analysis   helps you identify significant business risks.
• Failure Modes and Effects Analysis   helps you identify possible points of failure in your business process, so that you can fix these before problems arise.
• After Action Reviews   help you scan recent performance to identify things that can be done better in the future.
• Where you have several problems to solve, our articles on Prioritization   and Pareto Analysis   help you think about which ones you should focus on first.

Step 2: Find the Facts

(Questions 10, 14)

After identifying a potential problem, you need information. What factors contribute to the problem? Who is involved with it? What solutions have been tried before? What do others think about the problem?

If you move forward to find a solution too quickly, you risk relying on imperfect information that's based on assumptions and limited perspectives, so make sure that you research the problem thoroughly.

Step 3: Define the Problem

(Questions 3, 9)

Now that you understand the problem, define it clearly and completely. Writing a clear problem definition forces you to establish specific boundaries for the problem. This keeps the scope from growing too large, and it helps you stay focused on the main issues.

A great tool to use at this stage is CATWOE   . With this process, you analyze potential problems by looking at them from six perspectives, those of its Customers; Actors (people within the organization); the Transformation, or business process; the World-view, or top-down view of what's going on; the Owner; and the wider organizational Environment. By looking at a situation from these perspectives, you can open your mind and come to a much sharper and more comprehensive definition of the problem.

Cause and Effect Analysis   is another good tool to use here, as it helps you think about the many different factors that can contribute to a problem. This helps you separate the symptoms of a problem from its fundamental causes.

Step 4: Find Ideas

(Questions 4, 13)

With a clear problem definition, start generating ideas for a solution. The key here is to be flexible in the way you approach a problem. You want to be able to see it from as many perspectives as possible. Looking for patterns or common elements in different parts of the problem can sometimes help. You can also use metaphors   and analogies to help analyze the problem, discover similarities to other issues, and think of solutions based on those similarities.

Traditional brainstorming   and reverse brainstorming   are very useful here. By taking the time to generate a range of creative solutions to the problem, you'll significantly increase the likelihood that you'll find the best possible solution, not just a semi-adequate one. Where appropriate, involve people with different viewpoints to expand the volume of ideas generated.

Don't evaluate your ideas until step 5. If you do, this will limit your creativity at too early a stage.

Step 5: Select and Evaluate

(Questions 6, 15)

After finding ideas, you'll have many options that must be evaluated. It's tempting at this stage to charge in and start discarding ideas immediately. However, if you do this without first determining the criteria for a good solution, you risk rejecting an alternative that has real potential.

Decide what elements are needed for a realistic and practical solution, and think about the criteria you'll use to choose between potential solutions.

Paired Comparison Analysis   , Decision Matrix Analysis   and Risk Analysis   are useful techniques here, as are many of the specialist resources available within our Decision-Making section . Enjoy exploring these!

Step 6: Plan

(Questions 1, 16)

You might think that choosing a solution is the end of a problem-solving process. In fact, it's simply the start of the next phase in problem solving: implementation. This involves lots of planning and preparation. If you haven't already developed a full Risk Analysis   in the evaluation phase, do so now. It's important to know what to be prepared for as you begin to roll out your proposed solution.

The type of planning that you need to do depends on the size of the implementation project that you need to set up. For small projects, all you'll often need are Action Plans   that outline who will do what, when, and how. Larger projects need more sophisticated approaches – you'll find out more about these in the Mind Tools Project Management section. And for projects that affect many other people, you'll need to think about Change Management   as well.

Here, it can be useful to conduct an Impact Analysis   to help you identify potential resistance as well as alert you to problems you may not have anticipated. Force Field Analysis   will also help you uncover the various pressures for and against your proposed solution. Once you've done the detailed planning, it can also be useful at this stage to make a final Go/No-Go Decision   , making sure that it's actually worth going ahead with the selected option.

Step 7: Sell the Idea

(Questions 5, 8)

As part of the planning process, you must convince other stakeholders that your solution is the best one. You'll likely meet with resistance, so before you try to “sell” your idea, make sure you've considered all the consequences.

As you begin communicating your plan, listen to what people say, and make changes as necessary. The better the overall solution meets everyone's needs, the greater its positive impact will be! For more tips on selling your idea, read our article on Creating a Value Proposition   and use our Sell Your Idea   Bite-Sized Training session.

Step 8: Act

(Questions 2, 11)

Finally, once you've convinced your key stakeholders that your proposed solution is worth running with, you can move on to the implementation stage. This is the exciting and rewarding part of problem solving, which makes the whole process seem worthwhile.

This action stage is an end, but it's also a beginning: once you've completed your implementation, it's time to move into the next cycle of problem solving by returning to the scanning stage. By doing this, you'll continue improving your organization as you move into the future.

Problem solving is an exceptionally important workplace skill.

Being a competent and confident problem solver will create many opportunities for you. By using a well-developed model like Simplexity Thinking for solving problems, you can approach the process systematically, and be comfortable that the decisions you make are solid.

Given the unpredictable nature of problems, it's very reassuring to know that, by following a structured plan, you've done everything you can to resolve the problem to the best of your ability.

This site teaches you the skills you need for a happy and successful career; and this is just one of many tools and resources that you'll find here at Mind Tools. Subscribe to our free newsletter , or join the Mind Tools Club and really supercharge your career!

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Comments (220)

• Over a month ago Sonia_H wrote Hi PANGGA, This is great news! Thanks for sharing your experience. We hope these 8 steps outlined will help you in multiple ways. ~Sonia Mind Tools Coach
• Over a month ago PANGGA wrote Thank you for this mind tool. I got to know my skills in solving problem. It will serve as my guide on facing and solving problem that I might encounter.
• Over a month ago Sarah_H wrote Wow, thanks for your very detailed feedback HardipG. The Mind Tools team will take a look at your feedback and suggestions for improvement. Best wishes, Sarah Mind Tools Coach

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• 2,175 + 8,259 = 10,434
• 8,259 - 2,175 = 6,084
• 10,434 - 2,175 = 8,259
• 8,000 + 2,000 = 10,000
• $5 for each of the first 10 pages
• $1 for each remaining page
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• 3,254 - 4,567 = ?
• 7,821 - 4,567 = ?
• 3,254 + 4,567 = ?

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• 6 eggs are delivered to the Burke family.
• How many total eggs do the Martin and Burke families receive?
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4.3: Problem Solving

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In previous math courses, you’ve no doubt run into the infamous “word problems.” Unfortunately, these problems rarely resemble the type of problems we actually encounter in everyday life. In math books, you usually are told exactly which formula or procedure to use, and are given exactly the information you need to answer the question. In real life, problem solving requires identifying an appropriate formula or procedure, and determining what information you will need (and won’t need) to answer the question.

In this chapter, we will review several basic but powerful algebraic ideas: percents, rates, and proportions. We will then focus on the problem solving process, and explore how to use these ideas to solve problems where we don’t have perfect information.

In the 2004 vice-presidential debates, Edwards’s claimed that US forces have suffered “90% of the coalition casualties” in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies “have taken almost 50 percent” of the casualties. [1] Who is correct? How can we make sense of these numbers?

243 people out of 400 state that they like dogs. What percent is this?

Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.

Write each as a percent:

• 2.35 = 235%

To do the calculations, we write the percent as a decimal.

The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?

Here, $140 is the whole, and we want to find 9.4% of $140. We start by writing the percent as a decimal by moving the decimal point two places to the left (which is equivalent to dividing by 100). We can then compute: tax = 0.094(140) = $13.16 in tax.

In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?

The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition: $1200(1.07) = $1284.

Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84.

Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase , so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition: $1200 + $84 = $1284.

A TV originally priced at $799 is on sale for 30% off. There is then a 9.2% sales tax. Find the price after including the discount and sales tax.

The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?

To compute the percent change, we first need to find the dollar value change: $6800 – $7400 = –$600. Often we will take the absolute value of this amount, which is called the absolute change : |–600| = 600.

Since we are computing the decrease relative to the starting value, we compute this percent out of $7400:

Absolute and Relative Change

Given two quantities,

Absolute change has the same units as the original quantity.

Relative change gives a percent change.

The starting quantity is called the base of the percent change.

The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs. [2]

There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.

When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:

This tells us Albertsons is 186.7% larger than QFC.

This tells us QFC is 65.1% smaller than Albertsons.

Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?

To answer this question, suppose the value started at $100. After one week, the value dropped by 60%: $100 – $100(0.60) = $100 – $60 = $40.

In the next week, notice that base of the percent has changed to the new value, $40. Computing the 75% increase: $40 + $40(0.75) = $40 + $30 = $70.

The US federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to $8.45 trillion by the end of 2006. [3] Calculate the absolute and relative increase for 2001–2002 and 2005–2006. Which year saw a larger increase in federal debt?

A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years—sometimes referred to as “dropout factories”—decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.”

• Is the “decrease by 17” number a useful comparison?
• Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34?
• This number is hard to evaluate, since we have no basis for judging whether this is a larger or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that’d be less of an improvement.
• The last sentence provides relative change, which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate.

In the 2004 vice-presidential debates, Edwards’s claimed that US forces have suffered “90% of the coalition casualties” in Iraq. Cheney disputed this, saying that in fact Iraqi security forces and coalition allies “have taken almost 50 percent” of the casualties. Who is correct?

Without more information, it is hard for us to judge who is correct, but we can easily conclude that these two percents are talking about different things, so one does not necessarily contradict the other. Edward’s claim was a percent with coalition forces as the base of the percent, while Cheney’s claim was a percent with both coalition and Iraqi security forces as the base of the percent. It turns out both statistics are in fact fairly accurate.

In the 2012 presidential elections, one candidate argued that “the president’s plan will cut $716 billion from Medicare, leading to fewer services for seniors,” while the other candidate rebuts that “our plan does not cut current spending and actually expands benefits for seniors, while implementing cost saving measures.” Are these claims in conflict, in agreement, or not comparable because they’re talking about different things?

We’ll wrap up our review of percents with a couple cautions. First, when talking about a change of quantities that are already measured in percents, we have to be careful in how we describe the change.

A politician’s support increases from 40% of voters to 50% of voters. Describe the change.

Lastly, a caution against averaging percents.

A basketball player scores on 40% of 2-point field goal attempts, and on 30% of 3-point of field goal attempts. Find the player’s overall field goal percentage.

Proportions and Rates

If you wanted to power the city of Seattle using wind power, how many windmills would you need to install? Questions like these can be answered using rates and proportions.

A rate is the ratio (fraction) of two quantities.

A unit rate is a rate with a denominator of one.

Your car can drive 300 miles on a tank of 15 gallons. Express this as a rate.

Proportion Equation

A proportion equation is an equation showing the equivalence of two rates or ratios.

Many proportion problems can also be solved using dimensional analysis , the process of multiplying a quantity by rates to change the units.

Your car can drive 300 miles on a tank of 15 gallons. How far can it drive on 40 gallons?

However, we earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the gallons unit “cancels” and we’re left with a number of miles:

Notice if instead we were asked “how many gallons are needed to drive 50 miles?” we could answer this question by inverting the 20 mile per gallon rate so that the miles unit cancels and we’re left with gallons:

Dimensional analysis can also be used to do unit conversions. Here are some unit conversions for reference.

Unit Conversions

Weight and mass.

A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?

To answer this question, we need to convert 20 seconds into feet. If we know the speed of the bicycle in feet per second, this question would be simpler. Since we don’t, we will need to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might start by converting the 20 seconds into hours:

Now we can multiply by the 15 miles/hr

Now we can convert to feet

We could have also done this entire calculation in one long set of products:

A 1000 foot spool of bare 12-gauge copper wire weighs 19.8 pounds. How much will 18 inches of the wire weigh, in ounces?

Notice that with the miles per gallon example, if we double the miles driven, we double the gas used. Likewise, with the map distance example, if the map distance doubles, the real-life distance doubles. This is a key feature of proportional relationships, and one we must confirm before assuming two things are related proportionally.

Suppose you’re tiling the floor of a 10 ft by 10 ft room, and find that 100 tiles will be needed. How many tiles will be needed to tile the floor of a 20 ft by 20 ft room?

In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:

Other quantities just don’t scale proportionally at all.

Suppose a small company spends $1000 on an advertising campaign, and gains 100 new customers from it. How many new customers should they expect if they spend $10,000?

While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a fixed number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers.

Sometimes when working with rates, proportions, and percents, the process can be made more challenging by the magnitude of the numbers involved. Sometimes, large numbers are just difficult to comprehend.

Compare the 2010 U.S. military budget of $683.7 billion to other quantities.

Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.

If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000.

There are about 300 million people in the U.S. The military budget is about $2,200 per person.

If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.

Compare the electricity consumption per capita in China to the rate in Japan.

To address this question, we will first need data. From the CIA [5]  website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was 859,700,000,000, or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries.   From the World Bank, [6] we can find the population of China is 1,344,130,000, or 1.344 billion, and the population of Japan is 127,817,277, or 127.8 million.

Computing the consumption per capita for each country:

While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.

Geometric shapes, as well as area and volumes, can often be important in problem solving.

You are curious how tall a tree is, but don’t have any way to climb it. Describe a method for determining the height.

There are several approaches we could take. We’ll use one based on triangles, which requires that it’s a sunny day. Suppose the tree is casting a shadow, say 15 ft long. I can then have a friend help me measure my own shadow. Suppose I am 6 ft tall, and cast a 1.5 ft shadow. Since the triangle formed by the tree and its shadow has the same angles as the triangle formed by me and my shadow, these triangles are called similar triangles and their sides will scale proportionally. In other words, the ratio of height to width will be the same in both triangles. Using this, we can find the height of the tree, which we’ll denote by h :

Multiplying both sides by 15, we get h = 60. The tree is about 60 ft tall.

It may be helpful to recall some formulas for areas and volumes of a few basic shapes.

Area: L · W

Perimeter: 2 L + 2 W

Radius:  r

Area: π r 2

Circumference: 2π r

Rectangular Box

Volume: L·W·H

Volume: π r 2 H

If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?

Notice that if both pizzas were 1 inch thick, the volumes would be 113 in 3 and 201 in 3 respectively, which are at the same ratio as the areas. As mentioned earlier, since the thickness is the same for both pizzas, we can safely ignore it.

We can now set up a proportion to find the weight of the dough for a 16″ pizza:

Multiply both sides by 201

A company makes regular and jumbo marshmallows. The regular marshmallow has 25 calories. How many calories will the jumbo marshmallow have?

We would expect the calories to scale with volume. Since the marshmallows have cylindrical shapes, we can use that formula to find the volume. From the grid in the image, we can estimate the radius and height of each marshmallow.

The regular marshmallow appears to have a diameter of about 3.5 units, giving a radius of 1.75 units, and a height of about 3.5 units. The volume is about π(1.75) 2 (3.5) = 33.7 units 3 .

The jumbo marshmallow appears to have a diameter of about 5.5 units, giving a radius of 2.75 units, and a height of about 5 units. The volume is about π(2.75) 2 (5) = 118.8 units 3 .

We could now set up a proportion, or use rates. The regular marshmallow has 25 calories for 33.7 cubic units of volume. The jumbo marshmallow will have:

It is interesting to note that while the diameter and height are about 1.5 times larger for the jumbo marshmallow, the volume and calories are about 1.5 3 = 3.375 times larger.

A website says that you’ll need 48 fifty-pound bags of sand to fill a sandbox that measure 8ft by 8ft by 1ft. How many bags would you need for a sandbox 6ft by 4ft by 1ft?

Problem Solving and Estimating

Finally, we will bring together the mathematical tools we’ve reviewed, and use them to approach more complex problems. In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.

In most problems we work, we will be approximating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to approximate the solution using basic knowledge from our lives.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this beats per year:

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of 384 ft 2 .

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

This will allow us to estimate the material cost for the whole 384 ft 2 deck

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

• http://www.factcheck.org/cheney_edwards_mangle_facts.html ↵
• http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics ↵
• http://www.whitehouse.gov/sites/default/files/omb/budget/fy2013/assets/hist07z1.xls ↵
• Fluid ounces are a capacity measurement for liquids. 1 fluid ounce ≈ 1 ounce (weight) for water only. ↵
• https://www.cia.gov/library/publications/the-world-factbook/rankorder/2042rank.html ↵
• http://data.worldbank.org/indicator/SP.POP.TOTL ↵
• Problem Solving. Authored by : David Lippman. Located at : http://www.opentextbookstore.com/mathinsociety/ . Project : Math in Society. License : CC BY-SA: Attribution-ShareAlike

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Free Practice Aptitude Tests

Take 16 free practice aptitude tests . Each test comes with answers and fully explained solutions to each question.

What Is the Saville Wave Test?

The Saville Wave test is a personality questionnaire that comes in different versions to suit all sorts of job roles and levels of seniority.

Created by Saville Assessment, the Wave tests are designed to be the ‘best-in-class predictor of workplace performance and potential’.

They blend digital innovation and science to cover aspects of competency, potential, motivation, talent and preferred company culture in one short assessment.

The Wave tests are usually used in recruitment and selection, but they are also used for talent management and succession planning.

There are two main types of Wave tests used in recruitment:

• The Wave Focus Styles
• The Wave Professional Styles

It is becoming increasingly common for employers to ask potential new employees to complete a variety of assessments as part of their recruitment processes.

One example of this is the Thomas International PPA assessment , which is a personality test commonly used by employers who are looking for individuals with specific personality types or strengths in defined areas.

In this article, you’ll learn what the Thomas International PPA Test is, what you can expect if you are asked to take the assessment and how your test will be scored.

We have also included plenty of helpful tips and free Thomas International PPA sample test questions to help you prepare ahead of taking the Thomas PPA assessment.

The NWEA MAP Growth Test is used in thousands of schools across the US to assess children academically.

The test can be given three times in a school year and helps teachers to plan their lessons so that children can reach their potential and continue to grow throughout their time in education.

In this article, you will learn more about the different levels of the test and the way it is structured to suit different grades, what types of questions are on the tests and how best to prepare your child for success.

If your child is under the age of 12 and aims to get into a gifted school program, they will most likely be required to take the NNAT test to assess their skills.

If you know that your child is expected to take the exam, you can help them prepare for it by letting them know what to expect.

This article contains the most relevant information concerning the NNAT test, including its purpose, scoring system and levels.

You will also receive plenty of helpful tips on how to help your child prepare for the exam by completing a Naglieri Nonverbal Ability Test sample and working on their weaknesses in each question type.

The Hogan Business Reasoning Inventory (HBRI) is a scientifically-based aptitude test.

It is designed to assess cognitive ability and a candidate’s preferences for using qualitative and quantitative reasoning skills.

The HBRI is a popular tool for pre-employment screening, particularly in the management, sales and marketing sectors.

If you have been asked to take the HBRI, your test results will offer prospective employers insight into your decision-making skills, problem-solving abilities, approach to processing information and ability to learn from past experiences.

In this article, you can learn what to expect when taking the Hogan Business Reasoning Inventory HBRI test, see examples of the type of Hogan Business Reasoning Inventory questions you might be faced with and guidance on how to prepare for the assessment.

The MAP Kindergarten Test is a computer-adaptive test that measures your child’s progress throughout their academic career.

The grades K through 2 tests assess mathematics and reading abilities.

The results from these tests allow teachers to identify the gaps in your child’s knowledge and to better understand their ability to learn and retain information.

The MAP test is administered three times in the academic year to ensure your child’s progress is properly recorded.

The Ramsay Mechanical Aptitude Test is used as part of the pre-employment screening process for a number of roles in different industries.

It is an excellent indicator of how well you can learn on the job, as well as your ability to use basic physics principles and mechanical knowledge to solve problems.

In this article, you will learn more about which job roles require completion of the Ramsay MAT as part of the application process, and what different types of tests are available.

The format of the assessment, as well as the number of questions and the time limit, will also be discussed. You’ll also learn how the Ramsay MAT is scored and what happens next.

There will be example questions that are similar to those you are likely to find on the assessment, as well as some top tips for success.

The SHL OPQ32 test is the flagship personality test from SHL.

Used by major organizations all over the world, it is considered to be one of the best psychometric assessment tools currently available on the market.

It is a trait-based personality test that is designed to gauge a candidate’s personality attributes and behavioral preferences in the workplace.

A candidate’s test results are analyzed by recruiters, helping them to decide whether a candidate is a good match for the job role they have applied for.

OLSAT stands for Otis-Lennon School Ability Test.

Children take the test to help schools decide admissions into their gifted-and-talented programs.

Preparing your children for taking the OLSAT is a good idea as it could determine their eligibility and acceptance to extra academic programs that are offered by their schools.

If you are looking for a career working with the emergency services, then becoming a 911 call handler and dispatcher might be just the role for you.

If you can handle working under pressure, helping the public and are able to deal with difficult and uncomfortable situations with a calm head, then you might have what it takes to be the first port of call in an emergency.

Becoming a 911 call handler and dispatcher means you will need to have some very specific skills and abilities, and as part of the recruitment process, you will have the opportunity to demonstrate your suitability by taking the CritiCall 911 dispatch test .

Used throughout the US for recruiting people for 911 roles, the CritiCall test does not assess your previous experience or your knowledge of the role.

Instead, it is designed to assess candidates on the inherent aptitudes that are needed to be successful in the role.

This article will discuss what the CritiCall test is assessing, the types of questions that you are likely to face and what the recruiters will be looking for.

There will be some example questions, as well as details about what mark you will need to achieve to pass the test.

Finally, there are some CritiCall test prep tips to help you prepare for the assessment and what to think about on the day.

The McQuaig Word Survey is a type of personality assessment.

Survey responses are used to measure a candidate’s key personality traits and compare these with how they are currently behaving in the workplace.

The results from the survey indicate whether a candidate is behaving naturally in their current role, or whether they are making changes to their behaviour.

This article will help you pass the McQuaig Word Survey assessment test by giving you all the tools and practice questions you will need.

The Caliper test is an assessment used by employers to gain a better understanding of a candidate’s personality traits, cognitive abilities and motivations.

The Caliper test is used to help employers predict a candidate's suitability for a role.

In this article, we'll take a detailed look at what the Caliper test is and how it is scored.

We’ll also share some tips on how you can perform at your best when taking your Caliper assessment test.

More and more employers are choosing to use psychometric testing as a part of their recruitment processes as it helps to highlight those candidates who are most likely to be suited to the roles they are looking to fill.

Employers will often use this form of testing when recruiting for mid-to-high level managerial roles or positions that require a specific set of skills.

Using the results of assessments, recruiters and employers are able to see the strengths and skills of individuals as well as being able to predict future performance.

One of the most popular options for psychometric testing is the Criteria Cognitive Aptitude test – more commonly known as the CCAT.

The Bennett Mechanical Comprehension Test (BMCT) , also referred to as the Bennett Mechanical Aptitude Test, is considered the most popular mechanical aptitude test.

However, it is also believed to be the hardest one to pass.

The BMCT requires you to have a knowledge and understanding of physical principles and answer 55 questions about the application of these concepts within 25 minutes.

You typically need to score in the top 20% of candidates to progress to the next stage of recruitment.

If you have applied for a job, apprenticeship or college course in the UK or Australia, you may have been asked to complete a Basic and Key Skills Builder or BKSB assessment) .

The initial BKSB assessment determines suitable applicants for an apprenticeship or places a student in the correct class level on a college course. The assessment tests you on your maths and English skills to identify areas that need improvement.

This guide will explain the BKSB assessment in detail, provide example questions, and answer your queries about the test content and format.

There are a variety of tests and assessments that can be used by companies for candidates applying for jobs.

One of those is the Thomas GIA Test .

This article will define what the Thomas GIA Test is and who it is for, in addition to looking at what the test involves, how it is scored and tips for the next chance to pass the test.

You will also find Thomas GIA test examples and explanations for each answer.

The IE Global Admissions Test (ieGAT) is an entrance exam for the IE University (IEU) in Spain.

It covers numerical , logical and verbal reasoning .

Not every IE program requires an ieGAT score. However, as the programs that do are highly competitive, those who take the ieGAT Test must prepare themselves to achieve the best score possible.

This article will help you understand:

• What the ieGAT is
• The structure
• ieGAT scoring
• How to register for the ieGAT
• The best ways to prepare

More and more companies are introducing psychometric testing as a part of their recruitment processes.

This means that, if you are considering changing careers or applying for a new role within your existing industry, you may need to take an assessment.

One of the most popular tests for corporate employers is the test by Sova Assessment .

The United States Postal Service (USPS) provides extensive career opportunities and seemingly endless possibilities for professional development.

However, anyone looking to work at the USPS must pass a Virtual Entry Assessment designed to find suitable applicants for the role they are trying to fill.

This article covers the Postal Exam 474 , including its main parts, how to pass it and how to prepare for the Virtual Entry Assessment.

Let's start by looking at what exactly the 474 Virtual Entry Assessment is.

Developed by Drs Joyce and Robert Hogan in the 1980s, the Hogan assessment is a collection of tests designed to assess personality traits, leadership skills and cognitive abilities.

The Hogan assessment is generally used as a pre-employment test for management roles.

This article will guide you through the online Hogan tests , provide a range of sample questions, discuss how the Hogan Assessment results are calculated and recommend ways that you can prepare to take the Hogan assessment yourself.

If you’ve recently applied for a managerial or executive role, you may have been asked to take a Saville Analysis Aptitude Test , also known as the Swift Aptitude test.

The Swift Analysis Aptitude Test was created by Saville Assessment, which is a huge name in the test publishing market.

The CAT4 cognitive ability test is an examination designed to measure a student’s academic progress.

When the CAT4 test is scored, teachers and parents will be given a summary of the academic potential of the student.

Any student taking the test will be asked questions that will measure their non-verbal reasoning abilities, verbal reasoning skills, quantitative reasoning abilities and spatial awareness .

In this article, you’ll learn more about what types of questions are asked to examine these skills.

The Korn Ferry Leadership Potential Assessment (KFALP) is used to test candidates to see if they have the potential to become leaders and managers.

It uses seven different categories, known as Seven Signposts, to assess potential leaders:

• Learning Agility
• Leadership Traits
• Derailment Risks

This article will examine the theory behind the assessment, the different topics that are tested and how the assessment is scored.

There will also be example questions so that you know what to expect when you take the KFALP and some tips to help you score as highly as possible when you take the test.

Pymetrics tests identify specific behavioral characteristics and traits.

This article examines why pymetrics tests are used and what to expect in your assessment.

Tips are included to help you get the best results.

The McQuaig Mental Agility Test (MMAT) is a 15-minute timed test that is designed to assess your ability to think quickly.

In this short test, you will face questions that will allow you to demonstrate your speed of thought and general mental agility, which are useful aptitudes when it comes to many jobs in different industries.

In this article, find out more about the structure of the test, the different types of McQuaig Mental Agility test questions and what to expect on the day. You’ll also get some mental agility practice test questions and top tips to help you be successful in the MMAT.

The Federal Bureau of Investigation (FBI) is responsible for the enforcement of federal law and the protection of national security in the US.

Working for the FBI can be highly stressful. As a special agent for the FBI, the working week is likely to be 50 hours or more.

Special agents must be willing to be based anywhere in the world. They are expected to carry a firearm and work in potentially dangerous situations.

With this in mind, the FBI has a rigorous application and selection process for potential new recruits. It can take more than 20 months to complete the entire process and commence employment with the FBI.

This guide to the USPS postal exam 955 will take you through the different sections of the test, including example questions, provide tips on how you can prepare for the exam and answer several frequently asked questions.

The USPS postal exam 955 is used to screen applicants for mechanic and technician positions , such as electronic technicians or motor vehicle mechanics. It also sometimes referred to as the postal maintenance 955 exam, USPS maintenance mechanic 955 test or the 955 maintenance exam.

It tests applicants’ suitability by assessing personal characteristics, work experience, and electronic and technical knowledge and skills.

The USPS postal exam 955 replaced the previous 931, 932 and 933 exams .

The USPS postal exam 955 is free of charge , but you will need access to the internet and an email address.

If you are looking to work in the United States Postal Service, you will need to pass the USPS Postal Exam 476.

The USPS Postal Exam 476 is an online test that screens for the best candidates. The exam is used to find suitable candidates for a range of positions, including mail processing clerk, data conversion operator and clerk-related positions.

This article will outline what the USPS Postal Exam 476 includes, with particular attention to the separate sections of the examination.

In addition to this, how the exam is scored and how you can best prepare for it will be covered. There will also be a list of frequently asked questions for you to refer to if you have any doubts.

A List of Amazon Assessment Tests Available for Practice in 2024

• Amazon Work Simulation Assessment
• Amazon Maintenance Technician Test
• Amazon Coding Assessment
• Amazon Workstyle Assessment
• Amazon Area Manager Assessment
• Amazon Operations Manager Assessment
• Amazon Online MBA Assessment
• Amazon RME Apprenticeship Skills Battery Test
• Amazon Financial Analyst Assessment
• Amazon ATA Technical Assessment
• Amazon Control Systems Technician Test
• Amazon Warehouse Assessment Test

The Amazon assessment test is an essential way for the corporation to find the best-suited employees.

It is a series of challenges used to evaluate all its candidates during the recruitment process.

Amazon online assessments typically include both numerical and verbal reasoning tests.

These types of tests examine a potential candidate’s logical skills.

Candidates will also have to sit work-style assessments that simulate the working environment at Amazon.

Other Amazon exams include:

• The Amazon coding assessment (also known as the Amazon SDE online assessment)
• The work sample simulation
• An Amazon versant test

These last two, amongst others, will be discussed later in this article.

This Amazon reviewer job article will also discuss how to pass the Amazon assessment tests, some Amazon assessment answers you should know and what you need to do to best prepare yourself.

There is also a comprehensive list of frequently asked questions from those who are interested in taking these Amazon job tests to find employment with the company.

What Is the SHL Verbal Reasoning Test?

The SHL Verbal Reasoning Test is a graduate-level and above pre-employment aptitude test that is used in graduate and management recruitment for many roles across different industries.

The test is usually taken online, and it is designed to evaluate candidates on their ability to understand written information and make informed, reasoned and logical decisions based on that information.

SHL is a well-established test publisher, providing tests for more than 10,000 companies around the world. It offers a range of tests, including psychometric, behavioural and personality assessments that are based in occupational psychology and aptitude science. 

The tests have specific aims – and recruitment teams use SHL tests like the Verbal Reasoning Test to filter through similarly qualified candidates to find the applicants who have what it takes to be successful in a graduate or management level role.

When taking a verbal reasoing test, bear in mind that you might also be asked to take numerical reasoning tests, logical reasoning tests or personality tests along side.

IQ stands for intelligence quotient and is usually thought to represent the reasoning skills of individuals.

The idea of intelligence relates to how quickly people can solve problems or puzzles, use logic to answer questions, or quickly recall information and facts they’ve heard.

The first type of IQ test was created by a French psychologist named Alfred Binet.

The assessment that he made is still used and is known as the Stanford-Binet intelligence test.

Considering cheating on your GMAT (Graduate Management Admission Test) Exam?

Want to know how to do it, if you should do it and what the consequences will be?

Well you came to the right place!

Read on to find out more about cheating on the GMAT exam, but be warned...

... it's certainly not something I advise!

Do you have an upcoming online aptitude test ? 

Are you looking for the best aptitude test prep material to give you the very best chance of getting the highest possible grade? 

If so, this article will help you. 

Aptitude tests are a crucial part of your job search, and you usually only have one chance to showcase your skills. 

Psychometric aptitude tests can measure many different aptitudes and skill sets, in many different formats:

• Numerical reasoning
• Verbal reasoning
• Diagrammatic or inductive reasoning
• Mechanical reasoning
• Personality types
• Situational judgement and work environment tests
• Work style tests

Aptitude tests can be challenging and it is important to be fully prepared before you attend your job interview or assessment centre.

Several free and paid aptitude test preparation websites offer preparation packs to help you score the best you can.

Those dreaming of working for the TSA will most likely need to take a challenging exam called the TSA CBT Test during the hiring process. Here we’ll look at exactly what it involves and how you can make sure you pass it. Read on to find out more.

If you plan to work as an inspector, manager, marshal or security officer in any agency governed by the Transportation Security Administration, you must pass the TSA CBT test as part of your application process.

Read on to learn more about this assessment, including its purpose, what types of questions it has, how challenging it is and how to prepare for it.

You'll also be provided with a few example questions to help you get an idea of what this test looks like.

Let’s get started.

Aptitude tests are administered to understand your inherent abilities to reason and respond to specific tasks.

They are widely used in various forms to screen candidates or evaluate existing employees for a future job role.

The most generic and widely used aptitude tests are curated to measure different facets of your abilities, mainly on the following areas:

• Abstract Reasoning
• Numerical Reasoning
• Logical Reasoning
• Verbal Reasoning
• Attention to Detail

Apart from these base types, there are various other specialized aptitude tests which you may face in specific industries or based on your role in different career stages.

We have discussed each of the most common job related aptitude tests in detail.

Illustrative examples and helpful hints are provided throughout to aid your preparation.

Read on to find out more.

The Cognify test is a game-based cognitive assessment designed to measure an individual's cognitive aptitude to measure key job performance linked abilities and skills in a prospective candidate.

The Cognify test was once a product of Revelian, an Australian assessment company, but was later acquired by CriteriaCorp.

Moving away completely from the question-answer based template of traditional tests, Cognify uses an innovative approach where candidates don't face a series of questions on a screen.

Instead, the Cognify Assessment comprises 6-7 timed game-based mini-tests categorized into three cognitive abilities categories:

• Problem-Solving
• Verbal Knowledge

Well, before you start raising your eyebrows at the mention of ‘game-based’ and dismiss it as just another fad, pay attention!

Cognify assessment is credited as having brought a paradigm shift in the field of psychometric testing.

Many Tier-I graduate recruiters globally have started using this assessment in their candidate selection process.

The train driver test is used to establish whether a candidate is suitable for work as a train driver. This unique suite of tests includes psychometric assessment tools such as:

• The Group Bourdon Test (GBT)
• Test of Everyday Attention (TEA-OCC)
• Adaptive Tachistoscopic Traffic Perception Test (ATAVT)
• Situational judgement tests
• Vigilance tests
• Written communication tests

What Is the Train Driver Test?

In most countries, you will need to sit the train driver online test if you want to work as a train driver. If you have been asked to sit the assessments, there is no train driver psychometric test cost associated with the train driver exam.

Working as a train driver is a challenging and demanding role. As a train driver, you must be able to ensure the safety of passengers at all times.

The UK’s train driving tests are some of the most challenging. As well as testing aptitude for the job role, they are used to assess whether candidates have the mental abilities to cope with the stress and demands of the job role.

The train driver test is used to establish whether a candidate is suitable for work as a train driver. The train driver test is a unique group of psychometric tests for train drivers designed to assess the psychomotor and cognitive skills needed to work safely as a train driver.

The Predictive Index (PI) test is a popular type of pre-employment testing used to accurately measure an individual’s cognitive ability and behavioral profile during the hiring process in a wide range of industries and organizations. They are most commonly used during the early stages of the recruitment process. 

The PI cognitive test assesses verbal, numerical and analytical reasoning ability. 

The PI behavioral test creates a behavioral persona that describes character traits and tendencies.

A mechanical aptitude reasoning test is an important way to assess your knowledge on mechanical topics for potential roles in the army, emergency services and many other professions. Here, you will get all the information you need on what a mechanical comprehension test is and how to pass it.

Those applying for jobs related to the army, the emergency services engineering service, and similar occupations that require mechanical aptitude, are likely to be asked to take a mechanical reasoning test as part of the recruitment process.

Mechanical aptitude tests assess knowledge in electricity, optics, pressure and other fields of mechanics related to a specific industry.

From this article, you'll learn what mechanical reasoning tests look like, when to take them, what to expect from these assessment types, and how to practise and prepare for them.

Let’s get started!

If you would like to take a free practice Cognitive Ability Test before reading this article, click here .

If you would like to purchase an online Cognitive Ability Test prep pack, visit our partner website JobTestPrep .

The following tests are common cognitive ability tests:

• Spatial Reasoning
• Mechanical Reasoning
• Logical Ability Tests
• Space Visualization
• Information Processing
• Visual Pursuit
• Manual Speed and Accuracy

Have you been asked to take a Deductive Reasoning test as part of an upcoming interview process?

Continue reading to find out more about this type of test, including:

• Why employers use Deductive Reasoning Tests.
• How you can improve your performance at Deductive Tests.
• What types of questions you will be asked during the Test.

What Is A Deductive Reasoning Test?

Logical thinking or deductive reasoning tests are used by employers to measure an applicant’s ability to make logical arguments and form sound conclusions.

During this type of test, you will be presented with a variety of scenarios, statements and arguments for which you will need to apply a given set of rules to determine the validity of the corresponding conclusion.

Spacial Reasoning Definition

A spatial awareness test is a type of assessment that tests your ability to think in three dimensions and use your imagination to see movement through space.

Someone with good spatial awareness will be able to see in their mind how different shapes interact and be able to manipulate them to make a reasoned and logical decision.

The test is based on pictures, diagrams and shapes. You will need to mentally manipulate the presented image by disassembling or reassembling, rotating, seeing it in a mirror image or from different angles, or otherwise visualizing it differently to find the right answer to the question from the multiple-choice options provided.

Spatial awareness is something that we use to a greater or lesser degree every day, from understanding our position relative to other things around us to imagining the route we will take to get from one place to another.

Spatial reasoning tests are distinct from other similar assessments such as diagrammatic reasoning tests and abstract reasoning tests. It is important to understand how they differ as they are often included in aptitude tests and cognitive assessments alongside spatial reasoning tests.

Psychometric tests are often used by organizations as part of the recruitment process. Different types of psychometric tests are designed to measure various aspects of cognitive ability, reasoning capabilities and personality traits. Potential employers use the results to assess a candidate’s suitability for a role. A psychometric test is generally administered online; this helps hiring managers filter applicants quickly and easily. 

As Capp Assessment Tests become more common perhaps you have encountered one for the first time.

This can be a bit daunting and, since they look and feel a bit different to more traditional psychometric reasoning tests, it isn’t necessarily obvious what you need to do to be successful…

Don’t worry.

With the insight and tips we share with you below, you’ll be smashing your tests in no time.

FREE BONUS: Get free unlimited access to Capp test practice (for 30 minutes) on our partner website JobTestPrep.

What are Capp Assessment Tests?

Capp are a consultancy and psychometric test publisher who specialise in Strengths Based Assessments.

They also offer a number of different psychometric tests that are widely used many major organisations including Google, Atkins, Amazon and RBS.

Their Assessment Tests include critical reasoning, numerical reasoning, verbal reasoning .

Psychometric reasoning tests like these are very common.

This is because they are a cheap and efficient way for organisations to identify candidates who aren’t likely to be able to succeed in a particular job.

Because they are often used to filter candidates out of application processes, they are sometimes called screening tests or gateway tests.

Candidates like you have to achieve a particular level of performance in order to progress in the selection process.

With practice you can dramatically improve your performance. Practice is the best way to improve your test scores.

In the rest of this article we’ll show you how the tests work, suggest how you can prepare, and then direct you towards some practice tests so that when the big day comes you are ready.

Before you do anything else, take a look at the Capp website , where you can take free practice tests.

How do Psychometric Reasoning Tests Work?

In general, psychometric reasoning tests challenge users to answer a series of questions and compare their performance on a test with the average performance level of a reference or ‘norm’ group.

This is typically made up of individual with similar characteristics, such as education level, nationality or workplace seniority.

If you do better than most of the norm group you will receive a high score, whereas a low score suggests that your performance was weaker than most of the norm group.

Usually, a minimum standard of performance necessary for success in a role is identified at the start of an assessment process, and all candidates that don’t meet this level will be unable to progress through the process.

What makes Capp Assessment Tests Different?

Capp Assessments are ‘Next Generation’ psychometric aptitude tests ; this means they might look and feel a bit different to other psychometric tests you have completed in the past.

The main difference between the Capp tests and more traditional psychometric ability tests is that the Capp tests are responsive.

This means that the actual questions presented to a candidate will depend upon their performance on the previous questions.

Capp say that the responsiveness of their tests and the size of their question bank mean that the chances of two candidates taking exactly the same test is currently less than one in a billion .

In practice, this means that if you’ve been able to quickly and accurately solve the previous questions, you can expect to be presented with incrementally more challenging questions.

By contrast, if you have made a number of errors, the test will present questions at a lower level.

The aim of the tests is to work out what your maximum ability is. Or put another way, what the most challenging level you are capable of working at is.

Another thing that makes Capp Tests feel different is that they have no time limit (although the time you take to complete the test does effect the score so you still need to work as quickly as you can).

This takes a bit of the pressure off and can make taking these tests rather less stressful than others.

Finally, the variety of question types and the format of the questions in Capp Tests can be different to those used by more traditional test publishers.

Let’s take a closer look at this:

• Numerical Reasoning Tests

Traditionally numerical reasoning tests require candidates to select the correct answer from a number of potential options.

The Capp numerical reasoning test still does this, but it also requires candidates to rank potential answers or to type their answer into a free-text box.

This makes it harder to guess the correct answers.

• Verbal Reasoning Tests

Verbal reasoning tests typically give you a passage of text to read and then ask you whether a number of subsequent statements are true or false, based on the information contained in the passage.

This question type is included within the Capp Verbal Reasoning Test, but there are also a number of different question formats included.

This means that as well as testing verbal reasoning, the Capp test can also assess verbal dexterity, comprehension, interpretation, and adaptability.

As well as traditional multiple choice questions, the test also presents:

• Free text editing : This type of question requires you to type your answers directly into the question. You might be asked to correct spellings or grammar, or edit a passage of text.
• Bucket sort : You will be presented with two categories/styles of writing; your task is to place each item presented to the category/style of writing that it best fits.
• Drag and Drop : This type of question requires you to drag statements or words to the place that they best fit.
• Ranking : These questions can be quite subtle and require you to really understand the nuance of language and language use. You will be presented with a number of statements and asked to rank these based on some feature of the text, such as positivity.
• Selecting the most appropriate word to fill in the sentence : You will be presented with a passage of text with a number of blanks in it, for each blank space you must select the most appropriate word to fill the space from a drop down menu.

Critical Reasoning Tests

The Capp Critical Reasoning test evaluates your ability to think critically in a number of ways.

In each instance, a passage of information is presented followed by a series of statements, your task is to select the appropriate response from the drop down menu.

Questions focus around five areas:

• Inference: rating the probability of truth of inferences based on given information
• Recognition of assumptions: identifying unstated assumptions underlying given statements
• Deduction: determining whether conclusions follow logically from given information
• Interpretation: weighing evidence and deciding if generalisations or conclusions based on data are warranted
• Evaluation of arguments: evaluating the strength and relevance of arguments with respect to a particular question or issue.

Are you considering cheating on your upcoming SHL tests ?

In this full disclosure article, I’ll tell you why people cheat on tests, how people cheat, and whether or not it’s worth doing..

Don't cheat!

Practice... it's the only legitimate way to improve your scores, you'll sleep better at night and probably get better results in your tests too.

Still want to read about how to cheat on a test?

Numerical Reasoning Tests can be very tricky.

And when it comes to results, preparation and practice are key.

But that's easier said than done.

If you're researching this type of aptitude test for the first time or if you want to improve your numerical ability , perform better on tests and get more job offers this article will provide some practical strategies that you can use immediately .

For the best chance of success, read the article below slowly, work through the example questions , follow our tips and actionable advice and then start taking practice tests .

Ready to get started?

Let's go!...

Want to try a practice test before reading this article?

You can take our free numerical test right here:

Logical reasoning tests are a little different to your average psychometric test .

With this type of assessment, there are many different variations so it is sometimes difficult to determine which aspect of logical reasoning you will be assessed on.

With this guide, you’ll learn the difference between inductive and deductive reasoning tests , and some tips for maximising performance.

Designed to evaluate how you interpret patterns, shapes, numbers and other data to reach logical conclusions, the assessments are used across a number of different sectors at all levels of recruitment from entry right up to managerial positions.

The in tray exercise (also called an e-tray exercise ) is a popular assessment activity which employers use to evaluate the skills of applicants in a workplace situation.

If you have an In Tray exercise coming up as part of your interview process, this article will help you prepare.

Within these exercises, candidates will be presented with a given scenario, along with a set of tasks to complete which may include things like responding to email messages, reports or briefing documents.

Aptitude tests are short tests employers use to assess whether a candidate has the level of competency necessary for success in the role.

The tests are used to see if a candidate has the skills necessary to do the job.

Aptitude tests are standardized, for the most part, and the results of all the candidates are compared to each other to see which candidate may be the best for the job.

Aptitude tests provide employers with a quick way to assess a candidate’s ability to perform in high-pressure situations and think in critical ways as they would if they were on the job.

What Is a Situational Judgment Test?

A situational judgement test (SJT) is a psychometric test that is often used as part of the recruitment process for graduate and managerial positions as well as roles that are customer-facing in a wide range of industries.

The SJT is designed to assess how a candidate deals with work-related problems and situations, focusing on essential aptitudes , competencies and soft skills that are not always easy to evaluate in other ways.

Although SJTs are usually bespoke to the company (or in some cases, the specific role), they tend to follow the same basic structure.

Each question is formed by presenting a fictional yet realistic work-based scenario. This might be text-based, it may include some illustrations or it could be animated or acted out in a video.

Following the scenario, there will be several options that you can choose from, each giving a possible course of action to follow to solve the issue that is presented in the situation given.

The answer that you choose will be compared to the benchmark answers that the recruitment team is using – these represent the core competencies for the role, as well as alignment with company values.

SHL assessment tests are important steps in many job interviews and career advancement opportunities. Therefore, it is essential to have a comprehensive understanding of how the different types of SHL tests work and how you can prepare for them in order to get top scores.

In this article, we will provide an overview of how SHL assessments work, sample SHL test questions, tips on improving your test performance, and strategies for prepping and succeeding with any SHL test.

What Is an SHL Assessment Test?

SHL is a global assessment company that is well known and recognised as a leader in pre-employment psychometric tests; the tests that SHL publishes are used by 75% of the FTSE 100 and they are available in more than 40 languages.

So if you are applying for a new role (especially for a graduate position), you are likely to come across them in the recruitment process.

In addition, the company offers consultancy and management services via its TalentCentral platform.

The SHL assessment are a series of tests that can be delivered individually or in a battery, and some of them are bespoke to the company that is using them, making them an excellent way for the recruitment team to ensure that the applicants for a role have the basic competencies, personality traits, work behaviours and cognitive abilities to be successful.

Sind Sie auf der Suche nach kostenlosen psychometrischen Tests zur Übung?

Dann ist diese Seite genau das Richtige für Sie.

Was ist ein psychometrischer Test?

Psychometrische Tests (auch Eignungstests genannt) sind fester Bestandteil von Jobinterviews vieler Unternehmen auf der ganzen Welt.

Diese Tests bestehen normalerweise aus einer Reihe von zeitlich erfassten Fragen , die meist numerischen (mathematischen Fragen), verbalen (Fragen zum Leseverständnis) oder logischen (diagrammatischen Fragen) Ursprungs sind.

Testes psicométricos (também conhecidos como testes de aptidão) são uma parte comum do processo de entrevistas de emprego em muitas companhias no mundo todo.

Geralmente, esses testes consistem de uma série de questões com um certo tempo de resposta.

As questões costumam ser numéricas (questões matemáticas), verbais (compreensão textual) ou lógicas (questões de diagrama).

Testes SHL . Se você está lendo isso, há uma boa chance de você ter acabado de descobrir que fará um desses testes difíceis como parte de um processo de recrutamento em andamento.

Se você chegou tão longe e agora está se sentindo tenso para se sentar na frente de um ‘abstract quiz’, não se preocupe...

Nós cuidaremos de você.

El Razonamiento Inductivo está basado en patrones y es otra variante de las muchas pruebas psicométricas utilizadas por los empleadores como una forma de determinar la idoneidad de un candidato para sus roles.

En un nivel similar al del razonamiento esquemático , el razonamiento inductivo probará tu habilidad para aplicar la lógica y la razón para la resolución de problemas.

Cómo funcionan las pruebas inductivas

Dentro de la prueba se te presentará una serie de diagramas los cuales se vincularán mediante una regla subyacente.

Esta regla afectará el diseño del diagrama y tu tarea será identificar el patrón.

Bonificación: puedes obtener acceso ilimitado y gratuito a la práctica de prueba (durante 30 minutos) en nuestro sitio web asociado JobTestPrep: Clic aquí .

Por lo general, se espera que los candidatos seleccionen entre 4 y 6 posibles respuestas completas bajo condiciones de tiempo.

Las pruebas de razonamiento inductivo a menudo complementan otras pruebas como las de razonamiento verbal o numérico.

A veces las empresas requieren que complete una prueba de juicio situacional o un cuestionario de personalidad junto con la evaluación de razonamiento inductivo.

Los resultados de cada prueba se revisarán individualmente y luego colectivamente para determinar si tú serías una buena opción para la empresa.

¿Por qué los empleadores utilizan estas pruebas?

Algunas veces se las denomina prueba de razonamiento abstracto, las evaluaciones de razonamiento inductivo están diseñadas para evaluar tus habilidades en la resolución de problemas y el razonamiento lógico.

Cuando completes la prueba, los reclutadores buscarán tu capacidad para trabajar de manera efectiva con información desconocida para alcanzar una solución viable.

Las pruebas se utilizan a menudo para evaluar tu capacidad de pensar creativamente, aplicar habilidades analíticas y diseñar soluciones innovadoras, mientras que a menudo son un indicador de tu nivel general de inteligencia.

Como tal, es esencial que realices el trabajo preparatorio necesario antes de la prueba real para asegurarte de poder completarla exitosamente y crear una buena impresión.

La prueba de razonamiento inductivo es frecuentemente usada por empleadores corporativos; es común esperar que se complete al menos una prueba psicométrica como parte del proceso de reclutamiento.

Los empleadores utilizarán estas pruebas para ver la eficacia con la que trabajas bajo presión y tu enfoque de la evaluación.

Las pruebas de razonamiento inductivo son usadas predominantemente en los roles técnicos o aquellos que requieren una resolución frecuente de problemas y los empleadores las utilizan para evaluar cómo identificas patrones, con qué eficacia puedes identificar reglas y consistencias de datos y si puedes predecir la secuencia de objetos a medida que evolucionan.

En términos de evaluación psicométrica, el razonamiento inductivo, el razonamiento abstracto y el razonamiento esquemático son tres pruebas que a menudo se superponen con la evaluación. Los proveedores utilizan nombres diferentes para cada uno, lo que hace que las cosas sean un poco más confusas.

Estas pruebas ciertamente varían entre los empleadores y la etapa en el proceso de reclutamiento también será diferente.

Algunas empresas los utilizan como un ejercicio de selección previa a la entrevista para limitar un conjunto de candidatos, mientras que otras organizaciones pueden usarlos hacia el final del proceso de reclutamiento o como parte de los días de evaluación.

Contenido de la prueba de Razonamiento Inductivo

La mayoría de las pruebas de razonamiento inductivo presentan una serie de secuencia de palabras, ilustraciones o formas y te piden que decidas cuál es la siguiente.

Esto requiere prestar atención a los detalles, a la resolución de problemas y perseverancia para alcanzar la respuesta requerida, todo lo cual se evalúa en condiciones de tiempo, lo que agrega aún más presión.

La prueba en sí misma requerirá que compares varios elementos incluyendo colores y formas, o que los clasifiques basándote en cantidad o tamaño.

Como un ejemplo, se te proporcionará un juego de seis cuadros conteniendo una cantidad de formas y luego se te pedirá que elabores una secuencia lógica para cada cuadro.

Para obtener la respuesta correcta, deberías identificar un patrón tal como similitudes, diferencias o una combinación de ambos.

Estas tareas pueden parecer extremadamente complejas, por ello es importante realizar tantas prácticas de pruebas similares como sea posible antes de la prueba real y también tanta práctica como puedas antes de la entrevista o del día de evaluación.

Asegúrate de llegar a tiempo y haber dormido bien la noche anterior, de lo contrario, es posible que te falte la concentración y que parezca que no entiendes lo que te piden que hagas.

Una aproximación a las Pruebas de Razonamiento Inductivo

Cuando comienzas la prueba, lee la pregunta detenidamente y trata de observar solamente a un elemento de la forma a la vez.

Es muy fácil sentirse abrumado por el contenido de una evaluación de razonamiento inductivo, por lo que la mejor manera de abordarla es intentar y decidir el patrón, considerando específicamente el tamaño, la orientación y la ubicación de la forma interior.

Los patrones están diseñados para ser complicados en tomarte el tiempo y utilizar tu lógica para resolver el problema.

Si estás teniendo una particular dificultad en identificar un patrón, trata de observarlo desde el final en lugar del principio.

Esto puede resaltar de manera efectiva algo que quizás hayas omitido usando el método tradicional de revisar las formas.

Toma conciencia de la hora pero no mires el reloj, y no te asustes en la medida de lo posible; esto sólo hará las cosas más difíciles.

Las pruebas de razonamiento inductivo son creadas para ser completadas bajo presión, por lo que la práctica de completar las pruebas en condiciones de tiempo puede ayudar de manera significativa.

Practicar es una de las mejores maneras de prepararte mentalmente para cualquier prueba psicométrica y el razonamiento inductivo no es diferente a ello.

Nada te preparará mejor para la evaluación que realizar una cantidad de exámenes de práctica, muchos de las cuales puedes encontrar en línea gratuitamente.

Cuando te familiarizas con el formato de la prueba y te acostumbras a responder preguntas rápidamente y trabajar bajo presión, es mucho más probable que tengas éxito que si no realizas ningún trabajo de preparación o práctica anteriormente.

¿Qué son las pruebas psicométricas?

Las pruebas psicométricas (también conocidas como Pruebas de Aptitud ) son ahora una parte común de los procesos de selección y evanotluación, por lo tanto un requisito necesario para solicitar trabajo.

Si tú aún no has completado una, es muy probable que lo necesites en algún momento en el futuro. Con esto en mente, hemos preparado para ti la Guía actual para las pruebas psicométricas para explicar qué son, cómo se utilizan y cómo completarlas con éxito.

Antes de comenzar con el artículo a continuación, ten en cuenta que tenemos tres pruebas psicométricas de práctica disponibles para que las pruebes.

Las pruebas de razonamiento verbal están diseñadas para examinar tu nivel de comprensión del pasaje de un texto.

Estas pruebas son un ejemplo de una prueba de habilidad (a veces conocida como pruebas de aptitud) y son utilizadas por los empleadores en combinación con pruebas de razonamiento numérico y pruebas de razonamiento lógico .

Las pruebas de razonamiento verbal tienen como objetivo identificar tu capacidad máxima de comprensión, o en otras palabras, el párrafo de un texto más desafiante que tú podrás entender.

Numerische Tests können knifflig sein. Übung und die richtige Vorbereitung sind der Schlüssel zum Erfolg.

Aber das ist leichter gesagt als getan…

Wenn Du zum ersten Mal über diese Tests nachliest oder wenn Du nach Wegen suchst um deine Fähigkeiten zu verbessern, besser abzuschneiden und mehr Interviews und Jobangebote zu bekommen, ist dieser Artikel ideal für Dich.

Hier erfährst Du von Strategien die Du sofort praktisch einsetzen kannst.

Falls du einen Übungstest machen möchtest kannst du hier jederzeit einen der kostenlosen numerischen Tests ausprobieren. Dieser Test beinhaltet zehn Fragen (mit Antworten und ausführlichen Erklärungen).

Wie kann man sein Ergebnis so schnell und effektiv wie möglich verbessern , selbst bis in der 99% Bereich ?

Lies den Artikel am besten langsam durch, folge unseren Tipps und unseren Empfehlungen – so hast du die größten Erfolgschancen. Wenn du damit fertig bist kannst du einen unserer Übungstests kostenlos ausprobieren.

Bonus: Kostenloser uneingeschränkter Zugang zum Eignungs-Übungstest (für 30 Minuten) auf unserer Partner-Webseite JobTestPrep.

Microsoft is one of the world's most commonly used computer software.

If you're working in an office, you are almost certain to use applications such as Microsoft Word, Excel, Outlook or PowerPoint.

Therefore, it makes perfect sense that employers want to know that you are proficient in these applications as part of their hiring process.

If your job requires data analysis or compiling data streams, you will likely need to be adept at using Microsoft Excel.

In these circumstances, you may be asked to participate in an Excel assessment test so a hiring manager can confirm that you know how to make the most out of the program.

With this in mind, we will look at what you could expect from a Microsoft Excel test.

Then, we'll take you through a series of Microsoft Excel practice test questions, and we'll give you everything you need to know so you can prepare for the Excel assessment.

What Is a Cubiks Test?

The Cubiks tests were developed by the Cubiks assessment consultancy, which was founded in 2000.

In 2019, Cubiks was acquired by PSI Talent Management UK, an award-winning provider of psychometric assessments.

In 2022, PSI Services became Talogy.

Cubiks tests are available in more than 50 countries around the world. Many highly-regarded employers in the UK use Cubiks tests, including:

• The UK Civil Service
• National Audit Office
• National Health Service

Cubiks tests are designed to help employers and organisations with recruitment, employee development and talent management. They are well known for their intuitive interface and easy-to-interpret structure.

When applying for job roles, you may be asked to complete one or more types of Cubiks test as part of the screening and selection process.

If you are already working, your employer might ask you to sit a Cubiks test assessment as part of the career development programme or talent management process.

This article offers an overview of what to expect from the Cubiks test. It also includes some Cubiks online test example questions and tips on how to succeed when taking the Cubiks test.

The Korn Ferry assessment is a tool used in the recruiting process for leadership positions.

The tests assess candidates across a range of skills, including:

• Logic reasoning ability
• Numerical reasoning ability
• Verbal reasoning ability
• Personality traits

As a result, the Korn Ferry assessment allows businesses to secure the best talent and identify individuals to be promoted to management positions.

The Korn Ferry assessment is an evaluation tool used by companies across the globe to ensure they employ the best talent.

The assessment comprises a series of smaller tests focusing on:

• Reading comprehension
• Personality
• Leadership assessments

As well as a tool utilized during the interview process, the Korn Ferry assessments are often used when looking to promote team members into management positions.

This article will discuss the Korn Ferry assessment, explaining exactly what it involves and giving tips to enable the best chance of success.

If you are applying for an executive-level or management role, you might be expected to take an aptitude test as part of the recruitment process.

The Swift Executive Aptitude Test is a short assessment designed to measure specific aptitudes that are necessary for success in a leadership position.

In this article, you will discover more about the test, the structure of the assessment, and example questions.

You will also learn what you will need to bear in mind to be successful in the test, including tips about preparation and a breakdown of what to expect from the scoring.

This numerical reasoning practice test has 10 questions.

The test has a mixture of numerical questions that vary in difficulty. 

Answers and full explanations are provided after you have completed a question.  You should aim to complete the test within 10 minutes.

Make sure you read and fully understand each question before answering. Work quickly, but don't rush. You cannot afford to make mistakes on a real test.

What is a Verbal Reasoning Test?

A Verbal Reasoning Test is a type of cognitive assessment designed to evaluate an individual's ability to comprehend and analyze written information, make logical deductions and draw conclusions based on the presented text.

These tests are often used in various educational and employment settings to assess a person's verbal reasoning skills, which are essential for tasks that involve understanding and interpreting written or spoken language.

This inductive reasoning practice test has nine questions (and includes answers and full explanations).

This abstract reasoning practice test has 10 questions (and answers with full explanations).

For each question, choose which of the figures in the bottom line – A, B, C, D or E – completes the series in the top line.

The level of difficulty varies significantly, from easy to extremely hard. Items having the solution based on one rule are easy, while those with the solution based on four rules are extremely hard; the others are in between - medium and hard, respectively.

Your goal is to understand the logic of each question (the rules behind it). Do not despair if you can’t find the solution immediately, especially for the very hard questions!

What is a Cognitive Test?

A cognitive test is an assessment tool designed to measure an individual's cognitive abilities, which are the mental processes involved in acquiring, processing, storing and using information.

Cognitive assessments are used to evaluate various aspects of cognitive functioning, including memory, attention, problem-solving, reasoning, language comprehension, and more.

Cognitive function tests are commonly employed in several contexts, including education, clinical psychology, neuropsychology and employment assessment.

This cognitive ability practice test has been designed to help you prepare for the real thing.  

What is a Deductive Reasoning Test?

A deductive reasoning test is a type of cognitive assessment that measures a person's ability to draw logical conclusions based on given information or premises.

Deductive reasoning is a form of logical thinking that involves moving from general statements or principles to specific conclusions. In other words, it is the process of applying a general rule or premise to a specific situation to determine a particular outcome.

In a deductive reasoning test, you are typically presented with a set of premises or statements that establish certain conditions or facts. You are then asked to use these premises to determine a valid conclusion.

The conclusions you reach must follow logically from the given premises, and the test assesses your ability to make accurate deductions based on the provided information.

Deductive reasoning tests are often used in educational settings, as part of standardized testing, and in various employment assessments.

They are designed to evaluate an individual's problem-solving skills, critical thinking ability, and their capacity to analyze information and reach logical conclusions.

These tests can take various formats, including multiple-choice questions, true or false questions or scenario-based questions where you need to determine the correct outcome based on the information provided.

Success in deductive reasoning tests often requires a strong understanding of logical principles and the ability to apply them effectively to specific situations.

What is Logical Reasoning?

Logical reasoning, often referred to as logical thinking or critical thinking, is a cognitive process that involves the ability to analyze information, identify patterns, make sound judgments and draw valid conclusions.

It is a fundamental skill that plays a crucial role in problem-solving, decision-making and rational thinking.

Logical reasoning involves breaking down complex information or situations into smaller, more manageable parts. It requires examining details and understanding the relationships between various elements.

What are the Types of Logical Reasoning Tests?

Logical reasoning tests come in various forms and are used by employers, educational institutions, and standardized testing organizations to assess an individual's ability to think critically and solve problems.

Here are some common types of logical reasoning tests:

Reading Comprehension: These tests assess your ability to understand and analyze written information, make inferences, and draw conclusions from passages of text.

Critical Thinking Tests: These tests evaluate your ability to analyze and evaluate arguments, identify assumptions, and assess the validity of statements or claims.

Analogical Reasoning Tests: Analogical reasoning involves recognizing relationships between words or concepts and applying these relationships to solve problems. For example, you might be asked to complete an analogy like "A is to B as C is to what?"

Numerical Computation: These tests assess your basic arithmetic skills, including addition, subtraction, multiplication, and division.

Numerical Sequences: These tests require you to identify patterns and relationships within number sequences and use them to predict the next number.

Data Interpretation: In these tests, you are presented with data in the form of tables, graphs, or charts, and you must interpret the information to answer questions.

• Abstract Reasoning Tests

Non-Verbal Reasoning: Abstract reasoning tests evaluate your ability to recognize patterns, shapes, and relationships among visual elements. They often involve series of diagrams or figures, and you must identify the logical rules governing them.

Inductive Reasoning: Inductive reasoning tests present you with a series of visual or abstract patterns and require you to identify the underlying rules and predict the next pattern in the sequence.

• Spatial Reasoning Tests

Spatial Awareness: These tests measure your ability to visualize and manipulate objects in three-dimensional space. You may be asked to complete puzzles, identify rotated or mirrored images, or solve spatial problems. Diagrammatic Reasoning Tests:

Diagram Interpretation: Diagrammatic reasoning tests use diagrams or symbols to present problems. You must analyze the diagrams to draw conclusions or identify patterns.

Syllogism and Logic Tests

Syllogisms: Syllogism tests present logical statements and ask you to determine whether a conclusion is valid based on the given premises.

Symbolic Logic: These tests involve working with formal logic symbols to evaluate logical arguments.

Inference and Deduction Tests

Inference Tests: Inference tests assess your ability to make logical deductions and draw conclusions based on a set of statements or information.

Deductive Reasoning: Deductive reasoning tests require you to apply deductive logic principles to solve problems and make decisions.

• Mechanical Reasoning Tests

Mechanical Understanding: These tests evaluate your knowledge of mechanical and physical concepts, such as gears, pulleys, levers, and basic physics principles.

• Cognitive Ability Tests

Cognitive Ability Tests: These assessments often include a combination of various reasoning types and are designed to measure overall cognitive abilities.

What are the Common Logic Tests Employers Use?

Employers often use a variety of logic tests to assess the cognitive abilities and problem-solving skills of job applicants. The specific logic tests used can vary depending on the nature of the job and the industry.

Here are some common logic tests that employers may use during the hiring process:

• Logical Deduction and Syllogism Tests
• Data Interpretation Tests
• Diagrammatic Reasoning Tests

This is a 10 question practice logical reasoning test . 

After you have given an answer to a question, the correct answer (and a full explanation of that answer) will be given.

What are the Topics Covered by a Logical Reasoning Test?

Syllogism, statements and assumptions, logical deduction, cause and effect, statements and conclusions, logical problems.

Set of 10 questions, along with correct answers and explanations for each.

Topics Covered:

General concepts, levers, springs, pulleys, area and volume, gears, inclined plane, basic electrical circuitry.

Difficulty Level:

Take a free practice mechanical reasoning test.

Situational awareness, evaluation of alternatives.

Take a Free Practice Situational Judgement Test

Block counting, 3D rotation, 2D rotation, reflection, broken shapes, transforming 2D to 3D, isometric view, difference in 2D versus 3D viewing.

What is the Watson Glaser Critical Thinking Test?

The Watson-Glaser Critical Thinking Test, often referred to as the Watson-Glaser test, is a widely used assessment tool designed to evaluate an individual's critical thinking skills.

It is commonly administered as part of the hiring process for various professional and managerial positions, particularly in fields where critical thinking and problem-solving abilities are highly valued, such as law, finance and management.

This is a nine question diagrammatic reasoning practice test.

We recommend a time limit of nine minutes for this test. 

After you have given your answer to a question, you will be shown the correct answer and given a full explanation.

What is the Critical Thinking Test?

The Critical Thinking Test is a comprehensive evaluation designed to assess individuals' cognitive capacities and analytical prowess.

This formal examination, often referred to as the critical thinking assessment, is a benchmark for those aiming to demonstrate their proficiency in discernment and problem-solving.

In addition, this evaluative tool meticulously gauges a range of skills, including logical reasoning, analytical thinking, and the ability to evaluate and synthesize information.

This article will embark on an exploration of the Critical Thinking Test, elucidating its intricacies and elucidating its paramount importance. We will dissect the essential skills it measures and clarify its significance in gauging one's intellectual aptitude.

We will examine examples of critical thinking questions, illuminating the challenging scenarios that candidates encounter prompting them to navigate the complexities of thought with finesse.

Critical Thinking Practice Test

Before going ahead to take the critical thinking test, let's delve into the realm of preparation. This segment serves as a crucible for honing the skills assessed in the actual examination, offering candidates a chance to refine their analytical blades before facing the real challenge. Here are some skills that will help you with the critical thinking assessment: Logical Reasoning: The practice test meticulously evaluates your ability to deduce conclusions from given information, assess the validity of arguments, and recognize patterns in logic. Analytical Thinking: Prepare to dissect complex scenarios, identify key components, and synthesize information to draw insightful conclusions—a fundamental aspect of the critical thinking assessment. Problem-Solving Proficiency: Navigate through intricate problems that mirror real-world challenges, honing your capacity to approach issues systematically and derive effective solutions. What to Expect: The Critical Thinking Practice Test is crafted to mirror the format and complexity of the actual examination. Expect a series of scenarios, each accompanied by a set of questions that demand thoughtful analysis and logical deduction. These scenarios span diverse fields, from business and science to everyday scenarios, ensuring a comprehensive evaluation of your critical thinking skills. Examples of Critical Thinking Questions Scenario: In a business context, analyze the potential impacts of a proposed strategy on both short-term profitability and long-term sustainability. Question: What factors would you consider in determining the viability of the proposed strategy, and how might it affect the company's overall success? Scenario: Evaluate conflicting scientific studies on a pressing environmental issue.

Question: Identify the key methodologies and data points in each study. How would you reconcile the disparities to form an informed, unbiased conclusion?

Why Practice Matters

Engaging in the Critical Thinking Practice Test familiarizes you with the test format and cultivates a mindset geared towards agile and astute reasoning. This preparatory phase allows you to refine your cognitive toolkit, ensuring you approach the assessment with confidence and finesse.

We'll navigate through specific examples as we proceed, offering insights into effective strategies for tackling critical thinking questions. Prepare to embark on a journey of intellectual sharpening, where each practice question refines your analytical prowess for the challenges ahead.

This is a three question practice in-tray exercise.

If you get a question wrong, make sure you find out why and learn how to answer this type of question in the future. 

Take a Free Practice In-Tray Exercise

What Is the PwC Assessment Test?

When you apply for a coveted role at PwC, you will be asked to undertake a PwC assessment test as part of the recruitment process.

The PwC test are used to evaluate candidates on measurable skills, abilities, aptitudes and personality traits that are needed for success in the type (and level) of the role that you have applied for.

PwC is one of the Big Four accounting firms globally, and from their headquarters in London, England, they have offices in 157 countries, a presence in 742 locations, and they currently employ nearly 300,000 staff.

With roles available in various departments, from consulting to legal, operations to audit, and tax to technology, competition for advertised jobs is fierce, and the PwC assessments are recognised as being particularly challenging to help narrow down the candidate pool to those applicants who really have what it takes to be successful.

In fact, less than 50% of candidates will advance past the screening tests as the benchmark for a passing mark is very high.

'AON assessments' are the new name for the cut-e tests, and they are often used as pre-employment evaluations for different skills, aptitudes, competencies and personality traits for various roles across different industries.

The AON assessments are characterized by being very short online assessment tests, and in many cases, candidates will be required to take more than one as part of a recruitment process.

With so much content to cover in all the different types of tests, it can be difficult to know what to expect from the AON assessments, which is where this guide will help.

Below you will learn more about why AON assessments are used and which companies use them as part of their hiring process.

We will discuss some of the features that the assessments have in common, as well as the most popular tests that are used by recruiters.

There will be some example questions with answers to get you familiar with the type of content you will be facing in certain tests and some helpful information regarding the way the AON assessments are scored and how you can give yourself the best chance to demonstrate that you have what it takes to be successful.

What Is the AON Assessment Test?

AON is well-known as a global financial services firm, and they acquired the cut-e testing battery so that they can provide top-of-the-range candidate evaluation and personnel development tools based on a scientific framework and testing methodology.

The CogAT Grade 4 test is used to understand a student’s thinking and reasoning abilities. It is not a test of learned knowledge; rather, it is a diagnosis of how they learn.

The 4th Grade CogAT test measures reasoning ability in three key areas: verbal, non-verbal and quantitative.

The assessment is often used to identify students for gifted and talented education programs.

If your child has been selected to sit the CogAT test in 4th grade, it can be confusing to know what to do to help.

This article will help you to answer these questions:

• What is the CogAT test ?
• What skills is the test assessing?
• What is the format of the test?
• How can I help my child prepare?
• What skills can we practice?
• What is the scoring system?

When applying for a job application, you may find that, along with providing your CV and attending an interview, you will be required to complete an IKM assessment .

This assessment will serve as a supplement to your overall application. So, you must understand what it entails and how it contributes to your application.

This article will explain the specifics of the IKM assessment, why it is important and how you can prepare for it.

What Is IKM?

The International Knowledge Measurement Service (IKM) offers organizations various assessments for employees and candidates among various career disciplines.

Among other things, this assessment ensures that employees hold the necessary requirements to go through the organization’s recruitment process.

Employee candidates will take the IKM assessment online remotely (self-supervised) or with client-side supervision from the organization.

The IKM assessment uses adaptive testing, meaning the difficulty of questions is dynamically selected based on the employee candidate’s previous answers .

This ensures that the assessment questions are neither too difficult nor too easy, greatly reducing the testing time.

The CAT4 Level D is a cognitive ability test used by a number of UK secondary schools. Typically taken by pupils in Year 7, the CAT4 Level D tests a child’s verbal, non-verbal, quantitative and spatial reasoning skills to give an accurate picture of their learning potential.

The Delta Assessment Test is a group of online tests that forms part of the Delta Airlines hiring process.

If you are applying for job roles with Delta, you may be asked to complete one or more of the Delta Assessment Tests.

Your test results will help the hiring manager to decide whether you are suitable for the job role you have applied for.

The tests you are asked to take will vary according to the job role.

The Deloitte immersive online assessment is a psychometric aptitude-style test. It is used to identify a candidate’s strengths and weaknesses.

Questions vary but are likely to include situational judgment style questions that link to the roles at Deloitte.

Candidates are also tested on their numerical reasoning and presented with personality questions.

Competition is tough for jobs on the Crossover recruitment platform.

There are thousands of applicants for each role, and only the top 1% are offered a contract .

After a successful initial application, the first step is taking the Crossover Cognitive Aptitude Test (CCAT).

To help you prepare, this article covers the following:

• How Crossover works
• The recruitment process
• What to expect in the CCAT
• The scoring system
• Tips to help you prepare

The Federal Bureau of Investigation (FBI) is the domestic intelligence and security service of the USA.

The agency investigates serious offenses such as terrorism, public corruption, cyber-attacks, and violent and organized crime.

The FBI's mission is to protect the American people and uphold the American Constitution.

The FBI has over 37,000 employees across hundreds of locations in the US.

To work for the FBI, you must fulfill specific criteria which include:

• Be a US citizen
• Be able to obtain an FBI Top Secret clearance
• Pass the FBI polygraph examination
• Pass the FBI Phase 1 test
• Adhere to the FBI drug policy

Roles available at the FBI include computer scientists, nurses, engineers, technicians, contract specialists, and of course, police officers.

It is important to note that the recruitment process can take over one year, so you must be willing to wait several months for the chance of your dream role.

In this FBI Phase 1 test prep guide, we will delve into the role of FBI special agents – upholders of the law that seek out cybercrime and infiltrate organized attacks such as terrorism.

When applying to be a special agent, you are required to take the FBI Phase 1 test .

What Is the FBI Phase 1 Test?

The FBI Phase 1 test is an assessment that evaluates your personality and suitability for a role as a Special Agent at the FBI.

The test is conducted online and is split into five parts.

As the second stage of the process, the FBI Phase 1 test is done after the successful completion of a written application.

The test is designed to assess several skills and qualities that are required for a role as an FBI special agent.

These include critical thinking, logical reasoning and personality. The test will also assess your background experiences.

Your answers are then compared to the benchmark of what is suitable for an FBI agent.

The five sections of the FBI Phase 1 test are:

• Logical reasoning
• Figural reasoning
• Personality Test
• Preferences and interests
• Situational responses

The assessment takes three hours to complete.

When applying for roles at the FBI, long waiting times are typical. The full special agent recruitment process can take over 20 months to complete.

If this is your dream job, it is certainly worth the wait as it is one of the most attractive career paths within any government agency.

To reflect this, the recruitment process is challenging and designed to reduce the number of candidates who could move on to the next stage.

This ensures that only the very best move through the application phases. In fact, only 30% of candidates can pass the FBI Phase 1 test.

You may have taken a personality test before, but the FBI Phase 1 test questions are framed and marked in a different way to other assessments.

Therefore, you should ensure you use FBI Phase 1 test practice questions and prepare in advance of the test.

It can be hard to plan for, but this is essential to get into the top 30% of successful candidates.

If you pass the FBI Phase 1 test, you will undergo background checks and receive an invitation to a regional meet-and-greet interview.

The main purpose of the CogAT Test grade 3 is to find out if a third grader is showing signs of being very smart.

Most of the questions on the test are about verbal, numerical and non-verbal reasoning. It's meant to show how a child might compare to other kids his or her own age. The CogAT grade 3 test can also be used to make individualized learning plans for kids.

The CogAT (Cognitive Abilities Test) is a standardized test used to measure children's cognitive abilities in the 3rd grade – age 9.

This test assesses a range of cognitive abilities, including verbal, quantitative and nonverbal reasoning. The CogAT is often used to identify gifted children and help educators develop appropriate educational plans.

This article will give insights and tips into how your child could pass the CogAT Test for 3rd grade students.

The MAP Test 2nd grade is a computerized test taken by children in the 2nd grade. It is designed to evaluate what the children already know and what they are ready to learn.

The test includes three sections:

Schools may not administer all three sections and may instead focus on one or two sections to measure pupils’ progress in those subjects.

The NEO Personality Inventory is a psychometric tool used to evaluate personality traits.

It is acknowledged globally and is used by recruiters and employers before hiring and, more broadly, to evaluate career potential.

The NEO Personality Inventory test is heavily associated with the 'Five-Factor Model' (which you may also know as the 'Big Five Personality Test') to identify personality traits.

It is widely believed that each person's personality can be broken down into five main categories. The NEO PI personality test looks at each of these five categories separately to create an understanding of who you are.

In this article, we'll look at the NEO PI test, why employers use it, and what you could expect if invited to participate in a NEO Personality Inventory test.

The Air Traffic Controller (ATC) Test, also known as the Air Traffic Skills Assessment (ATSA) is an exam used as part of the air traffic controller hiring process. It is a challenging assessment consisting of seven subtests designed to evaluate an applicant's aptitude for the role.

Becoming an air traffic controller is a challenging and rewarding career that requires extensive training. The Air Traffic Controller Test (previously known as the Air Traffic Selection and Training (AT-SAT) exam) is an important part of the selection process. 

The Air Traffic Skills Assessment (ATSA) measures a candidate's ability to handle the demands of the job. 

In this article, you’ll find example questions, a guide and tips for preparing for the ATSA exam.

This article relates specifically to the ATC test used in the US. Candidates in other countries may be expected to take a different version of the test. 

What is the CliftonStrengths test? This online assessment analyzes your personality and strengths for personal and professional development. You can purchase the basic test from Gallup for $19.99 and get a basic understanding of your top five personality themes. Or take the comprehensive version for $59.99 and receive a report that ranks all 34 themes and highlights your areas of excellence as well as your blind spots.

When applying for a job, you may find that the recruitment process consists of many different steps. There is the initial application form to start and usually an interview to finish. In the middle, there may be an assessment – an aptitude, intelligence or personality test.

The CliftonStrengths test is one assessment used by employers during the onboarding process. It was previously known as the CliftonStrengthsFinder.

In this guide, you will learn about the CliftonStrengths personality test and how it is used in recruitment.

The police psychological exam is a crucial part of the hiring process for law enforcement agencies. It is a personality test that confirms how suitable an applicant is for working in the police. The police psych test is used by most law enforcement agencies across the United States, although key details may differ from state to state.

What Is the Police Psychological Exam?

The police psychological exam is a series of tests and assessments administered to individuals who are seeking to become police officers.

The purpose of the exam is to evaluate a candidate's psychological fitness for the job and identify any potential psychological issues that may interfere with the candidate's ability to perform police work.

In this comprehensive guide , you’ll discover everything you need to know about the Capital One assessment and interview process.

These are designed to help the company select the best candidates for its team. To increase your chance of getting hired, it's important to be prepared.

Find out what to expect, how to prepare and the skills and qualities Capital One hiring managers are looking for in a candidate.

What Is the Capital One Assessment Test?

Capital One is an established financial services company with a focus on technology and innovation.

To become an employee, or ‘associate’, at Capital One you'll need to pass a series of online assessments and interviews .

The Capital One hiring process is as follows:

CogAT stands for Cognitive Abilities Test. These tests are normally administered by a classroom teacher or instructor, although some schools employ a specialist or test proctor to administer the test.

Many parents are interested in learning more about helping their children to succeed academically.

Achieving a high CogAT score could mean your child is eligible to join gifted or talented programs designed to enhance their development and learning.

In other schools, it is used as a tool to identify a pupil’s individual strengths or predict their future academic performance.

The CogAT test for 2nd grade is a cognitive ability test aimed at children around the age of eight years old.

It is often used as a pre-admission exam by gifted and talented schools and programs. It is designed to evaluate pupils’ cognitive abilities, including basic linguistic and math skills.

The test is made up of three sections or batteries:

• Non-verbal battery
• Verbal battery
• Quantitative battery

On the CogAT test 2nd grade, candidates are required to read the test questions instead of listening to the questions being read by the test proctor.

If you are looking for ideas on how to prepare your child for the CogAT test 2nd grade, read on to learn more.

What Is CogAT Test 2nd Grade?

The CogAT (Cognitive Abilities Test) was developed by Riverside Publishing, which is part of Houghton Mifflin Harcourt.

It is designed to assess problem-solving and reasoning skills in the following areas:

• Quantitative

Research has shown that high levels of ability in these three areas is linked to academic success.

If your child is considered potentially talented or gifted, they may be asked to sit a CogAT as part of the program entrance process.

Different CogAT tests are available for different age groups, from Kindergarten (K) up to grade 12.

In this article, you can find more information on the CogAT test 2nd grade. The CogAT test is used by schools across the US to help them identify exceptionally gifted pupils.

Each of the test levels corresponds to the age of the pupil sitting the test. For example, if your child is in grade 6 (aged 12), they will be sitting the Level 12 version of the test. Occasionally, schools may choose to administer a higher level CogAT to talented or gifted pupils; however, this is unusual.

Second grade pupils being considered for gifted programs will usually sit the CogAT Level 8 test. This test is made up of 154 questions and takes 122 minutes to complete.

Many schools use the CogAT Test 6th Grade to assess the non-verbal, verbal and quantitative abilities of sixth-grade students.

The Level 12 CogAT test is a useful tool for checking a student’s individual academic strengths and weaknesses. It can also be used as a screening assessment for entry into the gifted and talented program.

What Is the CogAT Test 6th Grade?

'CogAT' is an acronym for Cognitive Aptitude Test .

CogAT tests are usually administered at school by a teacher or instructor, although some schools employ test proctors and specialists to administer the tests.

This guide is designed to support you and your child through the CogAT Test 6th Grade. You can use it to find out what to expect from the test and tips on how to prepare for it.

We have also included information on the purpose of the test and how to interpret your child’s results.

The Procter and Gamble Assessment Test describes a series of pre-employment screening tests used by Procter and Gamble (P&G).

If you have applied for a job at P&G, you will be expected to sit these tests as part of the hiring process.

Each of the different tests is designed to assess a specific aptitude that is required for a job role at P&G.

In this article, you can learn more about the different tests used by Procter and Gamble. We have also provided tips on how to prepare for the assessments.

This guide includes useful tips and Renaissance Star testing sample questions to help students prepare for the test and feel confident on test day.

You can find detailed information on interpreting and understanding your Renaissance Star Test scores in our dedicated article .

What Is the 6th Grade MAP Test?

The MAP Growth test system was created by educators from Oregon and Washington who established the Northwest Evaluation Association (NWEA) back in 1973.

Their goal was to create an assessment that could accurately measure and track academic progress in children to ensure they graduated high school with all the essential skills and knowledge they required.

In 2000, the first MAP Growth Test was published.

The test is administered in all grades and is based on a set of learning principles known as the Common Core Principles .

The CCAT test grade 3 is a standardized assessment administered to grade 3 students in Canada.

It measures verbal, quantitative and non-verbal reasoning skills and is used to identify a student's learning potential, typically for admission to gifted educational programs.

The CCAT test grade 3 is an assessment commonly used by schools in Canada.

If you’re the parent or guardian of a child preparing for the test, this CCAT grade 3 guide will tell you everything you need to know.

What Is the CCAT Test Grade 3?

The CCAT test (Canadian Cognitive Abilities Test) is a standardized assessment administered to students in grade levels K-12 in the Canadian educational system.

Rather than a measure of academic achievement, the test assesses a child's ability to learn, reason, and problem-solve.

The Independent School Entrance Examination (ISEE) test is used by many independent and magnet schools in the US and overseas as an admission test for children across the entire school age range, but more commonly from year five upwards.

It assesses a child’s academic levels of reasoning across math and literacy in comparison to children of the same age, the norm for that school grade and other applicants to the school.

Created and administered by the Educational Records Bureau (ERB), the ISEE test is available to be taken online or in a pen and paper format.

What Are the ISEE Levels?

There are four levels of the ISEE test.

• ISEE primary for entry into years two to four
• ISEE lower level for entry into years five to six
• ISEE middle level for entry into years seven to eight
• ISEE upper level for entry into years nine to 12

Each level of the ISEE test is created to be relevant to a specific school age group, increasing in complexity with each year and level.

An employer’s recruitment process can include a wide range of assessments and interviews for the candidate to take that indicate to the employer how an individual might fare in the job.

One common way to measure job performance though is by getting candidates to take the PI Cognitive Assessment, which measures mental ability and critical thinking skills.

This article will look in detail at the assessment, its format, who uses it, example questions and PI Cognitive Assessment tips on how to be successful when taking it.

The Raven’s Progressive Matrices is a test that is often used as part of the recruitment process for high-level management and analytical roles.

In this article, you will learn more about the test, its history and background, as well as the different types of tests that are available and what you can expect if you are going to be taking the test.

You will also find some example questions that you can expect to see in each type of test and get helpful pointers that you can use to prepare and do well in the assessment.

If you are applying for a role with the United States Postal Service (USPS) , you will usually be asked to complete at least one of four 477 Virtual Entry Assessments as part of the recruitment process.

These exams are used to evaluate various skills, aptitudes, personality traits and work preferences, which can show whether you have what it takes to be successful in the role in the future.

The USPS 477 Exam is sometimes referred to as the CS VEA, which relates to customer service.

An iReady level score of 3.00 or over means the student is working at or above the level required to meet the standard for their grade.

The level score is calculated in line with expectations when the test was administered, not in comparison to the expected score by the end of the school year.

What Are the iReady Diagnostic Scores?

The iReady diagnostic test is administered to US school children in grades K to eight.

The purpose of this school assessment test is to help parents and teachers check a student’s academic process at the beginning, middle and end of each school year.

It is a computer-adaptive test, which means the questions are adjusted to become more difficult if a series of correct answers is given.

As a result, the test is designed to challenge the skill level of the student sitting the test, as well as assess their strengths and opportunities for growth.

If a student answers a few questions in a row incorrectly, the questions that follow will be easier.

Many people find i-Ready Diagnostic scores difficult to interpret.

As a child progresses through each academic year and moves up the year groups, their expected score will change.

The average score increases year on year, too.

In this article, you can learn more about the different types of iReady diagnostic scores, how these scores are displayed, and how to interpret them to better understand a student’s iReady test performance.

There are two types of HESI Exam:

• The Admissions (A2) test
• The Exit exam

The minimum passing score for the Admissions test is usually between 75 and 80 for each section, although this varies between schools.

The composite score range for the Admissions (A2) test is 750 to 900, with 900 being the maximum possible score.

The HESI Exit Exam score ranges between 0 to 1,500. 850 is considered to be an acceptable score, although HESI recommends a minimum score of 900.

If you want to sit your NCLEX licensing exam, you will need to achieve a score of at least 850 on the HESI Exit Exam.

HESI is an acronym for Health Education Systems Incorporated .

As a company, HESI administers exams and provides study material to help prepare students for the NCLEX professional licensure exam.

If you want to work as a nurse in the US, many nursing and healthcare programs use HESI tests to screen prospective students and determine suitability and readiness for specific study routes.

In this article, you can learn more about the HESI score ranges and passing scores required for each of these tests and what impact your HESI results may have on acceptance into your preferred nursing program.

The CogAT Kindergarten Test is an assessment designed to measure a child's abilities in various cognitive areas.

It plays a critical role in identifying a child's strengths and weaknesses and determining their readiness for advanced academic programs.

In this comprehensive study guide for 2024, you will explore the purpose, format, and structure of the CogAT Kindergarten Test.

Additionally, you will get valuable insights on how to prepare your child for the test, sample questions to familiarize yourself with the test content, strategies for success and answers to frequently asked questions.

Understanding the CogAT Kindergarten Test: Purpose, Format, and Structure

The purpose of the CogAT Kindergarten Test is to assess a child's cognitive abilities in areas such as verbal, quantitative, and nonverbal reasoning.

By evaluating these different components, the test provides educators and parents with valuable information about a child's potential and can help guide educational decisions.

The ATI TEAS Test , also known as the Test of Essential Academic Skills, is an important exam for students looking to pursue a career in the healthcare field. The most recent version is the ATI TEAS 7.

This comprehensive exam assesses a student's knowledge in various areas, including reading, math, science and English language usage.

If you're planning to take the ATI TEAS Test in 2024, it's essential to understand what the exam entails and how to best prepare for it.

In this article, we'll cover everything you need to know about the 2024 ATI TEAS Test.

The ATI TEAS 7 Math Test is a crucial component of the ATI TEAS exam, which is widely used by nursing and allied health schools to assess prospective students' academic readiness for their programs.

In this comprehensive guide, you will delve into various aspects of the TEAS Maths 7 Test, including what it entails, when it is taken, ATI TEAS math practice test questions to help you prepare, and tips for success.

So, let's dive right in!

The 1st Grade CogAT test is an important assessment that measures a child's cognitive abilities. It is designed to identify a child's strengths and weaknesses in areas such as verbal, quantitative, and non-verbal reasoning.

This article will provide you with a comprehensive guide on understanding and preparing for the 1st Grade CogAT Test.

The CogAT (Cognitive Abilities Test) for Kindergarten is an assessment designed to evaluate the cognitive development and problem-solving abilities of young children.

Typically, this version of the test is tailored to children around five to six years old who are attending kindergarten.

The test is typically used for educational placement, identifying gifted and talented students, and understanding a child's cognitive strengths and weaknesses. It can be administered individually or in groups, and is often used by schools to tailor instruction to better meet the educational needs of their students.

Understanding the results of the CogAT can help educators and parents support the child's learning and development more effectively, by identifying areas where the child excels or may need additional focus.

In this article, you’ll find practice CoGAT Kindergarten practice test questions and tips to help your child prepare for the big day.

The Cognitive Abilities Test (CogAT) 5th Grade Level is a crucial assessment tool for students between 10 and 11 years old.

Designed to measure verbal, nonverbal, and quantitative abilities, this standardized test plays a pivotal role in identifying students for gifted programs.

In this article, you’ll learn what the CogAT 5th grade test is, which subjects are tested, along with example questions and how best to prepare.

What Is the CogAT 5 Grade Test?

The Cognitive Abilities Test (CogAT) is a widely used standardized test designed to assess your child’s cognitive abilities in various areas.

The CogAT 5th Grade Level is specifically tailored for students in the 5th grade and measures their abilities in three main cognitive areas:

• Quantitative Reasoning
• Non-Verbal Reasoning

The MAP Test Grade 7 tests students’ proficiency in mathematics, reading and language usage.

Developed by the Northwest Evaluation Association (NWEA), it measures individual growth over time, adapting question difficulty based on responses.

This online test lasts around two to three hours, and the results are used to inform teaching or gauge students' ability levels.

Scoring is based on the RIT (Rasch Unit) scale, indicating a student's instructional level and growth potential in each subject area.

MAP Grade 7 Sample Question

The State of Texas Assessments of Academic Readiness (STAAR) test is a standardized assessment issued to public school students in Texas in grades 3 to 12.

Below you’ll find a range of STAAR test practice questions to help you prepare – whether you’re a parent coaching a child through their exam prep or a high school student revising for a test of your own.

For more info on the STAAR Test, read our dedicated article.

The STAR Early Literacy Test is an assessment tool used to measure children’s early literacy skills. It forms part of the wider Renaissance STAR (Standardized Test for the Assessment of Reading) assessment system by Renaissance Learning.

The STAR Early Literacy Assessment is mostly used to test students from pre-kindergarten to grade 3.

The test is designed to assess the following areas of early literacy:

• Phonemic awareness
• General vocabulary
• Comprehension
• Reading ability
• Early numeracy skills

STAR Early Literacy is a computer-adaptive test. This means that the difficulty of the questions adjusts according to a student’s responses.

The adaptive element of the test allows for more precise results and a better insight into a student’s overall literacy skills.

Word games are a great way to help your child prepare for the STAR Early Literacy Test.

You should also encourage your child to read daily.

You may wish to build this into their routine at certain times of the day. For example, reading before going to bed is often a good way to unwind.

If you are looking for other ways to help your child prepare, you can help them practice their time management skills, talk to them about maintaining a positive attitude towards the test and ensure they are getting sufficient rest.

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Administered at college and university level, the Accuplacer test is used by some educational institutions to determine how prepared a student is for the next steps in their academic career.

This guide looks specifically at Accuplacer test scores – how they are awarded and what they mean – so you can better understand how your Accuplacer score might impact your learning experience.

Accuplacer test scores are a set of metrics that evaluate a student's knowledge and skills in specific subject areas including reading, writing and math.

The Wechsler Intelligence Scale for Children (WISC-V) is a commonly used assessment for judging a child's intelligence. More than that, it can help to understand their reasoning and thinking abilities to support their development.

Here’s everything you need to know about this test.

The Wechsler Intelligence Scale for Children - Fifth Edition (WISC-V) is an individually administered and extensive evaluation tool used to assess children's reasoning and general thinking abilities.

It's typically given to children between ages 6 and 16.

After completing a test, children are awarded a Full-Scale Intelligence Quotient (IQ) score, along with age-based scores and rankings in several cognitive function fields.

Here we’ll provide an all-around study guide for parents whose children are required or scheduled to take the WISC-V test.

We’ll also include a comprehensive explanation of how it is constructed, its key features, tips for preparing, and a few example questions.

Let’s take a look!

The STAR assessments utilize a scoring system comprising scaled scores ranging from 0 to 1,400.

These scores reflect a student's proficiency level in subjects such as reading and math.

Benchmark categories provide descriptive labels for performance levels, while percentile rank compares a student's performance to a national reference group.

Additionally, grade equivalent scores and domain scores offer insights into grade-level equivalence and specific skill areas.

The STAR Assessment can play a crucial role in evaluating your child’s academic ability and guiding educational strategies.

Understanding its scoring system, test format and significance is important for parents and educators alike.

This article aims to provide comprehensive insights into the STAR Assessment, including its purpose, score interpretation and effective strategies to help children excel in these standardized tests.

The CogAT raw score represents how many questions were answered correctly on the CogAT test. This information is used to create the Universal Scale Score (between 100 and 150), which you will see on your child’s CogAT score report.

Here is an image of a typical score report:

With the MAP Growth Test used in many schools across the United States, MAP (Measures of Academic Progress) scores are an important part of your child’s life.

The MAP testing scores chart a student’s academic growth in a way that highlights areas of excellence and improvement.

It is essential that you understand how NWEA MAP scores are calculated so you can best support your child throughout their learning journey.

This guide will explain how to find and improve your child’s NWEA Map Scores.

The main three sections for the Upper and Middle level tests have a maximum score of 800. They have a total scaled score that ranges between 1,500 to 2,400.

Navigating the SSAT involves understanding its scoring system.

In this guide, you can explore the SSAT Score Chart and understand score ranges and percentile ranking and how they matter in private school admissions.

It's a comprehensive resource for decoding SSAT scores and making informed decisions about your child’s education.

What Is the SSAT Test?

The SSAT stands for the Secondary School Admission Test. The SSAT was first administered in 1957.

It is a standardized test designed for students seeking admission to private middle and high schools.

The primary purpose of the SSAT is to assess the skills and knowledge of students applying to independent or private schools.

It aims to provide an accurate measure of a student's academic abilities and readiness for a challenging curriculum.

As with other careers, joining the US military comes with its own set of recruitment processes, one of which is taking the ASVAB test .

If you’ve been looking to pursue a career in the US military, then it might be a test you’ve become familiar with or heard of before. It is an exam a recruiter will advise you to take prior to joining the armed forces.

The ASVAB , otherwise known as the Armed Services Vocational Aptitude Battery , is a test the armed services use to determine which part of the US military you will be most suited to join.

Within this article we will discuss what your ASVAB score means and what score counts as a good ASVAB score .

Good ASVAB Score Defined

Whether you’re looking to join the Coast Guard, Army, Marine Corps or another sector within the US military, each branch will require its candidates to score a minimum amount to qualify for that specific area.

It is important to note that there is no single ASVAB score , and you will normally receive a variety of different scores on your final report.

The Microsoft Codility Test evaluates coding skills and algorithmic thinking.

Designed to streamline Microsoft’s recruitment process, the Microsoft Codility Test assesses candidates' ability to solve real-world problems efficiently. 

Candidates can prepare using coding practice platforms and mastering programming languages. It's an integral tool in selecting skilled software engineers for Microsoft's diverse roles.

The Smarter Balanced Assessment Consortium Test, known as the SBAC test, is a standardized assessment of English and math used by schools in participating states.

Administered to students in grades K to 12, it measures grade level proficiency and academic progress through computer-adaptive testing and performance tasks.

The Smarter Balanced Test is an educational tool developed and administered by the Smarter Balanced Assessment Consortium (SBAC), hence the abbreviation SBAC test.

In this article we explore what the test involves, what the results mean and how to help a student prepare for their SBAC assessment.

What Is the SBAC Test?

The SBAC assessment is a set of standardized tests that evaluate how well students are performing in the subjects of English Language Arts (ELA) and mathematics.

These assessments are taken by students ranging from elementary school to high school in multiple states across the US.

The tests are developed and managed by the Smarter Balanced Assessment Consortium (SBAC), a collaborative group of states working together.

FireTEAM Test Prep: Top Tips:

• Master time management
• Brush up on basic concepts
• Diversify your reading
• Play observational and memory games
• Assess your communication style
• Prioritize rest and sleep

If you're considering a career in firefighting, taking the FireTEAM test is a pivotal step that can open doors to various fire departments across the US.

This article covers everything you need to know to put in a strong performance, including an overview of its format, practice questions and FireTEAM test tips to help you create an effective study plan.

A career in the fire service is a challenging – but extremely rewarding – journey. Such an important, high-pressure job requires a high level of physical, mental and emotional skills.

As well as the necessary personality traits, you generally need a high school diploma or GED. If you have a college degree, you have a better chance of securing a role in the fire service.

You will also be required to take a series of assessments that evaluate your physical and mental strength. One of the assessments used by Californian fire departments is the FCTC Written Test. To become a firefighter in California, you must pass this entry-level test.

In this guide, we will explore what the FCTC Written Test includes and how you can prepare for success.

To successfully enlist in the US Marine Corps, certain standards must be met. Marines require both physical and mental strength as well as discipline, determination and the ability to overcome obstacles. This is sometimes referred to as the ‘Marine Mindset’.

One of the ways candidates who wish to enlist will be assessed is by taking a test known as the Armed Services Vocational Aptitude Battery (ASVAB).

A good score on the test suggests that a candidate possesses the mental skillset to be successful in the military.

Marines need to be able to make quick, accurate decisions and adapt to and overcome threats and obstacles on the battlefield.

The PiCAT test is a commonly used assessment tool for those applying to military positions, such as those in the US Navy or the US Army.

This article explores the PiCAT test in more detail. We look at the test format to familiarize individuals with what the Navy PiCAT and Army PiCAT test covers.

Preparation is vital to performing to the best of your ability in the PiCAT test.

The article includes PiCAT practice test questions, answers to help you prepare, and tips to give you the best opportunity to approach the test positively.

The Mettl tests are developed by the world's largest assessment provider, Mercer Mettl.

The tests have been designed to analyze various competencies, including verbal, logical and numerical reasoning.

Alongside, the Mettl assessments evaluate candidates' personalities and working styles, establishing whether they are an accurate fit for the role and the broader company.

The Mettl tests are a comprehensive recruitment tool provided by Mercer Mettl – the world's largest assessment provider.

Moreover, the Mettl tests are designed to assess various skills, including numerical , verbal and abstract reasoning.

The assessments are also constructed to understand candidates' behaviors and personality types.

This guide explains everything you need to know about the Mettl test, including tips on how to pass the test in 2024.

What Is the Mettl Test?

As mentioned, the Mettl test is a comprehensive recruitment tool designed to test a range of skills.

It allows employers to ensure they recruit the most suitable candidates for the role.

Mastering the Pipefitter Test is crucial for those entering the field.

This guide provides valuable insights, a pipefitter sample test and strategies to conquer the examination.

Discover expert tips to excel in your pipefitting career by navigating the challenges of this important assessment.

What Is the Pipefitter Assessment Test?

The Pipefitter test is an important evaluation tool for individuals aspiring to secure roles as pipefitters in the construction and industrial sectors.

Qualifications and certifications necessary for such positions can vary by state. This makes the pipefitter assessment test a valuable method of demonstrating skills and knowledge.

The National Center for Construction Education and Research (NCCER) administers the most popular pipefitter assessment test, designed to assess the potential skills of candidates.

It covers the principles related to the installation and maintenance of both high and low-pressure pipe systems.

In addition, it focuses on how these are used across various sectors, including manufacturing, electricity generation and climate control systems in buildings.

The i-Ready Diagnostic Test is an internet-based adaptive diagnostic test linked to the i-Ready educational learning program.

Students from kindergarten to grade 12 take the test three times each year. The test is divided into two subtests:

i-Ready test results are used to help teaching staff create a personalized learning plan according to a student’s strengths and weaknesses.

What Is the i-Ready Diagnostic Test?

The i-Ready Diagnostic Test is a computer-adaptive, untimed assessment for students between grades K and 12.

Administered by Curriculum Associates , teachers can use it to monitor a student’s ability and progress throughout the school year.

In most cases, the i-Ready Diagnostic Test is administered three times each year. It is split into two subtests: math and reading.

The HSBC Online Immersive Assessment contains 38 questions over five subtests. The test includes a combination of behavioural questions and cognitive ability exercises.

It is an untimed assessment, but most candidates can answer all test questions within 50 minutes.

Some people find the test difficult, but adequate preparation will stand you in good stead to pass the assessment.

What Is the HSBC Hiring Process Like?

HSBC is a major global bank and financial institution. It offers services via three global businesses and serves millions of customers daily.

The hiring process at HSBC comprises four key stages:

• Initial Screening and Application
• HSBC Online Immersive Assessment
• Online Job Simulation Assessment

What Is the Electronic Data Processing Test?

The Electronic Data Processing Test (EDPT) is a pre-employment test taken by military candidates who want to transfer to IT or computer programming roles within the Marine Corps or Air Force.

The EDPT test is one of the most challenging pre-employment tests currently on the market with a pass rate of around 10%.

It is 90 minutes long and has 120 multiple-choice questions. This means you have around 45 seconds to answer each question.

While the minimum ASVAB score varies between military branches, the minimum acceptable score is 31.

However, as the majority of candidates score between 30 and 70, you want to aim for a percentile rank of at least 60.

The ASVAB Test Score Report is a valuable document that provides detailed information about your aptitudes, skills, and qualifications for military service.

It includes Career Exploration Scores to guide career choices, individual scores on ASVAB subtests to assess specific abilities and the critical AFQT score that determines your eligibility for enlistment.

Understanding the information presented in this report is essential for making informed decisions about your military career options.

What Is in the ASVAB Test Score Report?

The ASVAB (Armed Services Vocational Aptitude Battery) Test Score Report provides a comprehensive overview of your performance on the ASVAB test, which is a critical step in the military enlistment process.

The report helps you and military recruiters assess your aptitudes, skills, and potential for various military occupations.

What Is the MMPI Assessment?

The Minnesota Multiphasic Personality Inventory (MMPI) is one of the most widely used assessment tools used to help clinically diagnose mental health disorders.

Originally developed in the late 1930s, it is used by mental health professionals, lawyers and even in some cases by employers when they are hiring for positions that are considered to be high-risk, such as working in the police, in nuclear power plants or in air traffic control.

The MMPI is a self-reporting tool that is administered by professionals, and during the assessment, you will be asked to answer hundreds of true/false questions, which help paint a picture of your mental health and your personality traits.

As a diagnosis tool, the MMPI is considered to be clinically accurate. It has been updated multiple times over the years to make it more relevant, especially in terms of cultural sensitivity.

The MMPI offers results that show on a scale what symptoms a person has, and what mental health problems that could be indicative of.

In addition, the MMPI is usually used in tandem with other diagnosis tools to provide a clear picture of a person's mental health.

What Is the ACCUPLACER Reading Comprehension Test?

The Accuplacer Reading Comprehension test is part of a suite of assessments that are used to evaluate students prior to entry at college.

While the Accuplacer test battery is not used to determine whether a student will achieve a placement at college, the results are used to ensure that the student is studying at an appropriate level and is ready for education at this level.

Created by the College Board, which is a not-for-profit organization that is also responsible for creating assessments like the SATs, the Accuplacer tests are designed to offer better opportunities to students and make entry to top colleges accessible to all.

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Common Core: 3rd Grade Math : Solving Problems Involving the Four Operations, and Identifying and Explaining Patterns in Arithmetic

Study concepts, example questions & explanations for common core: 3rd grade math, all common core: 3rd grade math resources, example questions, example question #1 : solve two step word problems using the four operations: ccss.math.content.3.oa.d.8.

Example Question #2 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Example Question #1 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Example Question #4 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Example Question #5 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Now we need to add the watermelon, strawberries, and raspberries together to find our total. 

Example Question #7 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

To find the total number of flyers that she hung we add the amount of flyers on the east side and the amount on the west side. 

Example Question #8 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Example Question #9 : Solving Problems Involving The Four Operations, And Identifying And Explaining Patterns In Arithmetic

Now we need to add up our number of balls, stuffed animals and ropes to find our total. 

To find the total amount of laps that he swam, we need to add up the laps that he did each day. 

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WIAT-4 (WIAT-IV) Achievement Testing – Mathematics

What are the wiat-4 (wiat-iv) mathematics subtests.

The Mathematics Subtests on the WIAT-4 (WIAT-IV) are designed to measure a student’s mathematics proficiency, as well as their fluency or how quickly and accurately they can solve simple math facts. They assess a student’s math skills in a variety of areas, including basic computation, math fluency, math reasoning, and math problem-solving. The test consists of both oral and written questions, and it assesses a student’s ability to apply mathematical concepts to real-world situations.

The WIAT-4 Mathematics Subtest can provide useful information about a student’s math abilities, which can be used to identify areas of strength and weakness, inform educational programming and intervention strategies, and provide important information for educational planning. The subtests for the WIAT-4 (WIAT-IV) have not changed and are the same as the WIAT-III.

WIAT-4 (WIAT-IV) Mathematic subtests consist of numerical operations, math problem solving, and math fluency.  Below are some of the sample questions. 

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Numerical Operations

This section measures the student’s math calculation skills which include solving written calculation problems and solving basic operations. Initially, the student will answer oral questions about number concepts and counting. Later, they will write answers to printed math problems ranging from basic operations to geometry and algebra.

Sample Question #1: Solve the following:

1  +  2  =  _____

Sample Question #2: Solve the following:

8  –  3  =  _____

Sample Question #3: Six children are playing football and 10 children are playing soccer. What is the ratio of the number of students who are playing football to those who are playing soccer?

Answer:  6:10; 6 children playing football to 10 children playing soccer is expressed as the ratio 6:10.

Sample Question #4:  The triangles in the figure below are similar.  What is the value of x?

Math – Problem Solving

This section assesses a student’s problem-solving skills, from reasoning, basic concepts to everyday mathematics applications, including geometry and algebra. Initially, students may be asked to point to an answer or respond orally. This is followed by problem solving questions that require the application of math principles to determine the answers. If needed, students can use empty spaces on the test sheet to work out the problems.

Sample Question #1:  Demi went to visit Sam in the hospital.  When she got to his room, 5 people were there.  3 left.  How many visitors were there while Demi was there?

Answer:  3; 5 – 3 = 2 (including Demi)

Sample Question #3: Which of the following expressions is 5 times smaller than the sum of 4 and 10?

•  4 + (10/5)
•  (4 + 10) x 5
•  (4 + 10)/5

Answer:  4.  (4 + 10)/5; If something is “5 times smaller than” then it is divided by 5.

Sample Question #4:  What is the diameter of the circle below?

Answer:  2. 12cm; The diameter of a circle is a straight line that passes through the center of the circle and whose endpoints lie on the circumference of the circle. 6 cm is the given radius which is half the diameter. Therefore, the diameter is 6 x 2 = 12 cm.

Math Fluency

This section measures math fact accuracy under timed conditions. The problems are presented in a vertical format.

In this 3-part math fluency test, there will be a list of math problems that consist of addition, subtraction, and multiplication.  The student has 60 seconds to answer as many addition problems as they can. Then they have 60 seconds to answer as many subtraction problems as they can.  Then they have 60 seconds to answer as many multiplication problems as they can.  Students are scored based on how many of each type of problem they complete accurately within 60 seconds.

1st and 2nd graders are given addition and subtraction, while 3rd graders and beyond are given all 3, addition, subtraction, and multiplication.

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McKinsey PST: Question Types, Study Plan & Mock Tests

The McKinsey Problem Solving Test (McKinsey PST) is a very crucial part of the McKinsey recruitment process. It is where most of the applicant pool is eliminated, and yet there are so few resources to help you prepare for it. Fortunately, you’ve found the ultimate guide to nail the test with an unbelievable level of detail!

Table of Contents

McKinsey PST overview

What is the mckinsey pst.

The McKinsey Problem Solving Test (or PST) is a paper-based test used at McKinsey & Company to select candidates for the case interviews. The PST is conducted after resume screening; it has 6 types of question, testing the candidate on 3 crucial problem-solving skills – data interpretation, mental calculations and logical reasoning.

McKinsey PST passing score / Acceptance rate

McKinsey has never officially stated the passing score or acceptance rate for the PST. However, these numbers can be estimated using reports from test-takers, with passing score being around 70%, and acceptance rate at roughly 30-35% (1 in every 3 candidates will pass).

Does McKinsey still use the PST?

Currently, the McKinsey PST is being replaced by the new Problem-Solving Game. However, the transition is not complete globally. In addition, the new Problem-Solving Game still retains the core principles from the old test, so preparing for the PST is still relevant.

Why do they need PST?

It’s very similar to what we have here! McKinsey believes that the gap between CV screening and in-person case interviews is too big. The firm may miss many good candidates with bad resumes or may interview too many candidates who don’t live up to their resumes.

At the end of the day, in-person interviews are expensive, and the Problem Solving Test provides a cost-effective solution.

Who has to take the McKinsey PST?

Every candidate who passes the resume screening round has to take the McKinsey PST – if they apply for the management consulting track. Some report indicates that MBA applicants might be waived from the test – however, this is rare and you should confirm with the target office HR.

What does PST look like?

26 questions, 1 hour, paper-based, and no calculator! The test has 26 multiple choice questions set within the context of 3 business cases. A candidate has exactly 60 minutes to finish the test. He will be provided with a watch, pencils, scratch paper, and the test is in a paper-based format. No calculator is allowed. No personal assistant is allowed. Just you and the test!

As the business landscape is changing, candidate recruitment has become increasingly complex. This requires another way of presenting PST content for McKinsey. They have changed the format into gamification and planned to implement this method to all McKinsey offices within 2020. Visit the mock game designed exclusively for MConsultingPrep followers!

PST vs. GMAT vs. SAT?

If you are new to PST, you may hear the myth that PST is similar to the Math section in GMAT or SAT. In fact, being excellent in SAT Math does help with quantitative calculation in PST, however, the context is different. The SAT Math section includes only simple calculations in simple context; meanwhile, logic in business problems is highly emphasized in PST.

Why is this test so challenging?

• You will not have enough time to properly think through each question.

If you are going to read every single word in the case background and do every calculation “asked”, you will not be able to finish the test. You will need to know how to work through stress and pressure, how to give out “high-probability” answers instead of “exactly-right” ones, and how to painlessly skip questions.

• You will be judged by a machine (or if by a person, they will try to be like a machine).

I myself feel much more comfortable in an in-person case interview, where I will be fine as long as I have the right tactics. The interviewers generally allow candidates to make a few mistakes here and there, to slow down the process if needed, and to ask for help when necessary. In the PST, the result is all that matters. There will be no mercy granted. If you don’t get enough correct answers, you are out.

• On top of those, the questions themselves are hard!

A huge amount of logical and analytical reasoning is required. You will need to really grasp the logical fundamentals of how management consultants solve problems, e.g: the difference between a conclusion vs a hypothesis; etc.

How to prepare for the McKinsey PST?

Step 1 : Seek confirmation from the target office if you must take the PST

Step 2: Get familiar with official sample tests from McKinsey

Step 3 : Learn the logic of each question type, common mistakes, and how to answer correctly

Step 4: Practice mental math to improve calculation speed and accuracy

Step 5 : Practice speed reading and data selection

Step 6 : Practice answering individual question type under time pressure

Step 7 : Do one mock test with simulated test conditions

Step 8 : Review the test and your performance

Step 9 : Return to step 3, 4, 5, 6 to for further practice on weakness

Step 10 : Do mock test again, repeat until you can confidently hit 90% or more

McKinsey PST question types

Some emailed me and asked what they should do if there were 3 days left until the PST. I would still suggest you follow our spirit of learning. When you have little time, choose which question you want to prioritize, tackle it carefully and decide how deep you will go into it. If I were in your shoes, I would use this prioritization table

In addition to the above, I would suggest you practice your math in these last days. The learning curve at the beginning is usually high for anything, including  Mental Math . Doing better math can significantly improve your test score. It reduces simple mistakes (which can still cost you points) and allows you to have more time for other questions.

McKinsey PST study plan

These are the same steps I took to pass the PST years ago, and the basis for my product – the PST Comprehensive Package which has helped countless candidates pass this notoriously difficult consulting test.

Step 1. Answer the questions correctly

You are recommended to first answer all the test questions correctly without time pressure. Before, you need to break down all the questions into 6 question types as below. Besides, it is necessary to understand how these questions are constructed, what are their logical foundation, and even how the wrong choices are made.

Step 2. Answer the questions quickly

There will be no turning point that indicates you it is about time to move to the second step, but you should gradually try to answer the questions both correctly and quickly. Once getting all the correct answers without clocking, you should put yourself under time constraint. If you don’t know how to increase your speed, you have better to start with 3 tips below:

#1. Increase your reading speed.

The PST contains 3 business cases with various number and case context, which requires you to read as fast as possible (of course correctly). Many candidates cannot finish their PST because of being overwhelmed in text. The Princeton intensive program is helpfulto increase your speed by 2 times faster without difficulties.

#2. Increase your calculator speed.

Half of the test involves math, which have no way to improve but practice rigorously. The more you practice, the better you gain. If you haven’t found any efficient tips, try our method to score well with Mental Math!

#3. Embrace test-hacking tips.

After years of coaching students to MBB, I have collected wonderful tips and tricks to nail your test with less effort! Find out some of those tricks as below or check out the McKinsey PST Comprehensive for more detail!

Reading facts

Reading-facts is the most common question type in the McKinsey PST (38%) and the BCG Potential Test (up to 100%). These questions test your ability to understand the facts/data itself. There will be no inferring, logic, hypothesizing, or creativity needed. Instead, proficiency in chart reading and calculations will be handy here. See the picture below for an illustration.

QUESTION FORMAT

The following are a few examples of typical question formats:

• Which of the following values is the best estimate of…?
• Which of the following statements is valid based on the data…?
• Which of the following can be concluded from Exhibit…?

Sometimes even though the word “conclude” is used, questions don’t require any logical reasoning, just your ability to read facts and perform basic calculations. In these cases, I still classify these questions into the reading-facts category.

SAMPLE QUESTION

This question is written based on an official McKinsey practice PST.

Which of the following statements is valid based on the data in Table 1?

A) Soccer revenue was more than $325 thousand five years ago

B) Tennis revenue grew by no less than 1.2% in each of the last five years

C) The total revenue of Saigon League did not grow at all in the last five years

D) If the growth rate in the last 5 years is maintained, Soccer revenue will be more than $420K 5 years from now.

You will see that no tricky logical reasoning is needed here. All you need in order to answer these questions is the ability to read the table and perform calculations correctly.

COMMON MISTAKES

A good way to determine the correct option is to investigate if the other three are wrong. Now there are two ways you can be wrong in this type of PST question: (1) Incorrect calculation and (2) Misread the facts/ data

Type #2 is harder to understand, so I will dive deeper into that here. Let’s look at the sample question above. Hope you got D, the correct choice.

Example 1: How you can misread the data – Why A is wrong

If you overlook the phrase “Average annual” on column 3′s title, then Soccer revenue 5 years ago would be: $342.8 k / (100% + 4.5%) = $328 k, which is more than $325 thousand. Revenue grew at an average rate of 4.5% in EACH of the last 5 years. It is NOT 4.5% over the whole period of 5 years.

Example 2: How you can misread the data – Why B is wrong

If you overlook the phrase “Average” in column 3′s title, then it seems like the growth rate for each of the last 5 years is exactly 1.2%, no more, no less. B, therefore, seems correct. However, as indicated in the table, 1.2% is just an average figure, which means there are years with a lower or higher growth rate.

Example 3: How you can misread the data – Why C is wrong

If you overlook the second column of the table (Revenue this year column), then it seems like the average overall growth rate for Saigon League is 0% (4.5% + 3.3% + 1.2% – 9% = 0%), which makes C correct. However, different lines have different sizes. Even though Golf had negative growth of 9%, it is a relatively small line so its impact on the overall rate is small as well.

Hope that you will not make this mistake in your real PST. Again, PST is a simple test… when you have enough time!

PREPARATION GUIDE

Skill #1: Calculation

We have a detailed article on Consulting Math and how to strengthen your quantitative proficiency.

Skill #2: Chart/exhibit/table reading

Always take a moment to read and understand every single chart or graph you encounter in your everyday life.After all, practice makes perfect.

You can also improve your reading speed through an amazing speed reading program by Princeton University .

Skill #3: Attention to details

The devil is in the details. It’s the little things that can make or break a project, and no true consultants would let themselves be caught unaware.

Develop a habit in daily life. Have the mindset that I am not going to miss any stupid details.

For every practice question you get in this type, make sure you understand not only why an answer is right, but also why an answer is wrong, exactly like what I did above.

PRACTICE QUESTIONS

Which of the following statements is valid based on the data provided on Graph 3 above?

A) The Service-to-Agriculture ratio increased by more than 3 times between 1995 and 2007

B) Service GDP in 1995 is more than Industry GDP in 2007

C) Agriculture is where GDP value dropped the most between 1995 and 2007

D) In 2007, Service GDP is no less than 6 times Agriculture GDP

Correct answer: D

If you want to practice more, check out my PST Comprehensive Package for questions and answers!

Fact-based conclusion

Once you get into consulting, you will probably hear the term “fact-based” a million times a day. Consulting is the business of making conclusions based on facts. Consultants face tons of different problems throughout the course of any project: from the top to the granular level, from function to function, from industry to industry, etc. Fact-based conclusion is such a fundamental aspect of consulting that it weighs in heavily on the PST.

Fact-based conclusion questions test your ability to draw and recognize sound and logical conclusions based on a set of data/facts provided. See the picture below for an illustration.

• Which of the following statements is a valid conclusion based on …?
• Which of the following statements can be concluded from …?

The McKinsey team has an interview with the Chief Operating Officer of the New Bingham Mine, Salt Lake City. During the interview, the following facts have been gathered:

• The factory must have at least one safety inspector 24 hours a day, seven days a week, in accordance with Federal and State labor regulations.
• To maximize operational efficiency, there must be exactly 10 line workers operating the mine.
• The mine operates from 8 am until 5 pm, Monday to Sunday.
• The mine employs 4 safety inspectors and 16 line workers to make 20 workers in total.
• The total weekly labor cost for the Bingham Mine is $16,000.

Which of the following statements is a valid conclusion?

A. One-fifth of the total labor cost for the mine is for safety inspectors.

B. At least one safety inspector must work more than 40 hours per week.

C. Line workers do not work more than 40 hours per week.

D. The majority of the mine’s labor cost is for line workers.

A – Fit-well but not fact-based

There are 4 inspectors out of 20 employees so it seems like the cost of the inspectors can very well be 1/5 of total labor cost. But a missing piece of data to conclude that is: does each person get a similar total income?

C – Fit well but not fact-based

The mine opens for 9 hours per day, 7 days per week, and there must be 10 line workers at a time, so it is 630 man-hours per week at the line positions. There are 16 line workers, so on average each of them only needs to work 39 hours per week. This seems to fit very well with the proposed conclusion: line workers do not work more than 40 hours per week. However, a missing piece of data to conclude is: does every line worker work the same amount of time (if not, there can be some who work over 40 hours while others work less)?

D – Fit well but not fact-based

Similar to A, there are more line workers, so it seems like the total cost for line workers is more than the total cost for safety inspectors. But a missing piece of data needed to conclude is: does each worker get paid the same amount?

=> Only B is proven true by the provided facts

There are 24 * 7 = 168 inspector hours needed in a week, equaling 42 hours per week per inspector. So there must be one who works more than 40 hours.

Identifying proven true conclusions is an important foundation to master all conclusion-related questions. However, most conclusion-related questions in the McKinsey Problem Solving Test will be given in other formats. In this section, we will learn about the two types of twists: (1) False conclusions and (2) Conclusions reversed . Let’s start with the first one.

TWIST TYPE 1: FALSE CONCLUSION

Any proposed conclusion must fall into one of the following three groups: Proven True, Proven False, and Unproven. This twist is when a question asks you to identify the False Conclusion instead of the True Conclusion.

• Which of the following statements is FALSE based on …?
• Which of the following statements is FALSE based on … ?

METHODOLOGY

A proposed conclusion is proven false when you can point out at least one instance where the conclusion is wrong. Similarly, with true conclusion questions, unproven conclusions should also not be selected.

Notice that proven FALSE conclusions are NOT conclusions not proven TRUE. A conclusion will stay unproven until it is proved to be TRUE or FALSE.

Which of the following statements is FALSE based on Table 1?

A. A, Inc. had lower average economic growth in the last five years than D, LTD.

B. A, Inc. had higher average economic growth in the last five years than D, LTD.

C. Investment risk rating is based on the difference between maximum and minimum revenue growth in the past five years.

D. Potential rating is based on the maximum recent revenue.

Of A and B, A seems to be false and B seems to be true. However, both of them are unproven. The maximum and minimum figures are not enough to conclude the average.

We don’t know if C is right or not, but we know that it is not proven false. In the provided data, there is no instance where the larger difference between maximum and minimum recent revenue growth indicates smaller risk (and vice versa).

With D, we know for sure that it is proven false because we can point out an instance where the assertion conflicts with the data (B Corp. vs. D, LTD.).

TWIST TYPE 2: CONCLUSIONS REVERSED

Very often, conclusion questions in the McKinsey Problem Solving Test are given in a reversed format. You will be given the conclusion first and asked to pick what facts/ data would be enough to come up with that conclusion.

The key to answering this type of question is to recognize which proposed fact makes the stated conclusion proven or unproven.

This question is written based on an official McKinsey practice PST:

FOCUS Travel is a premium Russian tourism company, offering tours to South East Asian countries. Facing the economic downturn, FOCUS revenue has been hurt badly. While the CFO (Chief Finance Officer) proposed an overall price cut to stay competitive, the CMO (Chief Marketing Officer) is concerned that a price reduction would negatively impact the premium perception of the brand, which drives a lot of sales.

Which of the following statements, if TRUE, would best support the CMO’s assertion?

A. In a recent survey, FOCUS’s customers quoted “price” as the most important indicator in choosing travel agencies in a list of ten factors.

B. In a recent survey, FOCUS’s customers quoted “price” as the most important indicator of quality in a list of ten factors.

C. In a recent survey, there were customers who said they would not buy FOCUS’s services if there was a 10% price increase.

D. In a recent survey, there were customers who said they would not buy FOCUS’s services if there was a 10% price decrease.

In this question, the “conclusion” has been given to us: Price reduction will negatively impact the premium perception, which will in turn negatively impact sales.

Of the four proposed answers, which facts are enough to prove the provided “conclusion” above?

A: This fact is only enough to conclude that price will impact sales. Not enough to prove that price reduction will negatively impact sales.

C: This fact is irrelevant.

D: This fact is not enough to conclude that price reduction will negatively impact sales because not all customers say so. The word “there were” can be understood as either a minority or a majority. It is only enough to conclude the proposed conclusions when “there were” is replaced with “the majority of” or “all“ .

With B, we can logically infer that price reduction will negatively impact the quality perception, which in turn will hurt to sales.

3.3. Root-cause reason

This question gives you a particular set of facts/data and asks you to identify what could be the cause for them. When doing a real consulting project, we consultants have to find out the root-cause reason. There may be various reasons that can cause the current situation, but the root-cause reason will help us tackle and solve it more efficiently. You can see the picture below for an illustration.

The following are a few examples of typical root-cause reason question format:

• Which of the following reasons, if TRUE, will help explain the Facts …?
• Which of the following does NOT explain the Facts …?
• Which of the following points is NOT a valid reason for the Facts …?

Only B is proven true by the provided facts

Facts provided: Visits to the website MConsultingPrep were relatively low last month.

Root-cause Reason Question: What reasons, if TRUE, would help explain the low traffic to MConsultingPrep last month?

The correct answers can be any of the following:

1. The quality of contents has been bad

2. Because of technical issues, some visitors could not access the website

3. Last month was December when the overall demand for job prep materials is lowest in the year

4. Other new consulting prep blogs opened recently

Fact-based Conclusion Question: What can be concluded from the data provided?

All of the statements above can be the reason for the stated fact, but NONE of them can be concluded from it.

An example of a statement that can be concluded: Because the conversion rate stayed constant over the years, revenue last month was relatively low.

What makes a statement NOT a potential reason for a particular fact?

There are two ways a statement cannot be the potential reason: (1) Wrong Subject and (2) Wrong Trend.

• A statement is (1) Wrong on Subject when the subject is irrelevant, which means the statement has zero effect on the phenomenon mentioned in the stated fact.
• A statement is (2) Wrong on Trend when the direction is reversed, which usually means the statement has a reversed effect on the phenomenon mentioned in the stated fact

Illustrative example Let’s continue with the simple example above. The Stated Fact: Visits to the MConsultingPrep blog were relatively low last month. (1) Example of a “Wrong Subject” statement: “Some new Investment Banking Prep blogs opened recently” Here the subject “Investment Banking Prep blogs” is irrelevant to the stated fact. The statement (1) will have zero effect on the stated fact. (2) Example of a “Wrong Trend” statement: “Some other existing Consulting Prep blogs closed recently” Here, even though the subject “Consulting Prep blogs” is relevant, the trend is reversed. The exit of Consulting Blogs will increase visits to MConsultingPrep. Therefore, statement (2) will have an opposite effect on the stated fact.

PRACTICE QUESTION

Fletcher is a major Steel producer in the Pacific continent. It has markets in New Zealand, Australia and other South East Asia countries. Of many types of steels, re-bar (reinforced bar) is typically used in high-rises and big construction projects.

There are three main groups of steel consumers in New Zealand:

• Homeowners purchase steel at retail sizes for purposes of self-constructing and self-renovating their homes
• Scaled private construction companies, who often contracts large construction projects and steel orders
• State-owned Enterprises (SOE), who build government’s projects such as roads, bridges, schools, hospitals, etc. (SOEs usually have bargaining power since steel providers need governments’ permissions in order to be legally used in particular countries.)

Table 1 below shows the size of re-bar steel market (in billions of $US)

Which of the following statements, if TRUE, best explains why future trends for South East Asia sales differ from sales in the other two markets?

A) South East Asia population is expected to grow strongest, which lead to high steel demand from individual homeowners.

B) South East Asia economy will be heavily based on SOE, so will the construction market.

C) South East Asia economy will shift toward privatization, so will the construction market.

D) Developed markets of New Zealand and Australia will have the most advanced steel production technology and facilities.

Correct answer: C

Word Problem

Word Problem is a quantitative question where the answer cannot be calculated directly from the data provided. Usually, we have to set up one or more equations in order to solve this kind of question. Word Problem questions in the McKinsey PST and Case Interviews are just the word problems we usually see in schools, GMAT… but put into business contexts. The method to solve them, therefore, is the same.

Table 1: Data on the Washing Room of Jean Valjean Restaurant

Suppose the restaurant opens 350 days a year. There are 3 meal shifts per day, 1 shift lasts 3 hours, 1 customer uses an average of 5 dishes per visit, and currently the restaurant hosts 530 customers on average daily.

What percentage of increase in the number of daily visits would be required in order to make purchasing the machines financially beneficial?

Step 1 . Convert data/facts into manageable and standardized format and units (only needed for complex questions).

Step 2 . Set up an equation with one (or more) unknown variables, i.e. X, Y, Z, etc.

Tips: Don’t worry about having to make the variable as the question asked. Just set up the equation in a way that makes the most sense to you as long as the variables can be easily converted to the asked variable. It will save much more time and helps you avoid silly mistakes.

Step 3 . Solve the equation and get the answer.

METHODOLOGY ILLUSTRATION

Let’s solve the sample question above together.

Step 1: This is a very complex question with many non-standardized and not ready-to-use data. If I am going to tackle this question on my PST, I would convert the provided figures and write them out on a table as follows.

Notice that I have converted all the necessary data points into the same unit of “Franc per day”.

The only data point not fully converted is the Labor cost in Manual Process (measured by the “per dish” variable), yet I want to make sure that I go as far as I can.

See how simple the problem is now!

Step 2: Now that we have very manageable data, let’s go ahead and set up an equation that will help us find the answer. The asked variable here is: what percent increase in current daily visits does Jean Valjean need?

As mentioned above, it is NOT necessary to put the variable question is looking for in the equation. In this case, doing so will result in a very awkward and complicated equation.

Instead, I set up the equation that makes the most sense to me (do note that there is more than one way to set up equations). Let Y be the “break-even size” (measured by people). I can easily calculate the percentage asked for after getting the break-even size.

Cost per day of Manual Process = Cost per day of Machine Process

Washing cost + Set up cost = Washing cost + Set up cost + other cost

Y x 5 dishes x 0.1 Franc + 30 Franc = 270 Franc + 90 Franc + 200 Franc

After executing step 1 and step 2, the problem becomes a lot easier. Now we have:

0.5 Y = 530

Y = 2 x 530

Once we have Y (the new “break-even” visit volume) of 2 * 530, we can quickly convert Y into the asked variable: What percent increase of 530 customers/day does Jean Valjean need?

The final answer is C.

Client Interpretation

In every consulting project, communication with the clients’ top-level (usually the Chairman or CEO) is always important. During my time with McKinsey, we usually hear an update every one or two weeks from our Project Director (usually a partner) on his meeting with the clients’ top level. Messages from those meetings are important on-going steers for the project. No surprise it makes up an entire question category in the Problem Solving Test.

Client Interpretation questions test your ability to read, understand, and interpret the messages the client is trying to convey in the case question or description. To some extent, this is very similar to GMAT verbal questions.

• Which of the following best summarizes the CEO’s concerns?
• Which of the following statements best describes the thoughts of the CEO regarding…?
• Based on the opinion of the head of the Department, which of the following statements is valid?
• Which of the following statements best describes the CEO’s aims for the McKinsey research?

Case context:

Mommy said she saw some dirty clothes on the dining table. She is also quite shocked to see Kevin’s toys in every room throughout the house. She even complains about how much time it takes her every night to clean up Kevin’s mess. “I will have to have a very straightforward conversation with Kevin tonight!”, said mom

Which of the following statements best describes the Mom’s concern?

A. Mom is not happy about too many of Kevin’s toys sitting on the dining table

B. Mom does not expect to see that many of Kevin’s toys in the house

C. Mom does not like to be responsible for anyone’s mess

D. Mom is too busy these days

E. Mom wants to talk to Kevin

F. Mom wants Kevin to be tidier

In this example, we continuously get small data points, all leading to one bottom-line, not explicitly mentioned but can be reasonably interpreted:

Kevin is too messy and mom doesn’t like that!

Notice that, the bottom-line here is not explicitly stated but it IS the bottom-line. All 4 sentences in the case context are small pieces of data leading to that final “so-what”. Having this “so-what” in mind, you can just skim through the answer and quickly pick F without concern about other choices.

In case you are curious about how other choices are “wrong-choice” …

Choices B and E are in fact right according to the case context, but not the bottom-line.

Choices A is simple wrong according to the case context (Tip #2)

Choices C and D are neither right or wrong according to the case context. There are not enough “evidence” to be reasonably interpreted using common sense.

TIPS AND TRICKS

Tip #1: Read the case description before going to multiple choices!

Normally the strategy of scanning through the answers first before going back to the case description works when you have a very long case description and don’t know where to look for the right information. Scanning through the answers helps you get a more focused read on the case description. However, the client’s assertion is typically found in a very short and specific part of the case description. So once you realize it’s a Client Interpretation question, go back to the case description and find that very specific part of the client’s assertion. Make sure you understand it very well. Then the rest of the work is just determining which of the four choices has the same meaning as the original assertion.

Tip #2: Cross out some obviously wrong choices …

… by recognizing a few words or short phrases that make a choice incorrectly reflect the client’s assertion. Sometimes, you can do this very quickly and effectively. If not, please see tips #3.

Tip #3: Catch the bottom-line, the “so-what” of client’s assertion

Client’s expression as quoted in the case context is always a bit blur and confusing. That is very realistic of what you may encounter in the real consulting work. It also makes these question types challenging. But in almost every situation, there is always one “so-what”, stated explicitly or implicitly. The trick here is to catch that so-what, ignore the noise, and go straight for the answer choice. Most of the time, the wrong choices DO contain a part of the client’s assertion, but either not the whole idea or the main, the bottom-line, the most important one!

Using this method, you can fly and land straight to the correct choice, not having to care too much about how wrong choices are made of. But if you are curious, some of the most common wrong-choice types:

• Choice that is simply wrong according to the client’s assertion (Tip #2 above).
• Choice that is in fact right, but is a minor point, NOT the bottom-line of the client’s assertion.
• Choice that seems to be right, but cannot be reasonably interpreted by common sense (not by scientifically supporting logic like in other question types).

Gangnam Market is a convenience-stores chain mainly in the Gangnam district, Seoul, Korea. Though it has been losing money almost every year since 2000, Gangnam Market secures a good deal of strategic locations in the highly populated Gangnam district. Recently, Gangnam Market was acquired by Lotte Mart in its aspiration to expand to the mini-market market. Lotte right away sets up a transformation project to get Gangnam Market back on track. The CEO of Gangnam Market states that aggressive transformation targets are fine for newly acquired stores with a similar operation model with Lotte’s big stores, but he hopes that the parent company is realistic about the convenience-stores model Gangnam has been operating with.

Which of the following statements best reflect the concerns of Gangnam Market’s CEO?

A) He is concerned that Gangnam Market will never be able to transform itself into Lotte system because Gangnam Market only presents in a specific geographic location

B) He is concerned that Lotte Mart sets transformation milestones that are too aggressive and not realistic for newly acquired companies like Gangnam Market

C) He is concerned that Lotte Mart’s transformation milestones are not realistic for companies with different operational model from Lotte Mart like Gangnam Market

D) He is concerned that Lotte Mart’s transformation targets are too high for Gangnam Market because it has been losing money for a while

Formulae questions are generally like word problems in PST where you don’t have to provide the actual numerical results, just the formulae containing letters representing input variables. Normally, the question will provide input variables in letter format and you will be asked to provide the right formulae in letter format (e.g. it takes the process center T hours to process each file. If the speed is doubled, it takes T/2 hours to process each file). This is one of the easiest PST question types on the McKinsey PST. Let’s make sure you don’t lose points on any question of this type in your exam!

Table 3.6.1: Labor Cost and Processing Data – Holcim Missouri plant

Which of the following formulae accurately calculates the annual cement output per worker?

A. (c x p) / (b + w)

B. (c x p) / [(b + w) x 12]

C. 144 x (c x p) / (b + w)

D. 12 x (c x p) / (b + w)

FORMULAE FOR SUCCESS IN FORMULAE QUESTIONS

Formula 1: Calculate first before looking at the given option

A popular technique for multiple-choice questions is to read the answers first before coming back to the facts. However, that technique would not help you with Formulae Questions. The reason for this is that, often, the end-result formula has already been simplified (e.g. canceling out the same variable on both numerator and denominator) as much as possible. It gives you neither the path to get there nor any hints on how to solve the problem. For instance, when you look at the four options in the example above, does any of them give you a sense of what it represents or how to get there? What does (c x p) represent? What do you get by multiplying Cement output by Monthly labor income?

Formula 2: Divide the problem into smaller pieces (take one step at a time)

This is the universal tip for everybody in the consulting industry, and it also works great here! Often, the result cannot be directly calculated from the provided variables. However, if you take an extra step in-between, the problem becomes a lot easier. Let’s solve the sample question above together to illustrate this point. I broke the problem into smaller steps as below:

• Step 1: Annual cement output per worker = Total annual cement output / Total number of workers
• Step 2: Since we already have Total annual cement output of (c), the next step is to calculate the total number of workers. Total number of workers = Total labor cost / Salary of 1 worker Both Total labor cost and salary are provided. Bingo!
• Step 3: Simplify the final formulae

Formula 3: Get the reading-facts tools right

In some aspects, the formulae question is also a tweaked version of reading-facts questions. You still need to read some facts and perform some calculations (with letters instead of real numbers). Therefore, it is important to master those reading-facts tools and apply them here.

Illustration of a usual mistake: Now come back to Step 2 above and explicitly solve it.

Step 2: Total number of workers = Total labor cost / Salary of 1 worker = (b + w) / p

Step 3: Annual cement output per worker = c / [(b + w) / p] = (c x p) / (b + w)]

Chosen choice: A

Unfortunately, A is NOT the correct answer, because the above calculation doesn’t take into account the difference in units – the salary is on a monthly basis whereas the total labor cost is on an annual basis. If you convert the unit, the final choice should be D.

No matter how beautifully you have tackled the problem, you will not get any credit if small mistakes like this slip through the crack. Make sure you don’t get blindsided by this kind of pitfall!

VICEM is a leading cement company in South East Asia. The following data regarding its business and production has been gathered.

Table 3.6.2: VICEM Business and Production data

Clinker factor is defined as the amount of clinker needed to produce 100 units of cement.

Which of the following formulas calculates the amount of clinker (in tons) needed to purchase in a year?

A) [(s x f) / 100] – c

B) [(p – s) / 100] – c

C) (p x f) – c

D) [(p x f) / 100] – c

 ANSWER KEY

McKinsey PST sample test

Download McKinsey PST practice test (PDF): TOYO case .

More free materials like this can be found in our Prospective Candidate Starter Pack – a collection of beginner’s materials to consulting resume, screening tests and case interviews. 

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This article will give a detailed explanation of failing the test and practical advice to minimize the chance of poor performance in the McKinsey game.

The McKinsey Solve (other names: PSG, Problem Solving Game, Digital Assessment, Imbellus) is a two-piece gamified test designed by Imbellus for McKinsey.

There are 2 techniques to boost your reading speed. Learn how to practice them effectively & overcome 3 common loopholes that limit your reading performance.

Creative Problem-Solving Test

Do you typically approach a problem from many perspectives or opt for the same old solution that worked in the past? In his work on human motivation, Robert E. Franken states that in order to be creative, you need to be able to view things from different perspectives.

Creativity is linked to fundamental qualities of thinking, such as flexibility and tolerance of ambiguity. This Creative Problem-solving Test was developed to evaluate whether your attitude towards problem-solving and the manner in which you approach a problem are conducive to creative thinking.

This test is made up of two types of questions: scenarios and self-assessment. For each scenario, answer according to how you would most likely behave in a similar situation. For the self-assessment questions, indicate the degree to which the given statements apply to you. In order to receive the most accurate results, please answer each question as honestly as possible.

After finishing this test you will receive a FREE snapshot report with a summary evaluation and graph. You will then have the option to purchase the full results for $6.95

This test is intended for informational and entertainment purposes only. It is not a substitute for professional diagnosis or for the treatment of any health condition. If you would like to seek the advice of a licensed mental health professional you can search Psychology Today's directory here .

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4th Grade Math Word Problems Worksheets

Math word problem worksheets for grade 4.

These word problem worksheets place 4th grade math concepts into real world problems that students can relate to.  

We encourage students to read and think about the problems carefully, by:

• providing  mixed word problem worksheets
• including irrelevant data within word problems so students must understand the context before applying a solution

The four operations

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Overview of the Problem-Solving Mental Process

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Rachel Goldman, PhD FTOS, is a licensed psychologist, clinical assistant professor, speaker, wellness expert specializing in eating behaviors, stress management, and health behavior change.

• Identify the Problem
• Define the Problem
• Form a Strategy
• Organize Information
• Allocate Resources
• Monitor Progress
• Evaluate the Results

Frequently Asked Questions

Problem-solving is a mental process that involves discovering, analyzing, and solving problems. The ultimate goal of problem-solving is to overcome obstacles and find a solution that best resolves the issue.

The best strategy for solving a problem depends largely on the unique situation. In some cases, people are better off learning everything they can about the issue and then using factual knowledge to come up with a solution. In other instances, creativity and insight are the best options.

It is not necessary to follow problem-solving steps sequentially, It is common to skip steps or even go back through steps multiple times until the desired solution is reached.

In order to correctly solve a problem, it is often important to follow a series of steps. Researchers sometimes refer to this as the problem-solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution.

The following steps include developing strategies and organizing knowledge.

1. Identifying the Problem

While it may seem like an obvious step, identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless.

Some strategies that you might use to figure out the source of a problem include :

• Asking questions about the problem
• Breaking the problem down into smaller pieces
• Looking at the problem from different perspectives
• Conducting research to figure out what relationships exist between different variables

2. Defining the Problem

After the problem has been identified, it is important to fully define the problem so that it can be solved. You can define a problem by operationally defining each aspect of the problem and setting goals for what aspects of the problem you will address

At this point, you should focus on figuring out which aspects of the problems are facts and which are opinions. State the problem clearly and identify the scope of the solution.

3. Forming a Strategy

After the problem has been identified, it is time to start brainstorming potential solutions. This step usually involves generating as many ideas as possible without judging their quality. Once several possibilities have been generated, they can be evaluated and narrowed down.

The next step is to develop a strategy to solve the problem. The approach used will vary depending upon the situation and the individual's unique preferences. Common problem-solving strategies include heuristics and algorithms.

• Heuristics are mental shortcuts that are often based on solutions that have worked in the past. They can work well if the problem is similar to something you have encountered before and are often the best choice if you need a fast solution.
• Algorithms are step-by-step strategies that are guaranteed to produce a correct result. While this approach is great for accuracy, it can also consume time and resources.

Heuristics are often best used when time is of the essence, while algorithms are a better choice when a decision needs to be as accurate as possible.

4. Organizing Information

Before coming up with a solution, you need to first organize the available information. What do you know about the problem? What do you not know? The more information that is available the better prepared you will be to come up with an accurate solution.

When approaching a problem, it is important to make sure that you have all the data you need. Making a decision without adequate information can lead to biased or inaccurate results.

5. Allocating Resources

Of course, we don't always have unlimited money, time, and other resources to solve a problem. Before you begin to solve a problem, you need to determine how high priority it is.

If it is an important problem, it is probably worth allocating more resources to solving it. If, however, it is a fairly unimportant problem, then you do not want to spend too much of your available resources on coming up with a solution.

At this stage, it is important to consider all of the factors that might affect the problem at hand. This includes looking at the available resources, deadlines that need to be met, and any possible risks involved in each solution. After careful evaluation, a decision can be made about which solution to pursue.

6. Monitoring Progress

After selecting a problem-solving strategy, it is time to put the plan into action and see if it works. This step might involve trying out different solutions to see which one is the most effective.

It is also important to monitor the situation after implementing a solution to ensure that the problem has been solved and that no new problems have arisen as a result of the proposed solution.

Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies .

7. Evaluating the Results

After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct, or it can be delayed, such as evaluating the success of a therapy program after several months of treatment.

Once a problem has been solved, it is important to take some time to reflect on the process that was used and evaluate the results. This will help you to improve your problem-solving skills and become more efficient at solving future problems.

A Word From Verywell​

It is important to remember that there are many different problem-solving processes with different steps, and this is just one example. Problem-solving in real-world situations requires a great deal of resourcefulness, flexibility, resilience, and continuous interaction with the environment.

Get Advice From The Verywell Mind Podcast

Hosted by therapist Amy Morin, LCSW, this episode of The Verywell Mind Podcast shares how you can stop dwelling in a negative mindset.

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You can become a better problem solving by:

• Practicing brainstorming and coming up with multiple potential solutions to problems
• Being open-minded and considering all possible options before making a decision
• Breaking down problems into smaller, more manageable pieces
• Asking for help when needed
• Researching different problem-solving techniques and trying out new ones
• Learning from mistakes and using them as opportunities to grow

It's important to communicate openly and honestly with your partner about what's going on. Try to see things from their perspective as well as your own. Work together to find a resolution that works for both of you. Be willing to compromise and accept that there may not be a perfect solution.

Take breaks if things are getting too heated, and come back to the problem when you feel calm and collected. Don't try to fix every problem on your own—consider asking a therapist or counselor for help and insight.

If you've tried everything and there doesn't seem to be a way to fix the problem, you may have to learn to accept it. This can be difficult, but try to focus on the positive aspects of your life and remember that every situation is temporary. Don't dwell on what's going wrong—instead, think about what's going right. Find support by talking to friends or family. Seek professional help if you're having trouble coping.

Davidson JE, Sternberg RJ, editors.  The Psychology of Problem Solving .  Cambridge University Press; 2003. doi:10.1017/CBO9780511615771

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. Published 2018 Jun 26. doi:10.3389/fnhum.2018.00261

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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The Four Fundamental Operations of Whole Numbers 4th Grade Math Worksheets

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Definition:

The four fundamental operations  in mathematics are addition, subtraction , multiplication and division. In this worksheet, we will learn about these operations involving whole numbers . 

• mathematical processes used to solve problems 
• the four basic operations are addition, subtraction, multiplication and division.

ADDITION –  It simply means that it is a combination of distinct sets of same things or quantities. A plus sign ( + ) is a symbols that indicates addition. The answer when two addends were added is called sum.

SUBTRACTION –  It is the opposite of addition. It means removing one quantity from the other. The minus sign ( – ) is used to represent subtraction. When a subtrahend is subtracted from the minuend, the answer is called difference.

MULTIPLICATION –  It is a repeated addition. It means that a certain number is being repeatedly added to itself by several times. The times sign ( x ) is used to represent multiplication . When a multiplicand is multiplied by its multiplier, the answer is called product. 

DIVISION –  It is the inverse of multiplication. It means splitting quantities into same or equal parts. The divide symbol (➗)  is used to represent division . When a dividend is to be split into a certain divisor, the answer is called quotient. Not all numbers when divided by a divisor produces equal amount. The extra amount is called remainder. 

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Optimization beyond data: Design thinking for SEO

One of the things I love about SEO is its inherent duality.

We get to leverage both sides of our brains:

• The right brain when it comes to on-page, content and even link building campaigns.
• The left brain when it comes to technical, data analysis, etc.

However, it’s easy to think of SEO as predominantly left-brained. SEO tactics tend to rely heavily on data and numerical figures. We lean into technical know-how and keyword optimization. Logically, we react to what the numbers are telling us and decide the next steps accordingly. It’s a proven approach.  

But what about the people driving that data? What about their intent? Can we use more creative thinking to pursue better optimization strategies? 

When it comes to SEO, our goal is not really to gain the coveted top blue link. It’s about reaching the right people and addressing their needs by giving what they want as quickly and as easily as possible.

So, how do we reach that goal?

Users are always looking to do something, whether it’s finding information, being entertained or purchasing a product. How do we tap into emotional and behavioral data to support them?

That’s where design thinking comes in. 

What is design thinking?

Design thinking is exactly what it sounds like: adopting a designer mindset. 

It’s a human-centered framework for problem-solving the way a designer would – by setting out to solve a problem using creativity rather than data alone.

The design thinking process is typically divided into five stages:

• Test and evaluate

With design thinking, the emphasis is not only on the solution but also on the end user.

SEO specifically focuses on providing the best solution for a specific audience. It’s about understanding user intent and adding value. 

Get the daily newsletter search marketers rely on.

A new optimization process 

1. empathize: get to know your audience.

Empathizing in this context is centered around identifying and understanding your audience.

The best SEO strategy optimizes for consumer discovery by understanding the people who are searching. This ensures you’re adding value to users and ultimately growing your audience through increased organic visibility. 

Conduct thorough research to learn about your existing audience and gain insights into search behaviors, motivations and pain points. 

There are a number of tools available to help research your target audience. Using Google Analytics to understand who is coming to your site is a great place to start.

In GA4, you can view audience reports under User > User Attributes to identify location, gender, age, language and even interests when available. 

You can also leverage Google Trends, Facebook Audience Manager and persona mapping or survey tools to learn more about your potential audience.

Gathering this information helps tailor efforts from keyword selection to content creation and off-page efforts. When your SEO strategy is anchored in reaching an audience you fully understand, you can reach them more efficiently.

Dig deeper: An SEO guide to audience research and content analysis

2. Define: What problem are we trying to solve 

The next step involves analyzing your audience data to define the SEO challenges you aim to address. This is critical to ensure you address the real wants and needs of the audience through SEO efforts rather than working from assumptions like search volume or clicks. 

Based on the unique audiences you have identified, you can better determine the specific challenges you need to address and how to reach users. Consider:

• What messaging and terminology is most likely to resonate with your target audience?
• Are users struggling to find relevant information on your website? 
• Are there gaps in your content that need to be filled?
• Based on location, what search engines are audiences using beyond Google? Yandex, Baidu, DuckDuckGo?
• Based on age and gender, what non-traditional search engines do you need to consider? TikTok, YouTube, Amazon, Pinterest? 

Clearly defining the problem allows you to focus your efforts on the areas that will impact your SEO performance most.

3. Ideate: How can we best solve that problem?

With a clear understanding of the SEO challenge, brainstorm as many potential solutions as possible. It’s easy to fall into the same pattern of optimizing your site based on analytics data and search trends, but with design thinking, we emphasize qualitative data over quantitative data.   

A few different brainstorming techniques to help transcend your left-brained thinking habits include:

• Brainwriting : Everyone in the group writes down three ideas related to the challenge. From there, everyone passes their ideas on to the person next to them to elaborate on the thought starters and add strategies or tactics. This process repeats until the ideas have been passed around the entire group. Ultimately, everyone has contributed to each idea.
• Starbursting : Given a specific idea or strategy, create a six-point star around the idea, posing the questions who, what, when, where, why and how. Focusing on these key elements for each idea encourages the team to think about value and execution.

Think outside the box and consider how to create value for your audience through content, website updates, or user experience. In this exercise, quantity is key – use cross-functional thinking to generate ample diverse ideas.

• Is content expansion something that could help address your challenge?
• Do you need additional content that addresses a pain point?
• Do you need to improve the ease with which users find existing content? 
• Should social media be considered to increase visibility and extend reach?
• Are there PR tactics that could help generate earned coverage (i.e., inbound links and brand mentions)?
• Would other media, such as video, webinars or podcasts, potentially help address the need? 

Maybe traditional SEO tactics will help solve your particular challenge. But oftentimes, by integrating cross-channel tactics, you can better tackle SEO challenges and add value for users.

Dig deeper: SEO planning: Your one-page SEO plan

4. Prototype: Build the thing 

Before implementing any SEO strategy or tactics at scale, create “prototypes” to visualize and test your ideas.

Visualization is crucial in understanding how a strategy may address the problem or challenge. However, your prototype does not have to be a high-fidelity visual asset.

This could involve:

• Updating keyword maps and topic clusters.
• Drafting sample content pieces.
• Creating mock-ups of new features.
• Developing wireframes for website updates. 

In many cases, fancy tools and a team of engineers aren’t necessary. You can use lo-fi tools like Figma or Google Sheets to build basic prototypes that clearly convey the solution.

Whatever shape a prototype takes, keeping the unique problem or challenge in mind and relating it back to the audience is essential.

When considering the effectiveness of your prototype, use role-playing to put yourself in the shoes of the target audience. 

Dig deeper: Conveying keyword insights to non-SEOs: A visual approach

5. Test and evaluate: Does this solution work?

Design thinking makes so much sense for SEO because, much like SEO, it is an iterative process. 

The final step is to gather feedback on “prototypes” and/or tactics to refine solutions and strategies.

Intentionally test your tactics and continuously monitor performance. Leverage a modern framework for running SEO tests . Embrace a culture of experimentation to evolve your approach and better understand pain points. 

• A/B test everything from metadata to messaging to content structure.
• Leverage heat mapping to better understand how the target audience is using your website.
• Test keywords and messaging using Google Ads.
• Consider usability testing through a tool such as Hotjar or UserTesting. 
• Actively seek feedback on your site’s design, layout and functionality via surveys.

Changes in user behavior are more directly and immediately measurable than traditional SEO KPIs.

By testing with real users, you can gather feedback early in the process and make necessary adjustments along the way.

Creative problem-solving for SEO

Remember, it all starts with redefining the problems we are trying to solve.

Reframing SEO challenges around the target audiences’ needs and challenges allows us to better give people what they want.

When your SEO efforts are focused on the right audience, it’s easier to reach them. Traffic increases, which leads to more conversions.

Use design thinking to balance the analytical and creative sides of SEO. It can help you better understand when to use data, ignore trends and take risks, ultimately letting you create more user-centric and impactful SEO campaigns. 

Learning search algorithm: framework and comprehensive performance for solving optimization problems

• Open access
• Published: 09 May 2024
• Volume 57 , article number  139 , ( 2024 )

Cite this article

You have full access to this open access article

• Chiwen Qu 1 ,
• Xiaoning Peng 2 &
• Qilan Zeng 3  

In this study, the Learning Search Algorithm (LSA) is introduced as an innovative optimization algorithm that draws inspiration from swarm intelligence principles and mimics the social learning behavior observed in humans. The LSA algorithm optimizes the search process by integrating historical experience and real-time social information, enabling it to effectively navigate complex problem spaces. By doing so, it enhances its global development capability and provides efficient solutions to challenging optimization tasks. Additionally, the algorithm improves the collective learning capacity by incorporating teaching and active learning behaviors within the population, leading to improved local development capabilities. Furthermore, a dynamic adaptive control factor is utilized to regulate the algorithm’s global exploration and local development abilities. The proposed algorithm is rigorously evaluated using 40 benchmark test functions from IEEE CEC 2014 and CEC 2020, and compared against nine established evolutionary algorithms as well as 11 recently improved algorithms. The experimental results demonstrate the superiority of the LSA algorithm, as it achieves the top rank in the Friedman rank-sum test, highlighting its power and competitiveness. Moreover, the LSA algorithm is successfully applied to solve six real-world engineering problems and 15 UCI datasets of feature selection problems, showcasing its significant advantages and potential for practical applications in engineering problems and feature selection problems.

Avoid common mistakes on your manuscript.

1 Introduction

All production or social activities that human beings are engaged in are purposeful. The activity is always under the control of specific values or aesthetic orientations, and it often faces a decision problem of the feasible or even optimal scheme, i.e., an optimization problem (Martello et al. 1984 ). In recent years, the importance of optimization in engineering design, disease identification, and other issues has been recognized. Specifically, taking the objective function as the predefined measure of decision quality, the best decision or solution to predefined design problems can be obtained by evaluating various methods. Optimization is a problem that people often encounter in the production practice of scientific research and social transformation. It has the characteristics of unknown search space, non-differentiability of the objective function, high-dimensional, and non-convex (Ezugwu 2022 ). Generally, optimization techniques can be roughly divided into deterministic and non-deterministic ones (Parsopoulos and Vrahatis 2002 ). Deterministic methods are usually gradient-based and are further divided into linear and nonlinear ones. Although these methods help to solve linear and nonlinear optimization problems, they will fall into local optimization when dealing with non-differentiable optimization problems, so they cannot solve such problems in the minimum time or with accurate complexity. The non-deterministic method uses a random generation strategy to find the near-optimal solution in the problem space, which has the advantage of simple implementation and no gradient-related information (Li et al. 2011 ; Liu et al. 2013 ).

With the continuous expansion of engineering application fields and the continuous improvement of the complexity and difficulty of optimization problems, the need for optimization technology is becoming more and more obvious. As a class of non-deterministic methods (random search methods), meta-heuristic algorithms have demonstrated excellent performance when tackling challenges involving multiple-peak, discontinuous, and non-differentiable problems. Therefore, meta-heuristic algorithms have gained significant popularity in efficiently tackling diverse practical optimization problems across numerous fields, such as function optimization (Seo et al. 2006 ; Pan et al. 2006 ), pipeline scheduling (Rejowski and Pinto 2003 ; Li et al. 2021 ), optimization of neural network parameters (Abdolrasol et al. 2021 ; Abd Elaziz et al. 2021 ), key gene identification (Mandal and Mukhopadhyay 2015 ), image segmentation (Chander et al. 2011 ; Pham et al. 2018 ), parameter identification of photovoltaic module models (Ibrahim et al. 2020 ; Liang et al. 2020 ), optimization of engineering design, etc. (Kundu and Garg 2022 ; Onay and Aydemı̇r, S. B. 2022 ).

In terms of formation principle, meta-heuristic methods can be categorized into four distinct groups, each offering unique approaches to solving optimization problems encountered in various disciplines: the methods based on the evolutionary mechanism, the methods based on physical principles, the methods based on swarm intelligence, and the methods based on human social activities (Sharma et al. 2022 ; Wilson et al. 2022 ; Tian et al. 2022 ; Ewees et al. 2022 ). Section  2 offers an extensive compilation of the comprehensive literature on the development and application of numerous novel metaheuristics across various domains.

Meanwhile, many scholars have optimized the original basic algorithm to solve the optimization problems of real-world engineering applications more efficiently. For example, due to the large randomness and uncertainty of the randomly generated initial population, many scholars have improved the initial population by using chaos mapping (Tutueva et al. 2020 ), reverse learning (Ruan et al. 2022 ), Sobol sequence (Sun et al. 2021 ), square neighborhood topology (Rachdi et al. 2020 ), and other strategies to achieve algorithm optimization and convergence performance improvement. Some scholars have adopted strategies such as sine–cosine optimization (Chen et al. 2020a ), Gaussian walk (Khalilpourazari et al. 2021 ), Levy flight (Kaidi et al. 2022 ), and quantum behavior (Zhang et al. 2021a ) to optimize individual iterative updates. Also, nonlinear inertia weight (Yu et al. 2020 ), horizontal cross (Chen et al. 2019 ), spiral update (Lin et al. 2022 ), and other approaches have been employed to achieve the balance between the global and local development of the algorithm. Moreover, some scholars have combined the advantages of two or more algorithms and proposed an improved hybrid strategy algorithm (Shi et al. 2005 ; Yaghini et al. 2013 ; Qiao et al. 2020 ), such as the exploratory cuckoo search (ECS) algorithm proposed by Abed-Alguni et al. (Abed-Alguni et al. 2021 ) and the improved SSA algorithm proposed by Dorian (Dorian Sidea 2024 ).

In metaheuristic algorithms, the exploration phase of the global search space has the ability to escape local optima, while the exploitation phase enhances the algorithm’s precise search capability within local regions. Balancing exploration and exploitation is a challenging task in every metaheuristic algorithm, as it affects whether the algorithm can find the global optimum solution. In general, metaheuristic algorithms offer different performances in solving optimization problems due to their different operations and mechanisms. According to the “No Free Lunch” (NFL) theorem (Qiao et al. 2020 ), all metaheuristic algorithms have the same average performance when solving optimization problems. In other words, there is no optimal algorithm that can solve all optimization problems, which means that the performance of different algorithms varies when providing prior knowledge to solve specific problems with metaheuristic algorithms. Finding the most suitable algorithm for each specific type of optimization problem remains a challenge. Each metaheuristic algorithm has its unique characteristics as they draw inspiration from different natural or biological behaviors. To evaluate performance and find suitable application areas, metaheuristic algorithms require comprehensive testing on various benchmark functions and real-world applications, and continual improvement. These reasons support the innovation and design of metaheuristic algorithms to solve a wide range of optimization problems.

This paper proposes a novel optimization algorithm, called the learning search algorithm (LSA), which is inspired by human learning behaviors and promotes both global exploration and local development phases simultaneously. The LSA algorithm involves dynamic adaptive global exploration to local development phase control parameters, which enhance its global search ability and avoid falling into local optima. In the local development phase, the algorithm exploits the teaching behavior of the model in the current population to actively learn behaviors from role models and improve the learning ability of the entire population. The proposed LSA algorithm is evaluated on 40 benchmark functions from IEEE CEC2014 and CEC2020, 6 real-world engineering optimization problems and 15 feature selection cases in the UCI dataset. Contrasted with nine high-performance algorithms and eleven recently proposed algorithms, the LSA algorithm shows promising results in terms of convergence speed, search accuracy, and scalability. The experiment results suggest that the LSA algorithm outperforms the selected comparison algorithms on most of the selected test problems. The statistical analysis further confirms the proposed algorithm’s superiority by conducting the Wilcoxon signed-rank test and the Friedman rank-sum test.

The paper presents several significant contributions:

In terms of learning mechanism, the LSA algorithm simulates the process of human knowledge acquisition. In this algorithm, a historical experience knowledge base is established, from which humans learn knowledge. Humans also possess the ability of active learning and knowledge acquisition from exemplary radiations. Additionally, the learning outcomes continuously update the historical experience knowledge base.

Regarding its adaptability, the LSA algorithm exhibits a high level of adaptability. In each iteration process, knowledge and information flow constantly between the historical experience knowledge base and the current individuals. This enables the LSA algorithm to adaptively adjust the search strategy based on the complexity and characteristics of the problem, thereby enhancing the search efficiency and solution quality.

The concept of idea transmission is another important aspect of the LSA algorithm. It improves the solutions of individuals through the transmission of ideas. The algorithm transfers excellent search ideas to learning individuals based on historical experiences and the most outstanding solutions of the population.

Furthermore, the LSA algorithm possesses interpretability and ease of implementation. Its ideas are relatively simple and intuitive, making it easy to understand and implement. The role switch between individuals and the process of knowledge transmission can be explained and analyzed, allowing users of the algorithm to better understand the optimization process.

The remaining sections of this paper are organized as follows: In Section  2 , an overview of the literature on metaheuristic algorithms is provided. Section  3 provides a detailed introduction to the principle and mathematical model of the LSA algorithm. In Section  4 , comprehensive experiments are conducted to demonstrate the superiority of the LSA algorithm over comparative optimization algorithms. Finally, Section  5 presents the conclusions of this paper.

2 Literature review

This section provides an overview of the current advancements in metaheuristics. In recent times, numerous metaheuristic algorithms have been introduced and extensively studied. These algorithms primarily fall into four categories: (1) swarm-based algorithms that emulate swarm intelligence, (2) evolutionary-based algorithms that draw inspiration from natural evolutionary processes, (3) physics or chemistry-based algorithms that are inspired by physical or chemical phenomena, and (4) social or human-based algorithms that are influenced by human or social behaviors. Table 1 presents a compilation of notable and recently developed metaheuristic algorithms.

Natural evolution algorithms are developed from biological phenomena such as natural evolution, and the representative algorithm is the GA algorithm (Mirjalili 2019 ); Other algorithms include the differential evolution (DE) algorithm (Das and Suganthan 2010 ), evolution strategy (ES) (Schwefel and Rudolph 1995 ), memetic algorithm (MA) (Moscato et al. 2004 ), and genetic programming (GP) (Sette and Boullart 2001 ). Swarm intelligence optimization algorithms, which simulate the social behavior of animals based on group foraging behaviors, have attracted increasing attention. Notable algorithms in this domain include the particle swarm optimization (PSO) algorithm (Poli et al. 2007 ), the crow search algorithm (CSA) (Askarzadeh 2016 ), cuckoo search (CS) algorithm (Yang and Deb 2014 ), the social spider algorithm (SSA) (James and Li 2015 ), the sparrow search algorithm (SSA) (Xue and Shen 2020 ), the red fox optimization (RFO) algorithm (Połap and Woźniak 2021 ), the salp swarm algorithm (SSA) (Mirjalili et al. 2017 ), dolphin partner optimization (DPO) (Dhanya and Arivudainambi 2019 ), Lion Optimization Algorithm (LOA) (Yazdani and Jolai 2016 ), dingoes hunting strategies (DHS) (Peraza-Vázquez et al. 2021 ), mycorrhiza tree optimization algorithm (MTOA) (Carreon-Ortiz and Valdez 2022 ), charged system search (CSS) (Kaveh and Talatahari 2010 ), chameleon swarm algorithm (CSA) (Braik 2021 ), wild horse optimizer (WHO) (Naruei and Keynia 2022 ), mayfly optimization algorithm (MOA) (Zervoudakis and Tsafarakis 2020 ), capuchin search (CSA) (Braik et al. 2021 ), Zebra Optimization Algorithm (ZOA) (Trojovská et al. 2022 ), Tasmanian Devil Optimization (TDO) (Dehghani et al. 2022a ), Artificial rabbits optimization (RSO) (Wang et al. 2022a ), Osprey Optimization Algorithm (OOA) (Dehghani and Trojovský 2023 ), Exponential distribution optimizer (EDO) (Abdel-Basset et al. 2023 ), and others. These algorithms have demonstrated promising performance in solving various complex optimization problems. Besides, there is a class of physical search algorithms based on the simulation of physical phenomena, such as the simulated annealing (SA) (Bertsimas and Tsitsiklis 1993 ), gravitational search algorithm (GSA) (Saremi et al. 2017 ), curved space optimization (CSO) (Moghaddam and Moghaddam 2012 ), lighting attachment procedure optimization (LAPO) (Nematollahi et al. 2017 ), black hole mechanics optimization (BHMO) (Kaveh et al. 2020a ), plasma generation optimization (PGO) (Kaveh et al. 2020b ), solid system algorithm (SSA) (Zitouni et al. 2020 ), atomic search algorithm (ASO) (Li et al. 2020a ), Heap-based Optimization (HBO) (Askari et al. 2020 ), Weighted mean of vectors algorithm(INFO) (Ahmadianfar et al. 2022 ), Exponential distribution optimizer(EDO) (Ayyarao et al. 2022 ), Subtraction-Average-Based Optimizer(SABO) (Trojovský and Dehghani 2023 ), etc. Some intelligent algorithms have been designed based on human social activities, such as the teaching–learning-based optimization (TLBO) (Rao et al. 2012 ), skill optimization algorithm (SOA) (Ramshanker and Chakraborty 2022 ), cooperative search algorithm (CSA) (Feng et al. 2021 ), human urbanization algorithm (HUA) (Kılkış and Kılkış 2019 ), heap-based optimization (HBO) (Askari et al. 2020 ), stock exchange trading optimization (SETO) (Emami 2022 ), arithmetical optimization algorithm (AOA) (Abualigah et al. 2021 ), Driving Training-Based Optimization (DTBO) (Dehghani et al. 2022b ), Chef-Based Optimization Algorithm (CBOA) (Trojovská and Dehghani 2022 ), War Strategy Optimization Algorithm (WSO), and so on. Additionally, in the past three years, many relatively new meta-heuristic algorithms have been proposed, and they are not classified into the mentioned categories. For example, inspired by the management strategy of the constitutional monarchy government, Ahmia et al., proposed the monarchy meta-heuristic (MN) optimization algorithm (Ahmia and Aider 2019 ). Brammya et al. utilized a simulation of human deer hunting behavior to propose a deer hunting optimization algorithm (DHOA) (Brammya et al. 2019 ). In a similar vein, Hayyolalam and Kazem devised a black widow optimization (BWO) algorithm (Hayyolalam and Kazem 2020 ), drawing inspiration from the mating behavior of black widow spiders. Nematollahi et al. introduced the Golden Ratio Optimization Method (GROM) as an optimization approach (Nematollahi et al. 2020 ). Li et al. developed the virus propagation optimization (VSO) algorithm (Li et al. 2020b ), which simulates the propagation process of the virus. Alsattar et al. proposed the bald eagle search (BES) algorithm (Alsattar et al. 2020 ) based on the hunting process of the bald eagle.

3 Learning search algorithm

This section provides the details and optimization procedure of the proposed learning search algorithm (LSA). The algorithm is inspired by human learning behaviors in the social environment, including the global learning behavior guided by historical experience and other social individuals, and the local learning behavior guided by role models. The analysis of the mathematical model and the realization process and time complexity of LSA is presented below.

3.1 The basics of MHSs and the proposed LSA method

The general framework of meta-heuristic search algorithms (MHSs) typically consists of three essential components: the selection guides mechanism, the search operators design, and the update mechanism design, as demonstrated in Fig.  1 (Kahraman et al. 2023 ).

General steps involved in the MHS process

The process of selecting candidate solutions from a population to guide the search process is a fundamental aspect of the MHS algorithm. Various methods exist for guiding this selection (Fig.  1 ), but the dominant approach is currently the survival theorem, which compares the fitness values of individuals within the population (Forrest 1993 ; Holland 1992 ). More recently, the Fitness-Distance Balance (FDB) has emerged as a promising new method for guiding selection in the MHS algorithm (Kahraman et al. 2022 ; Guvenc et al. 2021 ; Duman et al. 2023 ). Selecting excellent individuals as guidance is a critical step in MHS algorithms. Reasonable selection of individuals, balancing diversity and convergence, directly affects the efficiency and quality of search results. The driving force behind human progress is the ability to learn different perspectives and interpretations from different historical periods, cultivating critical thinking and analytical skills. By comparing and evaluating different historical events and interpretations, people can gain a better understanding of the complexity and diversity of history and draw their own conclusions. Additionally, role models serve as symbols of successful experiences, having achieved excellence in a particular field or skill. Humans can use the behavior, thinking, and decision-making of role models to guide the application of knowledge, avoiding some common mistakes and dilemmas. Following the mechanism of MHS algorithms, we can use historical experience and role models as guidance leaders in the search process, emphasizing the balance between diversity and convergence.

The design of search operators is a crucial element of MHS algorithms as it shapes the models simulating the distinct behaviors and survival skills unique to each population. The search operators for various MHS algorithms differ, including genetic-based crossover and mutation operators (Chen et al. 2020a ; Mirjalili 2019 ; Das and Suganthan 2010 ; Schwefel and Rudolph 1995 ; Holland 1992 ), operators based on swarm foraging behavior (Poli et al. 2007 ; Askarzadeh 2016 ; Yang and Deb 2014 ; James and Li 2015 ; Xue and Shen 2020 ; Połap and Woźniak 2021 ), operators based on physical natural phenomena (Bertsimas and Tsitsiklis 1993 ; Moghaddam and Moghaddam 2012 ; Li et al. 2020a ), and operators based on human social activities (Wilson et al. 2022 ; Tian et al. 2022 ; Ewees et al. 2022 ). A high capacity for summarizing experiences and imitative learning, as well as autonomous learning ability, is why human learning surpasses that of other organisms. The proposed algorithm embodies the subjective autonomy of human learning behavior and the diversity of learning approaches, fully embodying its potential for breakthroughs in MHS.

In MHS algorithms, the majority of update mechanisms employ a greedy approach based on fitness values (Yang and Deb 2014 ; Carreon-Ortiz and Valdez 2022 ; Saremi et al. 2017 ). This approach guarantees a balanced turnover of individuals within the population, ensuring that the introduction of a specific number of new individuals is accompanied by the removal of an equivalent number of existing individuals. An alternative approach for update mechanisms, referred to as the “direct” approach, is depicted in Fig.  1 , exemplified by the SCA (Trojovský and Dehghani 2022 ) and SMA (Li et al. 2020c ). In these algorithms, mutated individuals survive at each step of the search process, while previous individuals are eliminated. Furthermore, the NSM score-based approach has also proven to be an efficient method for update mechanisms (Kahraman et al. 2023 ). Human learning behavior can be characterized as “taking the essence, discarding the dross.” Unlike other organisms, humans possess the ability for reflection, critical thinking, and abstract reasoning. They can selectively choose valuable content that aids in personal learning and understanding, assimilating and integrating it into their own knowledge system. Thus, the update mechanism designed for the LSA algorithm adopts a greedy strategy based on fitness values.

3.2 Inspiration

A learning behavior encompasses the acquisition of behaviors by individuals, resulting from a combination of genetic and environmental factors and shaped by life experiences. For human beings, learning is not only an activity of simply adapting to the environment but also has social significance. Therefore, human learning has social characteristics, and this is mainly manifested in its indirect experience and positive initiatives.

Interaction with others allows individuals to acquire knowledge not only from their direct experiences but also from the collective historical experiences of human society. As human culture has evolved, society has accumulated a vast body of knowledge and experience, which has been transmitted through social inheritance. From birth, individuals in human society have the ability to assimilate the wisdom passed down by previous generations through interactions with teachers in educational institutions. Additionally, they also have the opportunity to acquire valuable social experiences through interactions with their contemporaries. This mode of indirect experiential learning is characterized by its rich and diverse content and form, setting it apart from learning processes observed in animals (McFarland et al. 1993 ; Bennett 2011 ), as depicted in Fig.  2 (a).

Depicts human learning behavior through two distinct avenues. Panel ( a ) highlights the role of historical experience and interaction with other individuals in the learning process. Panel ( b ) illustrates how active learning and mentorship from role models also contribute to effective learning outcomes

Animal learning is primarily an adaptive process driven by environmental factors, making it a passive endeavor. In contrast, human learning encompasses not only a desire to understand the world but also a determination to shape and alter it. Thus, humans engage in active interactions with their surroundings, learning through integration with the individuals they encounter. The purpose of human learning extends beyond merely satisfying physiological needs; it also encompasses the demands of social life. Consequently, humans possess a wide range of learning motivations and objectives. In their pursuit of these objectives, humans actively explore diverse and effective learning methods, a capability that surpasses the realm of animal learning (Schoenewolf 1990 ; Bruner 2009 , 1971 ), exemplified in Fig.  2 (b).

Inspired by the behaviors observed in human life and learning, this paper presents the Learning Search Algorithm (LSA) as a groundbreaking meta-heuristic approach. The LSA’s mathematical model is outlined below.

3.3 Mathematical model of the proposed algorithm

Human learning exhibits two distinct modes of behavior, characterized by different approaches to acquiring knowledge. One mode involves the utilization of historical experience and interactions with others, enabling a global search process. In this mode, individuals benefit from the indirect aspect of human learning, where accumulated wisdom and collective experiences guide their learning journey. The other mode involves the active participation of individuals, particularly the role model who represents the current optimal individual. This role model not only imparts knowledge to others but also actively engages in learning, thereby facilitating local search within the learning algorithm. This active aspect of human learning contributes to the refinement and fine-tuning of the individual’s knowledge. By incorporating these two modes of learning behavior, the Learning Search Algorithm (LSA) enriches and comprehensively expands the overall knowledge of the population. This algorithm integrates the global exploration facilitated by historical experiences and interactions, along with the localized refinement through active learning from the role model. Such integration leads to a synergistic effect, where the collective wisdom accumulated through historical experiences is combined with the adaptability and learning capabilities of individuals. Furthermore, the LSA incorporates autonomous control factor dynamics, ensuring a seamless wide-ranging exploration to precise refinement. This dynamic adaptation mechanism enables the algorithm to strike a balance between exploration and exploitation, allowing for efficient knowledge acquisition and optimization.

3.3.1 Initialization

In this investigation, we utilized a population-based swarm intelligence optimization technique referred to as the Learning Search Algorithm (LSA). LSA is designed to find optimal solutions by iteratively updating the individual candidate solutions within the population. The population’s position is modeled using a matrix, as demonstrated in Formula ( 1 ):

where, \(n\) represents the number of individuals, \(\dim\) indicates the dimensionality of the search space, and \(x_{i,j}\) represents j th dimension of individual i . It is noteworthy that each position is generated through uniform distribution, as illustrated in Formula ( 2 ):

where, \(rand(0,1)\) denotes a random number between 0 and 1, while \(ub_{j}\) and \(lb_{j}\) correspond to the upper and lower bound values, respectively.

Formula ( 3 ) provides a means of evaluating the fitness score for each individual in the search population. This score serves as a metric to assess their overall level of fitness within the context of the study.

In the LSA algorithm, the balance control factor \(\delta\) realizes the conversion from global exploration to fine-tuning in a dynamic and self-adaptive way, and the calculation method is:

where, the balance factors \({\delta}_{init}\) and \({\delta}_{final}\) refer to the initial and eventual values, respectively, while \({t}_{\max }\) signifies the maximum iteration count. Additionally, \(y^{t} \in ( - 1,1)\) is a chaotic sequence. The multiplication factors λ and γ are defined.

The selection of multiplication factors λ and γ is discussed herein. To ensure that the balance control factor δ remains within the interval (0, 1), we first analyze the selection of various values for γ (see Figs.  3 and 4 ) . Figure  4 indicates that the requirement is satisfied when 1.4 < γ < 2.2. Building on this observation, we further refine the selection of γ by testing values of 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, and 2.1 on 30 benchmark functions from CEC 2014. Test results show that the choice of γ minimally affects the LSA algorithm’s outcomes; however, slightly superior performance is observed when γ equals 2 (refer to Appendix Table  27 ). Therefore, for brevity, we set γ to 2 in this paper. Additionally, we explore the impact of λ on the LSA algorithm across various values. To ensure a balance between global exploration and local exploitation capabilities, the LSA algorithm should exhibit strong global exploration abilities in early iterations and potent local exploitation capabilities along with the ability to escape from local optima in later iterations. Consequently, some individuals in later iterations of LSA should conduct global exploration operations to prevent the algorithm from converging to local optima. As depicted in Fig.  3 , the value of λ should range between (0.5, 0.85) (when λ is too large, δ exceeds 1). To precisely determine the value of λ, experiments are conducted with λ set to 0.5, 0.6, 0.7, 0.75, and 0.8, respectively. Statistical analysis of the results in Appendix Table  26 reveals that setting λ to 0.75 yields 10 optimal results, making it the most favorable choice among these scenarios. In summary, setting λ to 0.75 and γ to 2 is deemed reasonable. Initially, \(\delta\) continually decreases, indicating the transition from wide-ranging exploration to focused searching. However, as the process proceeds, some individuals fall into local optimization, resulting in an increase of \(\delta\) . To mitigate this issue, \({\delta}_{init}\) and \({\delta}_{final}\) are assigned the values 1 and 0, respectively, to enable dynamic balancing between international and regional development in the proposed algorithm.

The range of the balance control factor δ when the multiplication factors λ take different values

The range of the balance control factor δ when the multiplication factors γ take different values

This approach ensures that the algorithm maintains a balance between wide-ranging exploration and local refinement during the entire iteration process, thus enabling it to achieve both objectives effectively. It increases the likelihood of conducting extensive global search in the initial stages of evolution and progressively shifts focus towards thorough local search in later stages. Consequently, this method optimally balances the algorithm’s capacity for broad-based exploration and targeted growth.

3.3.2 Exploration phase

The historical experiences of humanity offer valuable insights for the education of posterity and individual learning journeys. To make the algorithm have a strong global exploitation ability, this paper used the historical experience and other individual information to guide the search progress. The individuals in the population learn from past events and the current inhabitants at a random probability, and the corresponding updated schematic diagram is illustrated in Fig.  5 (c). The mathematical model is shown in Formula ( 6 ).

where, \({x}_{i}^{t}\) represents the i th learning individual, while \({x}_{j}^{t}\) denotes an individual selected at random from the identical population. The new individual is denoted by \({x}_{i}^{t + 1}\) , and \(rand\) is randomly generated with support [0,1]. Additionally, \({history}_{r}^{t}\) represents an individual randomly selected from the past experience database. In the initial phase of populating, Formula ( 2 ) is utilized to produce a population of equal size, with \(x\) being assigned random values. In every iteration, \(history\) is modified using Eq. ( 7 ), and the matrix of \(history\) suffers random variations according to Formula ( 8 ).

where, \({\text{a}}\) and \({\text{b}}\) are randomly generated with support [0,1]. The variable \(permuting\) represents a random permutation operation.

Different search patterns of the individuals in 2D search space (a) The individual's active learning mode (b) The radiation pattern of role models (c) Global exploration mode

3.3.3 Exploitation phase

The learning process involves learners acquiring knowledge from historical experiences, while simultaneously enhancing the overall learning ability of the population. This is achieved through active learning from role models, who are identified as optimal individuals exhibiting effective teaching behaviors. Nonetheless, equitable benefits from these role models are not experienced by all individuals within the population due to limited capacity to acquire knowledge. Thus, it is imperative to ascertain the ideal number of beneficiaries derived from the role models to attain optimal enhancement efficacy. Psychological research has indicated that the average attention span of an individual typically ranges from 7 to 9 (Schoenewolf 1990 ; Bruner 1971 , 2009 ). Consequently, an excessive number of teachers can hinder the teaching process, leading to challenges in fully utilizing the instructional potential of role models. Considering the aforementioned analysis, it is evident that certain individuals actively learn from role models, while role models specifically instruct select individuals on particular occasions. The learning schematic diagrams for the algorithm are presented in Fig.  5 (a) and (b), accompanied by the corresponding mathematical model illustrated in Formula ( 9 ).

where, \(rand\) and \(r\) are randomly generated with support [0,1]. \(x_{best}^{t}\) denote the position of the role model. The degree to which the learner acquires knowledge from the role model is modulated by the learning factor denoted as” \(\beta\) ”, which is computed using Formula ( 10 ). Moreover, we also compute the \(x_{average}^{t}\) using Formula ( 11 ).

where, we investigate the relationship between the number of objects taught by role models  \(\left(sub\right)\) , the algorithm effectiveness  \(\left(sub=3\right)\) , and a random integer  \(\left(randi\right)\) . The experimental measurement process reveals that the algorithm achieves optimal performance when \(sub = 3\) . Additionally, \(randi\) is a randomly selected integer within the range of 1 to sub , where sub represents a specific value.

During this stage, learners enhance their knowledge acquisition through the teaching method employed by role models. This directed learning from role models effectively enhances the overall learning ability of the learners.

3.3.4 Cross boundary processing

During the search process, individuals in the population may exceed the constraint of the problem domain, so it is necessary to handle individuals outside the boundary. Meanwhile, different boundary processing methods have the particular effect on the efficiency. The processing method adopted by this algorithm is shown in Formula ( 12 ):

where, \(lb\) and \(ub\) correspond to the upper and lower bound values, respectively.

3.4 The procedure of LSA

Based on the earlier theoretical analysis, the LSA consists of two primary stages: global discovery and regional growth. In the global discovery stage, the algorithm simulates learning behaviors by drawing upon previous learning and unique learning tendencies observed in contemporary society. Conversely, the regional growth stage emulates instruction and acquisition behaviors. In this section, we present a comprehensive overview of the key procedures employed by LSA. Additionally, to assist with implementation, we provide the pseudo-code of the algorithm in Fig.  6 and Algorithm 1.

Flow chart of LSA algorithm

The pseudo code of LSA

3.5 Time complexity analysis

The time complexity of the proposed LSA algorithm serves as a crucial performance indicator. The entire LSA process consists of three key steps: the initial setup, evaluation of fitness value, and refinement of the learning search process. The computation complexity of the initialization process is \(O(nPop)\) . During each iteration, approximately \(\delta \cdot nPop\) individuals engage in global development operations, \((1 - \delta ) \cdot nPop/2\) participants utilize the role model guidance strategy, and \((1 - \delta ) \cdot nPop/2\) individuals utilize the active learning strategy from role models. As a result, the time complexity of LSA can be estimated as approximately \(O(nPop) + O(t_{\max } \cdot \delta \cdot nPop) + O(t_{\max } \cdot (1 - \delta ) \cdot nPop/2) + O(t_{\max } \cdot (1 - \delta ) \cdot nPop/2) = O(n + t_{\max } \cdot nPop)\) .

A heatmap visually represents data using color gradients to illustrate variations in data magnitude, aiding in the comprehension of correlations and trends. In Fig.  7 , darker hues indicate longer algorithm runtimes. Here, we examine the time efficiency of the LSA algorithm in addressing optimization problems from two angles. Firstly, we analyze its overall time complexity, which hinges on factors such as population size, iteration count, and problem intricacy. Figure  7 (a) illustrates that, with a set number of iterations, larger populations incur greater time costs (as depicted by colors closer to red), albeit with improved algorithmic precision (as indicated in Appendix Table  28 ). Conversely, Fig.  7 (b) demonstrates that, with a fixed evaluation count, solving more complex problems consumes more time (evident in functions F26 and F27, represented by darker colors), albeit with marginal gains in precision (as shown in Appendix Table  29 ). Secondly, we scrutinize algorithm runtime and precision through specific execution strategies, comparing outcomes with varied balance control values (denoted by δ) using formula ( 5 ). Figure  7 reveals that setting δ to 0 results in maximal execution times (reflected by red hues), while δ set to 1 minimizes execution times (reflected by blue hues). This disparity arises from δ’s influence on the algorithm’s tendency towards local exploitation (Eqs. ( 10 )-( 11 )) or global exploration (Eq. ( 6 )), impacting time costs. However, employing a fixed δ value compromises search precision compared to formula ( 5 ) (as detailed in Appendix Table  30 ).

Time consumption of the LSA algorithm under different parameters. a Time spent executing CEC 2014 functions with fixed number of iterations. b Time spent executing CEC 2014 functions with fixed number of evaluations. c Time spent executing CEC 2014 functions with different values of δ

3.6 Convergence analysis

3.6.1 markov chain model of the lsa algorithm.

Definition 1: The state of a learning agent and state space of a learning agent.

The state of a learning agent is composed of the position \(x\) and the global best position \(best\) , denoted as \(I = (x,best)\) , where \(x \in A,best \in A\) and \(f(best) \le f(x)\) , \(A\) is a feasible solution within the spatial range. The possible states of all learning agents constitute the state space of a learning agent, denoted as:

Definition 2: The state of a learning swarm and state space of a learning swarm.

The states of all individual learners within a learning swarm constitute the state of the learning swarm. The state of the \(m\) learners in the learning swarm at time \(t\) is denoted as \(s_{t} = (I_{{_{1} }}^{t} ,I_{{_{2} }}^{t} ,...,I_{{_{i} }}^{t} ,...I_{{_{m} }}^{t} )\) , where \(I_{{_{i} }}^{t}\) represents the i -th learner in the population at time \(t\) , and \(m\) is the total number of learners in the population. The collection of all possible states of the learning swarm forms the state space of the learning swarm, denoted as:

Definition 3: The state transition of individual learners.

For \(\forall I_{i} = (x_{i} ,best_{i} ){|} \in I,\forall I_{j} = (x_{j} ,best_{i} ){|} \in I\) , the state \(I_{i}\) of an individual learner transitions to another state \(I_{j}\) in one step, denoted as \(T_{I} (I_{i} ) = I_{j}\) .

Theorem 1: In the LSA algorithm, the probability of the state transition from state \(I_{i}\) to state \(I_{j}\) of an individual learner can be expressed as:

In the LSA algorithm, the algorithm primarily consists of two phases: the exploration phase and the exploitation phase. The exploitation phase is further comprised of two distinct update modes: “Active learning” and “Role models teaching”.

In each position update strategy, the state of an individual learner is formed by the position \(x\) and the global best position \(best\) . Therefore, the corresponding one-step transition probability also varies. Considering that the learner’s vector is multidimensional and the population forms a set of points in a multidimensional hyperspace, the process of the learner’s position change can be regarded as the transformation between points in this hyperspace.

According to Definition 3 and the geometric interpretation of the LSA algorithm, the one-step transition probability from state \(I_{i}\) to another state \(I_{j}\) in the exploration phase is given by:

where, the one-step transition probability from the individual’s global best solution to another state is given by:

The probability of the learning individual transitioning from position \(x_{i}\) to position \(x_{j}\) through the exploration phase search strategy is:

The transition probability from state \(I_{i}\) to another state \(I_{j}\) through the active learning strategy is:

where, the probability of the learning individual transitioning from position \(x_{i}\) to position \(x_{j}\) in one step is:

The transition probability from state \(I_{i}\) to another state \(I_{j}\) through the Role models teaching strategy is:

Definition 4: The state transition of learning community.

For \(\forall s_{i} \in S,\forall s_{j} \in S\) , in the iteration of LSA algorithm, the learning community transition from state \(s_{i}\) to another state \(s_{j}\) in a single step, denoted as \(T_{S} (s_{i} ) = s_{j}\) . The transition probability for the learning community state \(s_{i}\) to transition to another state \(s_{j}\) in a single step is:

where, \(m\) represents the number of individuals in the population. This equation states that the one-step transition from learning group state \(s_{i}\) to state \(s_{j}\) is the transition of the individual states from all individuals in group space \(s_{i}\) to the corresponding individual states in \(s_{j}\) .

3.6.2 Convergence analysis of LSA algorithm

Definition 5: Markov chain.

In a stochastic process, let process \({\{x_{n} {,}n \in T\}}\) have parameter set T as a discrete time series, denoted as \(T = \{ 0,1,2,...\}\) , where the entire set of possible values \(x_{n}\) constitutes a discrete state space \(I = {\{i_{1} {,}i_{2} ,i_{3} ,...\}}\) . If for any integer \(n \in T\) and any \(i_{1} {,}i_{2} ,i_{3} ,...,i_{n + 1} \in I\) , the following conditional probability \(p{\{x_{n + 1} = i_{n + 1} {|}x_{n} = i_{n} \}}\) holds, then \({\{x_{n} {,}n \in \mathrm{T}\} }\) is termed as a Markov chain.

Definition 6: Finite Markov chain.

If the state space \(I\) is finite, then the Markov chain is referred to as a finite Markov chain.

Definition 7: Homogeneous Markov chain.

\(p{\{x_{n + 1} = i_{n + 1} {|}x_{n} = i_{n} \}}\) represents the conditional probability that the system, in state \(i_{n}\) at time \(n\) , transitions to a new state \(x_{n + 1}\) . If this probability depends only on the state at time \(n\) and not on time \(n\) , then the Markov chain is referred to as a homogeneous Markov chain.

Theorem 2: In the LSA algorithm, the state sequence \(\left\{ {s(n);n \ge 0} \right\}\) of the learning community is a finite homogeneous Markov chain.

According to Definition 4, it is known that in the state transition of the learning community, there exists a state \(\forall {\text{s}}(n) \in S,\forall {\text{s}}(n + 1) \in S\) in the sequence \({\{ \text{s(n);n}} \ge {0\}}\) , and its transition probability \(p(T_{S} (s(n)) = s(n + 1))\) is determined by the transition probabilities \(p(T_{S} (I_{i} (n)) = I_{j} (n + 1))\) of all learning individuals in the community. From Eqs. ( 15 ) to ( 22 ), it is known that the state transition probability of any individual in the learning community is only related to the control factors \(\delta\) , the states \((x_{i} ,best)\) at time \(n\) , \(x_{average}\) , a random number \(r,r_{1}\) between [0,1], and \(\beta\) , but is independent of time \(n\) . Based on the above analysis, the state transition probability \(p(T_{S} (s(n)) = s(n + 1))\) of the learning community only depends on the individual states at time \(n\) , therefore, the sequence \({\{ \text{s(n);n}} \ge {0\}}\) exhibits the Markov property.

From Eqs. ( 16 ) to ( 22 ), it can be seen that \(p(T_{S} (s(n)) = s(n + 1))\) is independent of the time \(n\) . Combining with (i), it can be inferred that the sequence \({\{ \text{s(n);n}} \ge {0\}}\) is a homogeneous Markov chain.

When optimizing any problem in a computer, the variables used to describe the optimization problem are represented with a certain precision, and the search space is finite. In any individual \(I_{i} = (x_{i} ,best)\) of the learners, the dimensions of \(x_{i}\) and \(best\) are finite, and \(x_{i}\) and \(best\) are also constrained by \([x_{\min } ,x_{\max } ]\) . Therefore, the space of learning individuals \(I\) is finite. And the learning community composed of \(m\) learning individuals is also finite.

Based on (i), (ii), and (iii), it can be inferred that the state sequence \({\{ \text{s(n);n}} \ge {0\}}\) of the learning community is a finite homogeneous Markov chain.

Convergence criterion

The LSA algorithm belongs to the category of stochastic search algorithms, thus in this paper, the convergence behavior of the LSA algorithm is determined using a convergence criterion based on random algorithms (Solis and Wets 1981 ).

For the optimization problem <  A , f  > , where A is the feasible solution space and f is the fitness function, if there is a stochastic optimization algorithm D and the result of the k -th iteration is \(x_{k}\) , then the result of the next iteration is \(x_{k} { = }D(x_{k} ,\zeta )\) , where \(\zeta\) represents solutions previously encountered during the iterative search process of algorithm D. The lower bound of the search is defined as:

where, \(v(x)\) represents the Lebesgue measure on the set \(x\) . The region of optimal solutions is defined as:

where, \(\varepsilon > 0\) and \(C\) are sufficiently large positive numbers. If the stochastic algorithm D finds one point in \(R_{\varepsilon ,M}\) , it can be considered that the algorithm has found an acceptable global optimal or approximately global optimal point.

Condition H1: If \(f(D(x_{k} ,\zeta ) \le f(x))\) , then \(\zeta \in A\) implies \(f(D(x_{k} ,\zeta ) \le f(\zeta ))\) .

Condition H2: For \(\forall B \in A\) , s.t. \(v(B) > 0\) , there exists \(\prod\limits_{k = 0}^{\infty } {(1 - u_{k} (B))} = 0\) , where \(u_{k} (B)\) is the probability measure of the k - th iteration search solution of algorithm D on set \(B\) .

Theorem 3: Let \(f\) be measurable, and let \(A\) be a measurable subset of \(R^{n}\) . Suppose algorithm D satisfies conditions H1 and H2, and \(\{ x_{k} \}_{k = 0}^{\infty }\) is the sequence generated by algorithm D. Then, there exists a probability measure \(\mathop {\lim }\limits_{k \to \infty } P(x_{k} \in R_{\varepsilon ,M} ) = 1\) ,where \(R_{\varepsilon ,M}\) is the optimal region, i.e., algorithm D globally converges.

LSA Algorithm Convergence

Theorem 4 : The LSA algorithm satisfies condition H1.

Proof: In the LSA algorithm, individual current optimal positions are updated at each iteration, denoted as follows.

Therefore, the LSA algorithm preserves the best position of the population at each iteration, satisfying condition H1.

Definition 8: Set of optimal states of learners, denoted as G.

Let \(g^{*}\) be the optimal solution of the optimization problem <  A , f  > , and let \(G = \{ s{ = (}x{)|}f(x) = f(g^{*} ),s \in S\}\) denote the set of optimal states of learners. If \(G = S\) , then each solution in the feasible solution space is not only feasible but also optimal. In this case, optimization is meaningless, and the following discussions are based on \(G \subset S\) .

Definition 9: Absorbing state Markov chain.

Based on the population sequence from the LSA algorithm, a Markov chain \(\{ s(t),t \ge 0\}\) and the set of optimal states \(G = S\) are defined. If \(\{ s(t),t \ge 0\}\) satisfies the conditional probability \(p\{ x_{k + 1} \notin G|x_{k} \in G\} = 0\) , then this Markov chain is called an absorbing state Markov chain.

Theorem 5: The Markov chain generated by the LSA algorithm is an absorbing state Markov chain.

In the optimization of LSA, the update of individuals adopts the mechanism of preserving the best individuals. That is, only when the fitness value of the current best individual is better than the fitness value of the original individual, will the original best individual be replaced, as shown in Eq. ( 26 ). This ensures that in each iteration of the algorithm’s evolution process, the newly generated individuals are not inferior to those generated before. Therefore, the conditional probability \(p\{ x_{k + 1} \notin G|x_{k} \in G\} = 0\) is satisfied, that is, the population sequence \(\{ s(t),t \ge 0\}\) of the LSA algorithm forms an absorbing state Markov chain.

Definition 10: Let set \(D\) be a non-empty subset of the state space \(S\) . If \(\forall i \in D,\forall j \notin D\) , there exists \(p(i \notin D|j \in D){ = 0}\) , then \(D\) is called a closed set.

Definition 11: Let the set of optimal learning individual states be denoted as \(M\) , the set of optimal learning group states be denoted as \(G\) , and the global optimal solution of the optimization problem be denoted as \(best^{*}\) , then

where, \(I_{{\text{t}}}^{*} \in S\) represents the optimal learning individual state.

where \({\text{S}}_{n}^{*} = {\text{(I}}_{{_{1} }}^{*} {\text{,I}}_{{_{2} }}^{*} {,}...{\text{,I}}_{{_{i} }}^{*} {,}...{\text{I}}_{{_{m} }}^{*} {)}\) represents the optimal state of the population.

Theorem 6: The set of optimal states for individual learning, denoted as \(M\) , is a closed set.

Let the learning individual state \(I_{{^{i} }}^{n} = (x_{i}^{n} ,best^{*} )\) be the optimal state. According to the execution strategy of the LSA algorithm, it is evident that the next moment’s state \(I_{{^{i} }}^{{n{ + 1}}} = (x_{i}^{{n{ + 1}}} ,best^{*} )\) is also the optimal state. This can be concluded based on formulas ( 16 )-( 22 ).

In other words, \(\forall I_{{^{i} }}^{n} \in M,I_{j}^{n + 1} \notin M\) , \(p(I_{{^{i} }}^{n} \to I_{{^{j} }}^{n + 1} ) = {0}\) . Therefore, the set \(M\) of optimal states for the individual learner is a closed set.

Theorem 7: In the LSA algorithm, the set of optimal states for the learning population, \(G\) , is a closed set.

\(\forall s_{i} \in G,\forall s_{j} \notin G,s_{j} \in S\) , for any step size \(l,l \ge 1\) , according to the Chapman-Kolmogorov equation, we can obtain:

where, \(s_{ru - 1} \in G,s_{ru} \notin G,1 \le u \le l\) , it can be inferred from Definition 4:

Due to \(\exists I_{k}^{ru - 1} \in M,I_{k}^{ru} \notin M\) , then \(f(I_{k}^{ru} ) > f(I_{k}^{ru - 1} ) = f(I^{*} )\) . According to Eq. ( 17 ), \(p_{*} (best_{k}^{ru - 1} \to best_{k}^{ru} ) = 0\) , so \(p(T_{S} (s_{ru - 1} ) = s_{ru} ){ = 0}\) , which means \(p(i \notin G|j \in G){ = 0}\) . Therefore, \(G\) is a closed set in the \(S\) space.

Definition 12: For any \(n \ge 1\) , when \(l \ge n + 1\) , if \(best^{l} { = }best^{n}\) always holds true, \(s(n) \in S\) is referred to as an absorbing state. A set \(H\) constructed solely from a single absorbing state is called a closed set.

Theorem 8: Let \({\{ \text{s(n),n}} \ge {1\}}\) be a Markov chain representing the state sequence of a learning population, and \(\Lambda { = }G \cup H\) be a closed set. When \(\sum\limits_{n = 1}^{\infty } {p_{S} {\{ \text{s(}}n{)\} <\, }\infty }\) , then \(p(\mathop {\lim }\limits_{n \to \infty } {\text{s(}}n{)} \in \Lambda ){ = }1\) .

Based on the fact that all \(G,H\) are closed sets, it follows that \(\Lambda { = }G \cup H\) is also a closed set. Assuming that the learning population is in set \(\Lambda\) at time \(n\) and in state \(S(n{ + 1})\) at time \(n{ + 1}\) while being in state \(S(n)\) , we can conclude \(p_{S} {\{ \text{s(n + 1)}} \notin \Lambda {\text{|s(n)}} \in \Lambda {\} \text{= 0}}\) . Therefore,

If the learning population is in state \(S(n){ = (}I_{1}^{n} {,}I_{2}^{n} {,}...{,}I_{m}^{n} {)} \notin \Lambda\) , then \(\forall i \in [1,m],I_{i}^{n} \notin G,and \, I_{i}^{n} \notin H\) holds.

By summing up \(n\) , we can obtain

So, when \(n \to \infty\) and \(\sum\limits_{n = 1}^{\infty } {p_{S} {\{\text{ s(}}n{)\}\text{ < }}\infty }\) , \(p(\mathop {\lim }\limits_{n \to \infty } {\text{s(}}n{)} \in \Lambda ) = {1 - }p(\mathop {\lim }\limits_{n \to \infty } {\text{s(}}n{)} \notin \Lambda ){ = }1\) .

Theorem 9 : The LSA algorithm converges to the global optimum.

Proof: According to Theorem 4, it is known that the LSA algorithm satisfies condition H1. By Theorem 8, it is known that the probability of the LSA algorithm continuously searching for the global optimum for an infinite number of times is zero, thus there exists an \(\prod\limits_{k = 0}^{\infty } {(1 - u_{k} (B))} = 0\) satisfying condition H2. According to Theorem 3, it can be concluded that the LSA algorithm is a globally convergent algorithm.

3.7 The difference between TLBO and LSA

Teaching–learning-based optimization (TLBO) and Learning Search Algorithm (LSA) share common features as population-based algorithms inspired by human learning behavior. However, they diverge significantly in several aspects. Firstly, TLBO draws inspiration from the classroom teaching model, where knowledge transfer occurs bidirectionally: students learn from teachers, and teachers impart knowledge to students through direct instruction. Conversely, LSA predominantly acquires knowledge through individual learning from historical experiences (including both past and contemporary sources) and individuals emulating role models around them, with these role models disseminating knowledge to those receptive to specific learning aspects. Hence, the learning mechanisms of the two algorithms differ. Secondly, TLBO adopts a uniform knowledge dissemination approach throughout the entire population, overlooking individual physiological traits, which can hinder knowledge acquisition. In contrast, LSA, during its developmental phase, fully integrates learners’ unique attributes into the acquisition process, leveraging their ability to absorb knowledge from role models and learn from exceptional individuals. Thirdly, TLBO lacks distinction between global exploration and local exploitation phases, with all individuals following uniform learning and teaching strategies. In contrast, LSA embodies both phases and achieves a harmonious balance between global exploration and local exploitation through adaptive adjustment of balancing factors. Lastly, owing to its diverse learning strategies, LSA surpasses TLBO in search results for optimization problems like Unimodal Functions, Multimodal Functions, Hybrid Functions, and Composition Functions. This comprehensive superiority stems from LSA’s multifaceted learning approaches.

4 Experiment and simulation

To estimate the adequacy of the proposed approach, we conducted a comparative analysis against other state-of-the-art algorithms. Our implementation of the LSA algorithm was conducted utilizing MATLAB R2016a. The experimental setup comprises a personal computer equipped with an AMD Ryzen 74700G with Radeon Graphics and 12 GB main memory. The study employed a diverse set of test problems, including 30 IEEE CEC2014 and 10 IEEE CEC2020 benchmark functions, 6 challenging real-world engineering design optimization problems, and 15 feature selection datasets from UCI. Table 2 presents a collection of 40 minimization functions called CEC2014 and CEC2020, which is a powerful set of real-parameter optimization benchmarks. These functions effectively mimic real-world optimization problems. To assess the performance of the LSA, we selected 9 prominent swarm intelligence optimization algorithms and 11 powerful recently developed algorithms as a comparison. The population size ( nPop ) was set at 50, and the number of evaluations (FEs) was set at 1000* nPop . The search dimension was fixed at 10, whereas the parameter values for the comparison algorithms are presented in Table  3 .

In order to ensure the reliability of our results, we executed each test 20 times independently and highlighted the best-performing outcomes in the data statistics tables. Furthermore, to investigate the statistical significance of our results, we employed the Wilcoxon test at a significance level of 0.05. The symbols “/ = /-” were adopted to indicate whether the LSA algorithm is superior, equal to, or inferior to the comparison algorithms in terms of performance.

4.1 Results of IEEE CEC 2014 and CEC 2020

This section presents a comprehensive analysis of the proposed LSA algorithm as well as various state-of-the-art original and enhanced algorithms with improved search performance. The IEEE CEC 2014 and CEC 2020 benchmark functions have been chosen as the evaluation benchmarks for this study.

4.1.1 The search process experiment of LSA

In this subsection, the proposed LSA algorithm is utilized to solve a range of benchmark functions, and a detailed analysis of the optimization process is conducted.

Figure  8 (a) illustrates the three-dimensional mathematical model of the benchmark function. Notably, the mathematical model of F2 exhibits relative simplicity, while the remaining four functions possess a higher level of complexity. Figure  8 (b) demonstrates the search trajectory of the proposed algorithm from a top-down perspective. The larger blue dots represent the best search positions during the search process, while the other colored dots denote the positions of the search individuals throughout iterations. Additionally, Fig.  8 (c) depicts the progression of the average fitness value for the entire population. It is evident from Fig.  8 (b) that for both simple and complex functions, numerous historical search trajectories of individuals are concentrated in proximity to the global optimum, effectively ensuring the thoroughness of local search. Moreover, several discrete colored points are scattered in other regions, demonstrating the capability of the algorithm to perform global exploration and avoid being trapped in local optima. The convergence of the average fitness curve in all mathematical models is evident in Fig.  8 (c), underscoring the robust search capability of the LSA algorithm. In Fig.  8 (e), it’s evident that the LSA algorithm achieves a balanced approach between Exploration and Exploitation over time (the computational methods for Exploration and Exploitation are based on literature (Cheng et al. 2014 ; Hussain et al. 2019 )). As iterations progress, Exploitation steadily approaches 100 while Exploration declines towards 0. Analysis of functions F2, F8, and F15 reveals a rapid decrease in population diversity, showcasing the algorithm’s robust exploitation abilities. Conversely, for hybrid and composite functions like F25 and F27, population diversity fluctuates notably (the computational methods for calculating population diversity are based on literature (Cheng et al. 2014 ; Hussain et al. 2019 ).), consistently remaining at a higher level, demonstrating the algorithm’s strong global exploration capabilities (as shown in Fig.  8 (d)).

Visualization of the algorithm’s search process. a 3-dimensional mathematical models of the function. b The search trajectory during the optimization process is displayed from a top view. c Changes in the average fitness value of the population. d Population diversity. e Exploration and exploitation capabilities of the algorithm

4.1.2 Comparison with well-known original algorithms

This subsection tests the performance of the target algorithm LSA and other well-known original algorithms, including GWO (Mirjalili et al. 2014 ; Long et al. 2020 ), HHO (Chen et al. 2020b ), HPO (Naruei et al. 2022 ), MFO (Mirjalili 2015b ), SSA (Mirjalili et al. 2017 ; Abualigah et al. 2020 ), BWOA (Hayyolalam and Kazem 2020 ), SOA (Ramshanker and Chakraborty 2022 ), TLBO (Rao et al. 2011 ), and TSA (Kaur et al. 2020 ). In the CEC 2014 benchmark functions, F1-F3 are single-mode functions, and they are used to test the development ability of the algorithm; F4-F15 are multi-mode functions, and they have multiple local optimal values. If the algorithm's exploration ability is not sufficient, it will converge prematurely and quickly, or even fall into a local optimum, resulting in low convergence accuracy. F16-F21 are mixing functions, and F22-F30 are composition functions. These two types of functions are more challenging than the previous two types of functions, so they can better reflect the search performance of the algorithm. F31-F40 are the CEC 2020 benchmark functions. Table 4 presents the average and variance results of each algorithm for solving the CEC 2014 and CEC 2020 benchmark functions, and the best results are highlighted in bold. Figure  9 shows the convergence effect of the LSA and the comparison algorithms in solving 12 benchmark functions. Figure  10 illustrates the Friedman rank sum sorting results of each algorithm. Table 5 and Fig.  11 show the results of the Wilcoxon rank-sum experiment with a 5% significance.

individual's active learning mode

The convergence curves of LSA and other original algorithms on 12 benchmark functions

The results of the Friedman test for the LSA and other original algorithms

The graphical representation of the Wilcoxon signed rank test results of the LSA algorithm compared to the original algorithms

It can be seen from Table  4 that LSA ranks first on 24 functions and ranks second on six functions, with an overall rank of #1 (average rank AVG of 2.1). The results of the p values in Table  5 indicate that compared with the algorithms GWO, HHO, HPO, SSA, BWOA, MFO, SOA, TSA, and TLBO, the LSA obtained 27, 37, 35, 33, 38, 36, 36, 40, and 24 victories, showing a significant difference. Although the LSA algorithm obtained 3, 5, and 7 optimal results when solving Unimodal Functions (F1-F3), Multimodal Functions (F4-F16), and Hybrid functions (F17-F22) problems respectively, its overall optimal result rate is only (3 + 5 + 7)/22 = 68.18%. Moreover, when solving Composition Functions (F23-F30) problems, LSA only achieved 2 optimal results, whereas SOA obtained 3 optimal results. This indicates that SOA has certain advantages in solving Composition Functions problems. This also confirms the “No Free Lunch” (NFL) theorem, demonstrating that there is no one-size-fits-all solution for all optimization problems. However, compared to other algorithms, the LSA algorithm obtains the most optimal results in solving these two types of problems. The bubble chart in Fig.  11 vividly demonstrates the statistical superiority of the proposed algorithm over these well-known original algorithms in terms of search results.

According to the results of the Friedman rank sum test shown in Fig.  10 , the LSA algorithm achieved the highest rank among all algorithms, with a rank mean of 2.09. Overall, these experimental results indicate that the LSA performs superiorly on the CEC 2014 and CEC 2020 functions and is stronger than other comparison algorithms.

Figure  9 illustrates the convergence curves of benchmark functions for GWO, HHO, HPO, SSA, BWOA, MFO, SOA, TSA, TLBO, and the proposed LSA algorithm. Based on the convergence curves of functions F1, F3, F7, F13, F17, F18, F20, F22, F25, F32, F35, and F37, the LSA algorithm achieves the highest fitness values and the fastest convergence speed among these unimodal functions, multimodal functions, hybrid functions, and composition functions. In contrast, other algorithms fail to obtain global solutions due to being trapped in local optima. Therefore, the experimental results demonstrate that LSA effectively utilizes its exploitation capability for unimodal functions and exploration capability for multimodal functions. The incorporation of exploration and exploitation stages ensures the global convergence of the LSA algorithm.

The stability of an algorithm is also an important indicator of whether it is good or bad. To examine the reliability, we selected F1, F3, F10, F18, F21, F26, F21, F30, and F34 as the test subjects. From the box diagram in Fig.  12 , it can be seen that the box diagram of LSA algorithm is the most flat. This indicates that LSA algorithm has good stability.

The box plot of the results by well-known original algorithms

4.1.3 Comparison of LSA with recently proposed advanced improved algorithms

In this subsection, LSA is compared with 6 high-performance improved algorithms proposed in recent years, including GNHGWO (Akbari et al. 2021 ), GWOCS (Wang et al. 2022b ), HPSOBOA (Zhang et al. 2020 ), NCHHO (Dehkordi et al. 2021 ), PSOBOA (Zhang et al. 2020 ), HFPSO (Aydilek 2018 ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC (Duman et al. 2022 ), TSALSHADE (Abdesslem layeb 2023 ), and ISSA (Dorian Sidea 2024 ).

Table 6 shows the experimental results of these 12 algorithms on CEC 2020 functions. Table 7 presents the p-value results of the Wilcoxon test. It can be seen from Table  6 that the LSA obtained an AVG value of 2.8, ranking first overall. Meanwhile, the p-values in Table  7 and Fig.  16 indicate that compared to the algorithms GNHGWO, GWOCS, HPSOBOA, NCHHO, HFPSO, PSOBOA, FDB-AGDE, dFDB-MRFO, AFDB-SFS, FDB-TLABC, and TSALSHADE, the LSA obtained 7, 6, 10,10,10,7,3,6,4,6, and 6 victories, respectively, showing a significant difference. The Friedman rank sum value of the LSA in Fig.  13 is 2.8, which is the smallest among all algorithms. The sorting results indicate that the LSA ranks first among all algorithms. These experimental results show that the LSA achieves superior performance on the CEC 2020 function and is stronger than other algorithms. The excellent performance of the LSA makes it a new optimizer that can be applied to solve complex and realistic optimization problems.

Friedman test was performed to compare the results of the LSA with other enhanced algorithms

Appendix Table  25 presents the results of LSA and ISSA algorithms in solving CEC 2014 problem. From appendix Table  25 , it can be observed that the LSA algorithm achieved victories in 28 test functions. Whether in single-modal, multi-modal, hybrid, or composite functions, the LSA algorithm outperformed the ISSA algorithm comprehensively.

The excellent convergence performance of the LSA algorithm is demonstrated in Fig.  14 , compared with GNHGWO, GWOCS, HPSOBOA, NCHHO, HFPSO, PSOBOA, FDB-AGDE, dFDB-MRFO, AFDB-SFS, FDB-TLABC, and TSALSHADE. The main reason for this achievement is the combined effect of the LSA algorithm’s local exploitation strategies (Active learning strategy and Role models teaching strategy) and global exploration strategy.

LSA and other improved algorithms’ convergence curves on 9 CEC 2020 benchmark functions

In summary, LSA shows strong competitiveness compared with the top-performing meta-heuristic algorithms put forth in recent years. This indicates that our proposed LSA obtains good results in dealing with numerical optimization problems.

To test the stability of the LSA algorithm, we selected F31, F34, F35, F36, F37, and F38 as the test subjects. According to Fig.  15 , compared with the distributions of optimal solutions, the box diagram of LSA algorithm is the most flat, indicating its good stability (Fig.  16 ).

The box plot of the results by recently proposed advanced improved algorithms for CEC 2020

The graphical representation of the Wilcoxon signed rank test results of the LSA algorithm compared to the other improved algorithms

4.1.4 Consumption time cost analysis

Tables 8 and 9 present the time consumption (unit: second) of each algorithm when solving the CEC2014 and CEC2020 functions. To more intuitively show the results, the proportion of time consumption is shown in Figs.  17 and 18 .

The proportion of cost time for LSA compared to the original algorithms

The proportion of cost time for LSA compared to the improved algorithms

It can be seen from Table  8 that compared with the original benchmark algorithm, the average time consumption of the LSA when solving the CEC 2014 and CEC 2020 problems is 0.305 s, ranking in the middle of the nine algorithms. Figure  17 illustrates the percentage of the cost time by each algorithm in executing different test functions, providing a more intuitive reflection of the performance of the proposed algorithm in terms of execution time. Meanwhile, Table  9 and Fig.  18 demonstrate that compared with the newly proposed improved algorithm, the average time consumption of the LSA in solving the CEC 2014 and CEC 2020 problems ranks 6th. The relatively high time expense of the target algorithm is mainly due to the computation cost of determining the number of beneficiaries in the exemplar instructing process, which is calculated using Formula ( 11 ).

4.1.5 Parameter sensitivity analysis

Different parameters may have different influences on the performance of the algorithm. To explore the influence of the parameter sub (the number of subjects taught by role models) on the performance of the LSA, this paper selected different values of sub, i.e., 2, 3, 4, 5, 6, 7, and 8, to conduct experiments, and the values of other parameters remained the same. In addition, we discussed the impact of different values of parameter \(y^{0}\) and parameter \(\delta_{init} > 1\) on the algorithm. The test case was obtained from CEC 2014 and CEC 2020. Each test case was independently run 20 times, and the final results are presented in Table  10 , where the optimal results are highlighted in bold.

It can be seen from Table  10 that when sub equals 3, the number of average optimal values obtained by the algorithm is the largest, indicating that the number of role models in teaching must be controlled within a certain range so that the teaching object effect can reach the best level. This is consistent with the analysis results in Sect.  3.3.3 .

From the statistical results in Appendix Table  23 , it can be observed that the best results were achieved when \(y^{0} = 0.75\) , with a total of 21 instances. On the other hand, the least favorable results were obtained when \(y^{0} = 0.5\) . Therefore, the value of \(y^{0}\) does have some impact on the target algorithm, although the fluctuation in the results of the problem-solving process is not significant.

According to the statistical results from Appendix Table  24 , it can be observed that when \(\delta_{init} = 1\) , the proposed algorithm achieves the best results with a count of 15, outperforming the cases where \(\delta_{init} > 1\) . The analysis suggests that this improvement can be attributed to the excessive global exploration carried out by the algorithm during the iterative process when \(\delta_{init} > 1\) , which consequently undermines the algorithm’s capability for local exploitation.

4.2 Results of real-world constrained optimization problems

The design of many real-world engineering structures is usually limited by various conditions. When solving such problems, engineers need to deal with additional constraints. To test the LSA algorithm in solving engineering real optimization problems, this paper selected 6 real-world engineering design problems, such as speed reducer design et al.

4.2.1 Speed Reducer Design (SRD)

The goal of SRD is to design a reducer with a minimum weight. SRD contains seven design variables and eleven inequality constraints. A detailed description of this problem can be found in the reference (Dhiman and Kumar 2017 ). The mathematical model of the problem is as follows:

where \(2.6 \le x_{1} \le 3.6,0.7 \le x_{2} \le 0.8,17 \le x_{3} \le 28,7.3 \le x_{4} \le 8.3,7.3 \le x_{5} \le 8.3,2.9 \le x_{6} \le 3.9,5 \le x_{7} \le 5.5\) .

When testing LSA to solve this problem, this paper selected some well-known meta-heuristic algorithms proposed in recent years including STOA (Dhiman and Kaur 2019 ), TQA (Chen et al. 2022 ), HS (Dhiman and Kumar 2017 ), ESMA (Örnek et al. 2022 ), GSA (Karami et al. 2021 ), EJAYA (Zhang et al. 2021b ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC (Duman et al. 2022 ),and TSALSHADE (Abdesslem layeb 2023 ) as comparison algorithms. Table 11 shows the statistical results of the LSA and comparison algorithms for solving this problem. It can be seen from Table  11 that the LSA obtained the result of 2986.064(consistent with the results of FDB-AGDE, dFDB-MRFO, AFDB-SFS, and FDB-TLABC), which is the best among all compared algorithms.

4.2.2 The Tension/Compression Spring Design (TCSD)

In the design of engineering problems, in addition to considering the optimal objective function of the mathematical model of the designed product, designers also need to consider the corresponding constraints. TCSD is a classic engineering design problem, and its goal is to minimize the weight of the designed product. In this problem, there are three variables and four inequality constraints (Faramarzi et al. 2020 ), and the mathematical model is as follows:

where \(0.05 \le x_{1} \le 2,0.25 \le x_{2} \le 1.3,2 \le x_{3} \le 15\) .

Various intelligent algorithms have been used to solve this engineering design problem, such as EO (Faramarzi et al. 2020 ), RL-BA (Meng et al. 2019 ), DDAO (Ghafil and Jármai 2020 ), SDO (Zhao et al. 2019 ), AFA (Dhrubajyoti et al. 2021 ), mGWO (Shubham and Kusum 2020 ), PFA (Yapici and Cetinkaya 2019 ), GCHHO (Song et al. 2021 ), VAGWO (Farshad et al. 2022 ), ExPSO (Khelil et al. 2022 ), TEO (Kaveh and Dadras 2017 ), QS (Zhang et al. 2018 ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC (Duman et al. 2022 ),and TSALSHADE (Abdesslem layeb 2023 ). Table 12 presents the solution results of the LSA and the above comparison algorithms to this problem, and the best optimization results are marked in bold. It can be seen from Table  12 that in terms of Best, Mean, Worst, and Std, the LSA obtains the best solution result. Table 13 indicates that the best result of the LSA for solving this engineering problem is 0.009872, and the values of \(x_{1} ,x_{2}\) and \(x_{3}\) are 0.05, 0.374433, and 8.546567, respectively.

4.2.3 Pressure Vessel Design (PVD)

PVD is another classic constrained optimization problem with four optimization variables and four constraints. Its goal is to minimize the total cost of materials in forming and welding cylindrical vessels. The mathematical model is as follows:

where \(0 \le x_{1} ,x_{2} \le 100,10 \le x_{3} ,x_{4} \le 200\) .

To explore the performance of the LSA in solving PVD, this paper selected some high-performance improved meta-heuristic algorithms recently proposed as comparison algorithms, including BIANCA (Montemurro et al. 2013 ), G-QPSO (Santos Coelho 2010 ), HAIS-GA (Coello and Cortés 2004 ), CB-ABC (Brajevic 2015 ), NHAIS-GA(Bernardino et al. 2008 ), DEC-PSO (Chun et al. 2013 ), T-Cell (Aragón et al. 2010 ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC(Duman et al. 2022 ),and TSALSHADE (Abdesslem layeb 2023 ). It can be seen from Tables 14 and 15 that the LSA, FDB-AGDE, dFDB-MRFO, AFDB-SFS, and FDB- TLABC achieve the best optimization result, and the objective function value is 5885.333, which is obviously better than those of other comparative algorithms.

4.2.4 Three-bar Truss Design (TTD)

The TTD problem is another classic minimization problem in engineering design, and its structure can be found in the reference (Ghasemi et al.  2022 ). It has 3 variants and 4 inequality constraints, and its goal is to minimize the weight of the three trusses.

where \(0 \le x_{1} \le 1,0 \le x_{2} \le 1,\) \(l = 100cm,P = kN/cm^{2} ,\sigma = 2kN/cm^{2}\) .

Table 16 shows the statistical results of each algorithm to solve the TTD problem, where the best results are highlighted in bold. According to the statistical results, LSA, FDB-AGDE, dFDB-MRFO, AFDB-SFS and FDB-TLABC obtained the best optimal value (263.8523) among all the 15 comparison algorithms.

4.2.5 Cantilever Beam Design (CBD)

The CBD is a civil engineering structural design problem consisting of 5 hollow elements, each of which has an equal thickness. Its objective function is to minimize the weight of the cantilever beam. The mathematical model of this problem is represented below:

where \(b = (b_{1} ,b_{2} ,b_{3} ,b_{4} ,b_{5} ) = (67,37,19,7,1)\) \(0.01 \le x_{i} \le 100,i = 1,...,5\) .

To test the ability of the proposed LSA to solve this problem, this paper selected 5 algorithms, STOA (Dhiman and Kaur 2019 ), TQA (Chen et al. 2022 ), GCA_I (Kumar et al. 2020 ), GCA_II (Kumar et al. 2020 ), SMA (Li et al. 2020c ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC (Duman et al. 2022 ),and TSALSHADE (Abdesslem layeb 2023 ) as comparison algorithms. Table 17 presents the statistical results of each algorithm’s performance in addressing this problem, with the best results highlighted in bold. From the statistical outcomes depicted in Table  17 , it is observed that LSA, FDB-AGDE, dFDB-MRFO, AFDB-SFS, and FDB-TLABC exhibit the most favorable optimization outcomes. In other words, these algorithms achieved the best optimal values among all the evaluated approaches.

4.2.6 Car Side Impact Design (CSID)

The goal of CSID is to minimize weight, which is related to 11 variables and 10 inequality constraints. Its detailed description can be found in the reference (Huang et al. 2015 ). Meanwhile, EJAYA (Zhang et al. 2021b ), TLCS (Huang et al. 2015 ), AOSMA (Naik et al. 2021 ), WOAGWO (Mohammed and Rashid 2020 ), PGJAYA (Yu et al. 2019 ), ERao-1 (Jian and Zhu 2021 ), CLJAYA (Zhang and Jin 2022 ), FDB-AGDE (Guvenc et al. 2021 ), dFDB-MRFO (Kahraman et al. 2022 ), AFDB-SFS (Duman et al. 2023 ), FDB-TLABC (Duman et al. 2022 ),and TSALSHADE (Abdesslem layeb 2023 ) were selected to test their performance in solving this problem. The mathematical model of this problem is as follows.

where \(0.5 \le x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} \le 1.5\) , \(0.192 \le x_{8} ,x_{9} \le 0.345\) , \(- 30 \le x_{10} ,x_{10} \le 30\) .

The statistical results of LSA on this problem are presented in Table  18 . It can be seen from this table that the optimal solution of LSA is 22.842, which is consistent with the same results by FDB-AGDE, dFDB-MRFO, AFDB-SFS, and FDB-TLABC algorithms. The values corresponding to each variable are 0.5, 1.116, 0.5, 1.302, 0.5, 1.5, 0.5, 0.964338, 1.000, -19.577, and 3.73E-07. These results indicate that the LSA algorithm has strong competitiveness.

4.3 Application to real-world optimization of feature selection

In this subsection, the proposed LSA is applied to the feature selection problem to verify the performance of the algorithm in solving optimization problems in this domain. Feature selection refers to selecting a feature subset with good distinguishing characteristics from a feature set according to a target. Therefore, feature selection requires a specific feature evaluation function. The K-nearest neighbor (KNN) algorithm is a classification technique based on supervised machine learning. Because of its characteristics of easy implementation and fast operation, it is often selected as a wrapper-based feature selection method. The fitness function used in feature selection problems has two main goals: a small number of features and the minimum classification error. The most ideal solution to the feature selection problem is to achieve the minimum error by selecting the fewest features. In this paper, the following objective function is adopted to calculate the objective function:

where, \(acc\) represents the accuracy of KNN classification, \(N\) represents the total number of features, and \(N_{i}\) represents the number of features selected by the i-th candidate solution. \(\alpha \in [0,1]\) denotes the weight, \(\beta = 1 - \alpha\) . According to the literature (Xu et al. 2023 ), the values of \(\alpha\) and \(\beta\) are set to 0.99 and 0.01, respectively.

In the above tests, it is shown that the LSA is suitable for solving continuous optimization problems. To this end, it is necessary to convert the value of the search individual in the algorithm into a discrete value of 0 or 1. Denoting the j-th dimension value of the i th individual is \(x_{i,j}\) , the conversion is shown below:

where, \(rand \in (0,1)\) is a random number.

To verify the effect of the LSA in feature selection, this paper experimented with the LSA on 15 UCI public datasets. These 15 test cases have different sample numbers (from 72 to 846), and different feature numbers (from 8 to 30). Table 19 presents the basic information of these 15 datasets. The original dataset can be downloaded from the UCI machine learning website http://archive.ics.uci.edu/ml/index.php .

Five algorithms were selected for comparison, including BGWO (Emary et al. 2016 ), CCSA (Zamani et al. 2019 ), SCA (Mirjalili 2016 ), SSA (Xue and Shen 2020 ), and WOA (Hayyolalam and Kazem 2020 ). The population size was set to 30, each test case was run 20 times independently, and the average value was taken as the statistical value. All other parameters were kept the same as their corresponding references. Tables 20 , 21 , and 22 present the feature selection results of each algorithm on the UCI dataset.

Table 20 shows the average fitness value of each algorithm, and the optimal result is shown in bold. It can be seen from Table  20 that except for the four data sets of Breast, Fertility, WDBC, and Vehicle, LSA obtained the best average fitness value among all the other 11 feature selection algorithms. On these 4 datasets, the performance of the LSA is second only to BGWO. Although the feature selection capabilities of CCSA, SCA, SSA, and WOA are very powerful, their results are still significantly worse than those of the LSA. The rankings of these six algorithms in terms of fitness value are LSA, BGWO, WOA, SCA, CCSA, and SSA

Table 21 shows the classification error rate of the LSA and other comparison algorithms on each dataset. It can be seen from this table that the classification error rate of the LSA on Ceramic and Audit-2 is 0 datasets. In the 15 test datasets, LSA ranked first on 12 and ranked second on the other 3 datasets. This proves the absolute superiority of the LSA algorithm. The features of all algorithms on the BreastTissue and Vehicle datasets are difficult to distinguish because the error rate of all algorithms exceeds 25%.

Table 22 presents the average number of features selected by each algorithm. It was found that no algorithm obtained an absolute advantage in the number of features. The main reason is that the weight of the selected feature number in the fitness function is relatively small. As a result, although some algorithms select a few features, their classification accuracy is low. The above analysis indicates that the LSA has strong competitiveness in feature selection.

Figure  19 shows the convergence curves of each algorithm when solving the feature selection problem. It can be seen from Fig.  19 that the LSA has high convergence accuracy and speed when solving such problems.

The convergence curves of all algorithms for the feature section on 9 UCI datasets

5 Conclusion

This paper introduces a novel learning search algorithm (LSA) designed to efficiently and accurately address optimization problems. In the global expansion stage, the algorithm leverages historical knowledge and up-to-date community information to guide the search direction, thereby enhancing its global search capability. In the local development phase, the algorithm employs the teaching behavior and direction of the role model within the population to enhance the learning capability of the entire population. By dynamically adapting the control factor, the algorithm strikes a balance between exploration and exploitation, thereby avoiding local optima and improving convergence speed. Experimental results vividly demonstrate the LSA’s search process for the optimal solution. Initially, 40 CEC 2014 and CEC 2020 benchmark functions are subjected to comparative testing using well-known original algorithms and recently proposed high-performing improved algorithms. Statistical analysis and the Wilcoxon signed rank test substantiate the LSA’s commendable performance and robust competitiveness vis-à-vis other meta-heuristic algorithms. Furthermore, six subsequent engineering design experiments underscore the LSA’s efficacy in solving real-world engineering applications with constraints. Finally, the LSA is used to solve the feature selection problem, and the experimental results on 15 UCI datasets further verify that the proposed algorithm performs significantly better than other methods in terms of classification accuracy and fitness value.

In this study, despite utilizing the LSA algorithm for solving continuous single-objective optimization problems, real-world constrained optimization problems, and real-world optimization of feature selection, limited research has been conducted on solving multi-objective problems. Many practical decision-making problems involve multiple criteria. For example, resource scheduling problems in cloud computing encompass objectives such as minimizing completion time and cost, and maximizing profit. Therefore, in the near future, we intend to further develop and enhance the LSA algorithm to tackle multi-objective optimization problems. Additionally, we aim to incorporate discretization methods into the LSA algorithm to enable it to handle discrete optimization problems, such as resource scheduling problems.

In our future work, we can employ adaptive mechanisms to adjust the parameters and operations of algorithms, enabling them to automatically adapt and improve their performance to different problems. Additionally, we can combine or cooperate metaheuristic algorithms with other optimization algorithms, machine learning methods, etc., to enhance the performance and adaptability of the algorithms. Moreover, LSA can be expanded to solve different optimization problems in various domains, such as neural networks, gene feature selection, shop floor scheduling, big data applications, and more.

Data availability

Data is provided within the manuscript.

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This study was jointly supported by the National Natural Science Foundation of China (Grant Number 62341210), Science and Technology Development Plan for Baise City (Grant Number 20233654).

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