one sample t hypothesis test calculator

t-test Calculator

Table of contents

Welcome to our t-test calculator! Here you can not only easily perform one-sample t-tests , but also two-sample t-tests , as well as paired t-tests .

Do you prefer to find the p-value from t-test, or would you rather find the t-test critical values? Well, this t-test calculator can do both! 😊

What does a t-test tell you? Take a look at the text below, where we explain what actually gets tested when various types of t-tests are performed. Also, we explain when to use t-tests (in particular, whether to use the z-test vs. t-test) and what assumptions your data should satisfy for the results of a t-test to be valid. If you've ever wanted to know how to do a t-test by hand, we provide the necessary t-test formula, as well as tell you how to determine the number of degrees of freedom in a t-test.

When to use a t-test?

A t-test is one of the most popular statistical tests for location , i.e., it deals with the population(s) mean value(s).

There are different types of t-tests that you can perform:

• A one-sample t-test;
• A two-sample t-test; and
• A paired t-test.

In the next section , we explain when to use which. Remember that a t-test can only be used for one or two groups . If you need to compare three (or more) means, use the analysis of variance ( ANOVA ) method.

The t-test is a parametric test, meaning that your data has to fulfill some assumptions :

• The data points are independent; AND
• The data, at least approximately, follow a normal distribution .

If your sample doesn't fit these assumptions, you can resort to nonparametric alternatives. Visit our Mann–Whitney U test calculator or the Wilcoxon rank-sum test calculator to learn more. Other possibilities include the Wilcoxon signed-rank test or the sign test.

Which t-test?

Your choice of t-test depends on whether you are studying one group or two groups:

One sample t-test

Choose the one-sample t-test to check if the mean of a population is equal to some pre-set hypothesized value .

The average volume of a drink sold in 0.33 l cans — is it really equal to 330 ml?

The average weight of people from a specific city — is it different from the national average?

Two-sample t-test

Choose the two-sample t-test to check if the difference between the means of two populations is equal to some pre-determined value when the two samples have been chosen independently of each other.

In particular, you can use this test to check whether the two groups are different from one another .

The average difference in weight gain in two groups of people: one group was on a high-carb diet and the other on a high-fat diet.

The average difference in the results of a math test from students at two different universities.

This test is sometimes referred to as an independent samples t-test , or an unpaired samples t-test .

Paired t-test

A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment.

In particular, you can use this test to check whether, on average, the treatment has had any effect on the population .

The change in student test performance before and after taking a course.

The change in blood pressure in patients before and after administering some drug.

How to do a t-test?

So, you've decided which t-test to perform. These next steps will tell you how to calculate the p-value from t-test or its critical values, and then which decision to make about the null hypothesis.

Decide on the alternative hypothesis :

Use a two-tailed t-test if you only care whether the population's mean (or, in the case of two populations, the difference between the populations' means) agrees or disagrees with the pre-set value.

Use a one-tailed t-test if you want to test whether this mean (or difference in means) is greater/less than the pre-set value.

Compute your T-score value :

Formulas for the test statistic in t-tests include the sample size , as well as its mean and standard deviation . The exact formula depends on the t-test type — check the sections dedicated to each particular test for more details.

Determine the degrees of freedom for the t-test:

The degrees of freedom are the number of observations in a sample that are free to vary as we estimate statistical parameters. In the simplest case, the number of degrees of freedom equals your sample size minus the number of parameters you need to estimate . Again, the exact formula depends on the t-test you want to perform — check the sections below for details.

The degrees of freedom are essential, as they determine the distribution followed by your T-score (under the null hypothesis). If there are d degrees of freedom, then the distribution of the test statistics is the t-Student distribution with d degrees of freedom . This distribution has a shape similar to N(0,1) (bell-shaped and symmetric) but has heavier tails . If the number of degrees of freedom is large (>30), which generically happens for large samples, the t-Student distribution is practically indistinguishable from N(0,1).

💡 The t-Student distribution owes its name to William Sealy Gosset, who, in 1908, published his paper on the t-test under the pseudonym "Student". Gosset worked at the famous Guinness Brewery in Dublin, Ireland, and devised the t-test as an economical way to monitor the quality of beer. Cheers! 🍺🍺🍺

p-value from t-test

Recall that the p-value is the probability (calculated under the assumption that the null hypothesis is true) that the test statistic will produce values at least as extreme as the T-score produced for your sample . As probabilities correspond to areas under the density function, p-value from t-test can be nicely illustrated with the help of the following pictures:

The following formulae say how to calculate p-value from t-test. By cdf t,d we denote the cumulative distribution function of the t-Student distribution with d degrees of freedom:

p-value from left-tailed t-test:

p-value = cdf t,d (t score )

p-value from right-tailed t-test:

p-value = 1 − cdf t,d (t score )

p-value from two-tailed t-test:

p-value = 2 × cdf t,d (−|t score |)

or, equivalently: p-value = 2 − 2 × cdf t,d (|t score |)

However, the cdf of the t-distribution is given by a somewhat complicated formula. To find the p-value by hand, you would need to resort to statistical tables, where approximate cdf values are collected, or to specialized statistical software. Fortunately, our t-test calculator determines the p-value from t-test for you in the blink of an eye!

t-test critical values

Recall, that in the critical values approach to hypothesis testing, you need to set a significance level, α, before computing the critical values , which in turn give rise to critical regions (a.k.a. rejection regions).

Formulas for critical values employ the quantile function of t-distribution, i.e., the inverse of the cdf :

Critical value for left-tailed t-test: cdf t,d -1 (α)

critical region:

(-∞, cdf t,d -1 (α)]

Critical value for right-tailed t-test: cdf t,d -1 (1-α)

[cdf t,d -1 (1-α), ∞)

Critical values for two-tailed t-test: ±cdf t,d -1 (1-α/2)

(-∞, -cdf t,d -1 (1-α/2)] ∪ [cdf t,d -1 (1-α/2), ∞)

To decide the fate of the null hypothesis, just check if your T-score lies within the critical region:

If your T-score belongs to the critical region , reject the null hypothesis and accept the alternative hypothesis.

If your T-score is outside the critical region , then you don't have enough evidence to reject the null hypothesis.

How to use our t-test calculator

Choose the type of t-test you wish to perform:

A one-sample t-test (to test the mean of a single group against a hypothesized mean);

A two-sample t-test (to compare the means for two groups); or

A paired t-test (to check how the mean from the same group changes after some intervention).

Two-tailed;

Left-tailed; or

Right-tailed.

This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing!

Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and degrees of freedom for you .

Once all the parameters are present, the p-value, or critical region, will immediately appear underneath the t-test calculator, along with an interpretation!

One-sample t-test

The null hypothesis is that the population mean is equal to some value μ 0 \mu_0 μ 0 ​ .

The alternative hypothesis is that the population mean is:

• different from μ 0 \mu_0 μ 0 ​ ;
• smaller than μ 0 \mu_0 μ 0 ​ ; or
• greater than μ 0 \mu_0 μ 0 ​ .

One-sample t-test formula :

• μ 0 \mu_0 μ 0 ​ — Mean postulated in the null hypothesis;
• n n n — Sample size;
• x ˉ \bar{x} x ˉ — Sample mean; and
• s s s — Sample standard deviation.

Number of degrees of freedom in t-test (one-sample) = n − 1 n-1 n − 1 .

The null hypothesis is that the actual difference between these groups' means, μ 1 \mu_1 μ 1 ​ , and μ 2 \mu_2 μ 2 ​ , is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the difference μ 1 − μ 2 \mu_1 - \mu_2 μ 1 ​ − μ 2 ​ is:

• Different from Δ \Delta Δ ;
• Smaller than Δ \Delta Δ ; or
• Greater than Δ \Delta Δ .

In particular, if this pre-determined difference is zero ( Δ = 0 \Delta = 0 Δ = 0 ):

The null hypothesis is that the population means are equal.

The alternate hypothesis is that the population means are:

• μ 1 \mu_1 μ 1 ​ and μ 2 \mu_2 μ 2 ​ are different from one another;
• μ 1 \mu_1 μ 1 ​ is smaller than μ 2 \mu_2 μ 2 ​ ; and
• μ 1 \mu_1 μ 1 ​ is greater than μ 2 \mu_2 μ 2 ​ .

Formally, to perform a t-test, we should additionally assume that the variances of the two populations are equal (this assumption is called the homogeneity of variance ).

There is a version of a t-test that can be applied without the assumption of homogeneity of variance: it is called a Welch's t-test . For your convenience, we describe both versions.

Two-sample t-test if variances are equal

Use this test if you know that the two populations' variances are the same (or very similar).

Two-sample t-test formula (with equal variances) :

where s p s_p s p ​ is the so-called pooled standard deviation , which we compute as:

• Δ \Delta Δ — Mean difference postulated in the null hypothesis;
• n 1 n_1 n 1 ​ — First sample size;
• x ˉ 1 \bar{x}_1 x ˉ 1 ​ — Mean for the first sample;
• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• n 2 n_2 n 2 ​ — Second sample size;
• x ˉ 2 \bar{x}_2 x ˉ 2 ​ — Mean for the second sample; and
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

Number of degrees of freedom in t-test (two samples, equal variances) = n 1 + n 2 − 2 n_1 + n_2 - 2 n 1 ​ + n 2 ​ − 2 .

Two-sample t-test if variances are unequal (Welch's t-test)

Use this test if the variances of your populations are different.

Two-sample Welch's t-test formula if variances are unequal:

• s 1 s_1 s 1 ​ — Standard deviation in the first sample;
• s 2 s_2 s 2 ​ — Standard deviation in the second sample.

The number of degrees of freedom in a Welch's t-test (two-sample t-test with unequal variances) is very difficult to count. We can approximate it with the help of the following Satterthwaite formula :

Alternatively, you can take the smaller of n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 as a conservative estimate for the number of degrees of freedom.

🔎 The Satterthwaite formula for the degrees of freedom can be rewritten as a scaled weighted harmonic mean of the degrees of freedom of the respective samples: n 1 − 1 n_1 - 1 n 1 ​ − 1 and n 2 − 1 n_2 - 1 n 2 ​ − 1 , and the weights are proportional to the standard deviations of the corresponding samples.

As we commonly perform a paired t-test when we have data about the same subjects measured twice (before and after some treatment), let us adopt the convention of referring to the samples as the pre-group and post-group.

The null hypothesis is that the true difference between the means of pre- and post-populations is equal to some pre-set value, Δ \Delta Δ .

The alternative hypothesis is that the actual difference between these means is:

Typically, this pre-determined difference is zero. We can then reformulate the hypotheses as follows:

The null hypothesis is that the pre- and post-means are the same, i.e., the treatment has no impact on the population .

The alternative hypothesis:

• The pre- and post-means are different from one another (treatment has some effect);
• The pre-mean is smaller than the post-mean (treatment increases the result); or
• The pre-mean is greater than the post-mean (treatment decreases the result).

Paired t-test formula

In fact, a paired t-test is technically the same as a one-sample t-test! Let us see why it is so. Let x 1 , . . . , x n x_1, ... , x_n x 1 ​ , ... , x n ​ be the pre observations and y 1 , . . . , y n y_1, ... , y_n y 1 ​ , ... , y n ​ the respective post observations. That is, x i , y i x_i, y_i x i ​ , y i ​ are the before and after measurements of the i -th subject.

For each subject, compute the difference, d i : = x i − y i d_i := x_i - y_i d i ​ := x i ​ − y i ​ . All that happens next is just a one-sample t-test performed on the sample of differences d 1 , . . . , d n d_1, ... , d_n d 1 ​ , ... , d n ​ . Take a look at the formula for the T-score :

Δ \Delta Δ — Mean difference postulated in the null hypothesis;

n n n — Size of the sample of differences, i.e., the number of pairs;

x ˉ \bar{x} x ˉ — Mean of the sample of differences; and

s s s  — Standard deviation of the sample of differences.

Number of degrees of freedom in t-test (paired): n − 1 n - 1 n − 1

t-test vs Z-test

We use a Z-test when we want to test the population mean of a normally distributed dataset, which has a known population variance . If the number of degrees of freedom is large, then the t-Student distribution is very close to N(0,1).

Hence, if there are many data points (at least 30), you may swap a t-test for a Z-test, and the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test because, in such cases, the t-Student distribution differs significantly from the N(0,1)!

🙋 Have you concluded you need to perform the z-test? Head straight to our z-test calculator !

What is a t-test?

A t-test is a widely used statistical test that analyzes the means of one or two groups of data. For instance, a t-test is performed on medical data to determine whether a new drug really helps.

What are different types of t-tests?

Different types of t-tests are:

• One-sample t-test;
• Two-sample t-test; and
• Paired t-test.

How to find the t value in a one sample t-test?

To find the t-value:

• Subtract the null hypothesis mean from the sample mean value.
• Divide the difference by the standard deviation of the sample.
• Multiply the resultant with the square root of the sample size.

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ol{padding-top:0px;}.css-4okk7a ul:not(:first-child),.css-4okk7a ol:not(:first-child){padding-top:4px;} Test setup

Choose test type

t-test for the population mean, μ, based on one independent sample . Null hypothesis H 0 : μ = μ 0  

Alternative hypothesis H 1

Test details

Significance level α

The probability that we reject a true H 0 (type I error).

Degrees of freedom

Calculated as sample size minus one.

Test results

One Sample T Test Calculator

Enter sample data, reporting results in apa style, one sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators.

Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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Free statistics calculators designed for data scientists. This One Sample t test Calculator:

• Compares Sample to Expected Mean
• Assesses if significant difference
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Using The One Sample t test Calculator

For the details about designing your test, read the guidance below. To use the calculator, enter the data from your sample as a string of numbers, separated by commas. Adjust the calculator's settings (expected population mean, significance level, one or two tailed test) to match the test goals. Hit calculate. It will compute the t-statistic, p-value, and evaluate if we should accept or reject the proposed hypothesis.

For easy entry, you can copy and paste your data into the entry box from Excel. You can save your data for use with this calculator and other calculators on this site. Just hit the "save data" button. It will save the data in your browser (not on our server, it remains private). Saved data sets will appear on the list of saved datasets below the data entry panel. To retrieve it, click the "load data" button next to it.

One Sample t test Calculator

• Statistics and Histogram Graph
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Histogram of Sample

Test results, observations.

Click To Clear; enter values seperated by commas or new lines. Cut & Paste from Excel also works.

Significance Level

One or two tailed.

Can be comma separated or one line per data point; you can also cut and paste from Excel.

Saved Datasets - Click to Restore

Saved in your browser; you can retrieve these and use them in other calculators on this site.

Sharing Results of The One Sample t-test Calculator

Need to pass an answer to a friend? It's easy to link and share the results of this calculator. Hit calculate - then simply cut and paste the url after hitting calculate - it will retain the values you enter so you can share them via email or social media.

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Interpreting One Sample t test Results

This calculator is designed to evaluate a statement about the mean of a population using a sample drawn from that population. We refer to this statement as the null hypothesis, a claim we would accept in the absence of other evidence. This occurs by accepting the alternate hypothesis, which should be a mutually exclusive claim. For example, in quality control, we may test a batch of product to ensure it meets a particular standard about the strength of the product. Our null hypothesis would be that the strength rating is at least 150 lbs; our test will confirm this (accept the null hypothesis) or indicate we should accept the alternative hypothesis (strength under 150 lbs, reject).

One of the parameters in the calculator asks you to select if you want to run a one sided or two-sided test. A one sided test can be used to test if the sample mean is significantly below the expected mean for the population. The example above was a one-sample test. A two sided test looks for any significant deviation (up or down) relative to the null hypothesis. The two sided test is best when screening for differences, the one side test is useful if checking for a particular defect.

Mathematically, the t-statistic is a composite of several basic metrics from the descriptive statistics panel. We compare the sample mean with the expected value and compare the difference with the sample standard deviation, adjusted for sample size. The sample size is also used to calculate the degrees of freedom for the statistical distribution. The t-statistic is converted into a probability value based on Student's t-distribution, which is used to make the final assessment about the null hypothesis.

It is critical to remember some fundamental assumptions about the underlying population and sample process, particularly if you regularly sample. Increasing the sample size will inevitably make any result appear more significant, through increasing the degrees of freedom reflected in the statistic. This can be problematic if subtle factors in the underlying population change in the process (shift changes, time of day, operating conditions). It often makes sense to split your experiment into parts and seek to replicate results across different periods and operators to ensure your determination is accurate.

Descriptive Statistics

Hypothesis test, one sample t-test calculator.

To calculate a one-sample t test online, simply select a metric variable and enter the test value, then a one-sample t test will be calculated automatically.

One sample t-Test

A one-sample t-test is a statistical hypothesis test that is used to determine if the mean of a single sample is significantly different from a known or hypothesized population mean. It is a commonly used test when you want to compare the mean of a sample to a specific value.

The hypothesis for a one-sample t-test can be stated as follows:

• Null hypothesis (H0): The mean of the sample is equal to the population mean.
• Alternative hypothesis (H1): The mean of the sample is significantly different from the population mean.

The test calculates the t-value, which measures the difference between the sample mean and the population mean in terms of standard error units. It also calculates the p-value, which indicates the probability of obtaining the observed difference or a more extreme difference if the null hypothesis is true.

To perform a one-sample t-test, you need the following information:

• Sample data: The data collected from a single sample, usually a numerical variable.
• Population mean: The known or hypothesized mean of the population you want to compare the sample to.

If the p-value is below a predetermined significance level (commonly 0.05), the null hypothesis is rejected, suggesting that there is a significant difference between the sample mean and the population mean. If the p-value is above the significance level, the null hypothesis is not rejected, indicating that there is insufficient evidence to conclude a significant difference.

One-Sample t-Test Calculator

Compute a complete one-sample t-test, given the sample size, the observed the sample mean, the hypothesized mean, and the sample standard deviation. The calculator computes the t-value, the degrees of freedom, the critical t-value and p-value for a one-tailed (directional) hypothesis, and the critical t-value and p-value for a two-tailed (non-directional) hypothesis. Conducting one-sample t-tests is very common in a wide variety of analytics studies. Please provide the necessary values, and then click 'Calculate'.

Calculator: One-Sample t-Test

One-Sample t-Test Calculator

This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability values associated with the test. Please enter the necessary parameter values, and then click 'Calculate'.

For optimal use, please visit DATAtab on your desktop PC!

Metric Variables:

Ordinal variables:, nominal variables:, t-test calculator.

You want to calculate a t-test? It's easy with DATAtab, just copy your data into the table above and select your variables for which you want to calculate the t-test! DATAtab will automatically use the appropriate test and interpret your results.

If you want to calculate a t-test with your own data online, empty the upper table (click on Empty Table), copy your own data into it and make sure that the variable name is in the first row. Afterwards the variables are displayed below the table. Now click on the variables you want to evaluate. After selecting your variables, the t-test calculator will suggest which t-test you should use. You can choose from the following options:

• Simple t-Test
• t-Test for paired samples
• t-Test for independent Samples

Calculate t-Test

In the results section of the online t-test calculator you will find the mean and standard deviation of the samples and of course the calculated t-value and p-value. Which t-test you have to use is determined by the type of your sample or samples and how they are related to each other.

Simple t-Test Calculator

You want to test whether the mean of a sample is equal to that of the population? Then select a metric variable and specify the test value.

Independent t-Test Calculator

You want to compare the means of two independent groups? Then select two metric variables or one metric variable and one nominal variable with two values.

Dependent t-Test Calculator

You want to compare two groups where the measured values belong together in pairs? Then select two metric variables.

p-value Calculator

Of course you also get the p-value calculated and displayed in a table.

You can specify the significance level right at the beginning of the calculation. If you want to calculate a one-sided t-test, you can either specify this as well or you simply divide your p-value by two at the end.

More information about the theory behind the t-test and detailed examples can be found here:

• One sample t-test
• Paired t-test
• Independent sample t-test

t-Value Calculator

In order to calculate the p-value, the t-value must first be calculated. The p-value is then calculated from the t-value and the degrees of freedom.

A t-test is a type of inferential statistical test that determines if there is a significant difference between the means of two groups. It can be used when the populations are normally distributed and samples are independent and randomly selected. The t-test compares the means of two groups and determines if they are statistically different from each other.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

• Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
• Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
• Enter data for the test, based on the format you chose in Step 1.
• Click Calculate Now and View the results. All options will perform a two-tailed test .

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Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

• Exactly two groups
• Sample is normally distributed
• Independent observations
• Unequal or equal variance?
• Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

• Clear guidance to pick the right t test and detailed results summaries
• Custom, publication quality t test graphics, violin plots, and more
• More t test options, including normality testing as well as nested and multiple t tests
• Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

Enter a value for the null hypothesis. This value should indicate the absence of an effect in your data. Indicate whether your alternative hypothesis involves one-tail or two-tails. If it is a one-tailed test, then you need to indicate whether it is a positive (right tail) test or a negative (left tail) test.

Enter an \(\alpha\) value for the hypothesis test. This is the Type I error rate for your hypothesis test. It also determines the confidence level \(100 \times (1-\alpha)\) for a confidence interval.

Press the Run Test button and a table summarizing the computations and conclusions will appear below.

T-Test Calculator

Compare the means of two samples using a single-sample or two-sample t-test below.

• Single Sample
• Two Sample (Unpaired)

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How to do a t-test, types of t-tests, how to calculate t using a one-sample t-test, how to calculate t using a student’s t-test, how to calculate t using welch’s t-test, find the p-value.

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A t-test calculates how significant the difference between the means of two groups are. The results let you know if those differences could have occurred by chance, or rather, whether the difference is statistically significant.

A t-test uses the test statistic, sometimes called a t-value or t-score, the t-distribution values, and the degrees of freedom to calculate the statistical significance of the difference.

Since a t-test is a parametric test, it relies on assumptions about the process that generated the underlying data. In particular, the likelihood or unlikelihood that the t-test provides for a difference being due to chance depends on the assumption that the data are normally distributed and each data point’s values are independent of one another.

Depending on how plausible those assumptions are, the analysis that follows will be more or less useful. If your data is continuous and comes from a relatively large random sample from some population, the central limit theorem implies that these assumptions will likely be approximately satisfied.

The first part of doing a t-test is determining which type of t-test you need to do.

There are three different types of t-tests:

• one-sample t-test: used to compare the mean of a sample to the known mean of a population
• two-sample t-test: used to compare the mean of two different independent samples
• paired t-test: used to compare the mean of two different samples after an intervention or change

A one-sample t-test, or single-sample test, is used to compare a sample mean to a population mean when the null hypothesis is that the sample mean is equal to the population mean.

Those who first encounter this test often wonder why they would use it, since the population mean is often not known (and the data is often collected to determine the population mean in the first place).

It often does make sense to use a one-sample t-test if you have a particular interest in whether a sample’s mean is different from some reference value that is determined to be substantively important for other reasons.

For example, let’s suppose that 5 micrograms of lead per liter of blood is the maximum safe amount, according to most medical references. Then, you may well consider doing a one-sample t-test to examine whether the average blood lead level of a sample of individuals was above that medically acceptable limit.

One-Sample T-Test Formula

To calculate the t value using a one-sample t-test, use the following formula:

Where: x̄ = sample mean μ = population mean s = sample standard deviation n = sample size

Thus, the test statistic t is equal to the difference between the sample mean x̄ and the population mean μ , divided by the standard error s / √n .

A Student’s t-test is used for test statistics that follow a Student’s t-distribution under the null hypothesis that two populations have equal means.

The name “Student” refers to the pseudonym of the author who first proposed the test in an academic journal, and does not refer to the fact it is one of the most commonly taught tests in statistics courses (although the latter is also true).

The Student’s t-test assumes that the variances of two populations are equal and asks whether their means differ significantly.

This is a type of two-sample test used to compare two sample means, where a large t-value suggests that the samples are very different, and a small t-value suggests that they are similar.

Similar to the one-sample t-test, individuals who first encounter this test may wonder about the plausibility of its assumptions. In particular, you might question how the variances in two samples could possibly be equal if the means are different.

In some contexts (for example, the industrial experiments that motivated Student’s efforts), there might be substantive reasons to assume equal variances. More informally, if you calculate the standard deviations in each sample and sees that they are close, you might proceed to calculate Student’s t-test.

More formally, some analysts would recommend that you initially conduct an F-test to determine whether variances are different, and then proceed to consider the means. But many analysts would also simply not make the equal variances assumption and proceed directly to Welch’s t-test.

Student’s T-Test Formula

The formula for a Student’s t-test is:

Given the formula to calculate the pooled standard deviation s p :

Where: x̄ 1 = first sample mean x̄ 2 = second sample mean n 1 = first sample size n 2 = second sample size s 1 = first sample standard deviation s 2 = second sample standard deviation n 1 + n 2 – 2 = degrees of freedom ν

In a Student’s t-test, the test statistic t is equal to the difference between sample means x̄ 1 and x̄ 2 , divided by the pooled standard deviation s p times the square root of 1 divided by the first sample size n 1 plus 1 divided by the second sample size n 2 .

The pooled standard deviation s p is equal to the first sample size n 1 minus 1 times the first sample standard deviation s 1 plus the second sample size n 2 minus 1 times the second sample standard deviation s 2 , divided by the degrees of freedom, in this case the sum of the sample sizes minus two.

It is called the “pooled” standard deviation because it combines or “pools” the data between both samples to determine the overall variability of the data.

This formula can be broken down into a few simple steps.

Step One: Calculate the Degrees of Freedom

Step two: calculate the pooled standard deviation, step three: calculate the test statistic.

Recall that the Student’s t-test assumes that the variances of two populations are equal. As was mentioned above, this is often a questionable assumption, and ultimately unverifiable.

In this case, you can use Welch’s t-test, which is sometimes also called an unequal variances t-test or an “unpooled” t-test. Like before, the null hypothesis with this test is that two populations have equal means.

Welch’s T-Test Formula

The formula for Welch’s t-test is:

Degrees of Freedom Formula

To find the degrees of freedom when using Welch’s t-test, use the Satterthwaite formula:

The next step is to find the p-value for the test statistic. The p-value is a measure of how “surprising” or “unlikely” some statistic would be given the particular assumptions that the analyst makes.

In the case of these t-tests for differences in means, the p-value is the probability of calculating a t-statistic that is as large or larger than what was actually calculated from the observed data if, in fact, the population means were identical.

More generally, a p-value is used to determine whether to reject the null hypothesis. In formal hypothesis testing, you would specify beforehand the p-value that would lead you to conclude that the two samples came from different populations.

What is the Right P-Value?

These standards differ by field and disciplines a lot, for example, in social and biological sciences, a p-value of 0.05 or smaller (implying 5% or lower chance of observing the data under the null hypothesis) is common, although in some cases 0.1 or 0.01 might be the standard.

In the physical sciences, it is not uncommon to pre-specify a “6 sigma” standard for certain kinds of evidence, which requires an astronomically small p-value.

How to Calculate the P-Value

To calculate the p-value from a t-statistic, use a t-table and locate the degrees of freedom in the leftmost column. Then, locate the desired p-value in the heading row, 0.05 is most commonly used for a 95% confidence level.

Then, find the intersection of the row and column to find the critical value.

Drawing Conclusions Using the P-Value

If the calculated t-value is larger than the critical value, then you can reject the null hypothesis. If it is less than the critical value, then you fail to reject the null hypothesis.

The t-distribution is related to the normal distribution; indeed, it can be thought of as the normal distribution’s “heavy-tailed” cousin. The degrees of freedom in the t-distribution determines how heavy the tails are, with fewer degrees of freedom resulting in greater departures from normality.

As the degrees of freedom increase, it becomes harder and harder to tell the differences between the associated t-distribution and the normal distribution.

Because of this fact, experienced statistical analysts are often able to approximately estimate the p-value of a particular t-statistic through their familiarity with the normal distribution.

A t-statistic of 2 or greater is typically enough to confirm statistical significance in the social and biological contexts.

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T-test for One Population Mean

Instructions: This calculator conducts a t-test for one population mean (\(\sigma\)), with unknown population standard deviation (\(\sigma\)), for which reason the sample standard deviation (s) is used instead. Please select the null and alternative hypotheses, type the hypothesized mean, the significance level, the sample mean, the sample standard deviation, and the sample size, and the results of the t-test will be displayed for you:

How to use this t-test calculator for One Sample

More about the t-test for one mean so you can better interpret the results obtained by this solver: A t-test for one mean is a hypothesis test that attempts to make a claim about the population mean (\(\sigma\)). This t-test, unlike the z-test, does not need to know the population standard deviation \(\sigma\).

How to Conduct a T-test for One Population Mean?

The test has two complementary hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population mean, under the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis. The main properties of a one sample t-test for one population mean are:

• For a t-test for one mean, the sampling distribution used for the t-test statistic (which is the distribution of the test statistic under the assumption that the null hypothesis is true) corresponds to the t-distribution, with n-1 degrees of freedom (instead of being the standard normal distribution, as in the case of a z-test for one mean)
• Depending on our knowledge about the "no effect" situation, the t-test can be two-tailed, left-tailed or right-tailed
• The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
• The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true
• In a hypothesis tests there are two types of errors. Type I error occurs when we reject a true null hypothesis, and the Type II error occurs when we fail to reject a false null hypothesis

How to Compute the t-statistic for one sample?

So, what is the one sample t test formula? In this case, for this t-test formula for the t-statistic is

The null hypothesis is rejected when the t-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) the type of tail (two-tailed, left-tailed or right-tailed) and the number of degrees of freedom \(df = n - 1\)

What happens with the t-test when I have 2 samples

Notice that this is a one sample t test calculator. If instead you need to compare two means, you should use a t-test for independent samples , instead.

In a similar way, you may have two samples but they are paired, matched or repeated, which case the appropriate tool to use is this paired t test calculator , when that is the case.

Decision for a one sample t-test

How do you make a decision on a one-sample t-test? First, you need to know the t-statistic, which we call \(t_{obs}\), and the degrees of freedom df, so that you can compute the p-value.

The process of calculation of the p-value will depend on the type of tails defined. For a two-tailed test, the p-value is computed as \(p = \Pr(|t_{df}| > |t_{obs}|)\). Then, for a left-tailed test, the p-value is computed as \(p = \Pr(t_{df} t_{obs})\).

One sample t-test example

A vendor has records showing that the average customer spends $80 dollars in her store on average, but as of recent, she feels that amount has increased. She collects a random sample of n = 30 customers, and she finds that the mean amount spent on the store was $85.4, with a sample standard deviation of $12.4. Does she have enough evidence to claim that the average spent on her store has increased significantly, at the .05 significance level?

The following information has been provided:

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

(2) Rejection Region

Based on the information provided, the significance level is \(\alpha = 0.05\), and the critical value for a right-tailed test is \(t_c = 1.699\).

The rejection region for this right-tailed test is \(R = \{t: t > 1.699\}\)

(3) Test Statistics

The t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that \(t = 2.385 > t_c = 1.699\), it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is \(p = 0.0119\), and since \(p = 0.0119 < 0.05\), it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is not enough evidence to claim that the population mean \(\mu\) is greater than 80, at the \(\alpha = 0.05\) significance level.

Confidence Interval

The 95% confidence interval is \(80.77 < \mu < 90.03\).

Graphically

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Statistics Made Easy

How to Perform a One Sample t-test on a TI-84 Calculator

A  one sample t-test  is used to test whether or not the mean of a population is equal to some value.

This tutorial explains how to conduct a one sample t-test on a TI-84 calculator.

Example: One Sample t-test on a TI-84 Calculator

Researchers want to know if a certain type of car gets 20 miles per gallon or not. They obtain a random sample of 74 cars and find that the mean is 21.29 mpg while the standard deviation is 5.78 mpg. Use this data to perform a one sample t-test to determine if the true mpg for this type of car is equal to 20 mpg.

Step 1: Select T-Test.

Press  Stat . Scroll over to TESTS. Scroll down to T-Test and press ENTER .

Step 2: Fill in the necessary info.

The calculator will ask for the following information:

• Inpt:  Choose whether you are working with raw data (Data) or summary statistics (Stats). In this case, we will highlight Stats and press  ENTER .
• μ 0 : The mean to be used in the null hypothesis. We will type 20 and press   ENTER .
• x : The sample mean. We will type 21.29 and press   ENTER .
• s x : The sample standard deviation. We will type 5.78 and press  ENTER .
• n : The sample size. We will type 74 and press  ENTER .
• μ :The alternative hypothesis to be used. Since we are performing a two-tailed test, we will highlight  ≠ μ 0  and press  ENTER . This indicates that our alternative hypothesis is μ≠20. The other two options would be used for left-tailed tests (<μ 0 ) and right-tailed tests (>μ 0 ) .

Lastly, highlight Calculate and press  ENTER .

Step 3: Interpret the results.

Our calculator will automatically produce the results of the one-sample t-test:

Here is how to interpret the results:

• μ≠20 : This is the alternative hypothesis for the test.
• t=1.919896124 : This is the t test-statistic. 
• p=0.0587785895 : This is the p-value that corresponds to the test-statistic.
• x =21.59 . This is the sample mean that we entered.
• s x =5.78 . This is the sample standard deviation that we entered.
• n=74 : This is the sample size that we entered.

Because the p-value of the test (0.0587785895) is not less than 0.05, we fail to reject the null hypothesis. This means we do not have sufficient evidence to say that the mean mpg for this type of car is different from 20 mpg.

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

One Reply to “How to Perform a One Sample t-test on a TI-84 Calculator”

I am working on my thesis about migration. I found some data on permits and declarations and I want to perform a T-test. So can I perform a T-test based on data in table using Data as input?

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T-Test Calculator for 2 Independent Means

This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation.

Further Information

A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females).

Requirements

• Two independent samples
• Data should be normally distributed
• The two samples should have the same variance

Null Hypothesis

H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second.

As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of males and females).