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When analyzing data, it's essential to understand its underlying distribution. One common distribution that arises in statistical analysis is the normal distribution . The Jarque-Bera test is a statistical test used to assess whether a dataset follows a normal distribution.

Named after its developers, Carlos Jarque and Anil Barre , the Jarque-Bera test is a parametric test that relies on the assumption that the data is normally distributed. Like other normality tests, the Jarque-Bera test is particularly useful when analyzing large datasets .

The test works by calculating the skewness and kurtosis of the dataset, which are measures of the shape of the distribution. These values are then compared to what would be expected under a normal distribution. If the dataset is significantly different from a normal distribution, the Jarque-Bera test will flag it.

The test statistic for the Jarque-Bera test is based on the difference between the sample skewness and kurtosis and their expected values under a normal distribution.

This test statistic is then compared to a critical value to determine whether the dataset is significantly different from a normal distribution.

The Jarque-Bera test is a powerful tool in data analysis , and understanding its mathematical basis is essential for effective use. In this beginner's guide , we will provide a comprehensive introduction to the Jarque-Bera test, its mathematical basis, and how it works. 

We will also discuss its strengths and limitations and practical applications , including how it can be used in hypothesis testing and assessing normality in datasets.

Whether you're a beginner looking to understand statistical analysis or a data scientist seeking to expand your statistical toolkit, this guide will equip you with the knowledge and skills to use the Jarque-Bera test with statistical accuracy.

Introudction to Jarque-Bera Test

In statistical analysis, understanding the underlying distribution of the data is essential to draw meaningful conclusions and make accurate predictions . The Jarque-Bera test is a statistical test used to determine whether a given dataset follows a normal distribution. 

It was first introduced by Carlos Jarque and Anil Bera in 1980 and has since become a standard method in statistical analysis.

Normality testing is an important aspect of statistical analysis as it allows us to make inferences about the data. The normal distribution is a symmetrical bell-shaped curve where the majority of data is clustered around the mean. 

Many statistical methods assume that the data is normally distributed, so it's essential to check whether this assumption holds true.

Assuming normality when the data is not actually normally distributed can lead to incorrect conclusions and predictions. Normality testing helps us identify whether our data fits a normal distribution or if we need to use different statistical methods to analyze it.

Why is normality testing important in statistical analysis?

The Jarque-Bera test is a powerful tool in determining whether the data fits a normal distribution or not. The test is based on the skewness and kurtosis of the dataset, which are measures of the shape of the distribution. 

Skewness is a measure of the asymmetry of the distribution, while kurtosis measures the peakedness of the distribution. A normal distribution has a skewness of zero and a kurtosis of three.

The Jarque-Bera test works by calculating the deviation of the sample skewness and kurtosis from what would be expected under a normal distribution. If the deviation is too large, then the data is not normally distributed. 

The test statistic, called the Jarque-Bera statistic, is then compared to a critical value to determine whether the dataset is significantly different from a normal distribution.

What is Normality 

Normality refers to the distribution of data that follows a normal distribution. A normal distribution is a bell-shaped curve where the majority of the data is clustered around the mean. 

It is a symmetrical distribution, meaning that the data on both sides of the mean is similar. A normal distribution is also characterized by two parameters, the mean and the standard deviation.

What is a normal distribution?

A normal distribution is a probability distribution that is characterized by its shape, which is bell-shaped and symmetrical around the mean. In a normal distribution, the mean, median, and mode are equal, and the majority of the data falls within one standard deviation of the mean. 

Many real-world phenomena, such as heights and weights of individuals, follow a normal distribution. The normal distribution is an essential concept in statistics as many statistical methods assume that the data is normally distributed.

Why is normality important in statistical analysis?

Normality is important in statistical analysis because many statistical methods, such as the t-test and ANOVA, assume that the data is normally distributed. Assuming normality when the data is not normally distributed can lead to incorrect conclusions and predictions. 

Normality testing is, therefore, necessary to ensure that the data meets the assumptions required for the chosen statistical method.

How is normality assessed?

Several statistical tests, including the Shapiro-Wilk test, the Anderson-Darling test, and the Jarque-Bera test, can be used to evaluate normality. These tests look at the data's distribution and compare it to what a normal distribution would predict. They offer a statistical assessment of the data's departure from normality.

A common technique for determining normality is the Shapiro-Wilk test . Based on the discrepancy between the data's observed distribution and expected normal distribution, a test statistic is calculated. The data is regarded as being normally distributed if the test statistic is below a specific cutoff.

Another statistical test for determining normality is the Anderson-Darling test .  It is similar to the Shapiro-Wilk test but can be more sensitive in detecting deviations from normality in the tails of the distribution.

The Jarque-Bera test is a test that assesses normality based on the skewness and kurtosis of the data. It calculates the difference between the observed skewness and kurtosis and what would be expected under a normal distribution. If the difference is too large, then the data is not normally distributed.

How Jarque-Bera Test Works

The Jarque-Bera test is a statistical test used to determine whether a dataset follows a normal distribution. It is a parametric test that relies on the assumption that the data is normally distributed.

How to Use Jarque-Bera Test

The test works by comparing the skewness and kurtosis of the data to what would be expected under a normal distribution. Skewness measures the degree of asymmetry in the distribution of the data, while kurtosis measures the degree of peakedness of the distribution. A normal distribution has a skewness of zero and a kurtosis of three.

The test statistic, called the Jarque-Bera statistic, is calculated using the sample skewness and kurtosis. The test statistic follows a chi-squared distribution with two degrees of freedom. If the test statistic is greater than the critical value at a given significance level, then the null hypothesis that the data is normally distributed is rejected.

Assumptions of Jarque-Bera Test?

The data must be assumed to be normally distributed in order for the Jarque-Bera test to be valid. As a parametric test, it is presumptive that the data originates from a particular distribution with well-known parameters and shape. As a result, it is inappropriate for data that violates the assumption of normality.

It's also crucial to keep in mind that with small sample sizes, the test might not be reliable. The test becomes more accurate at identifying deviations from normality as sample size rises.

Mathematics of Jarque-Bera Test

The Jarque-Bera Test is a statistical test that checks if a given dataset has the skewness and kurtosis corresponding to a normal distribution.

Skewness measures the asymmetry of the data around the sample mean, while kurtosis measures the tail behavior of the distribution. The Jarque-Bera Test is particularly useful for large sample sizes.

Jarque-Bera Test Formula

Interpretation of the Test Statistic

Under the null hypothesis of the data coming from a normal distribution, ( JB ) will have a chi-squared distribution with two degrees of freedom. Hence, if the computed ( JB ) value is significantly different from the chi-squared distribution, we can reject the null hypothesis. This indicates that our data might not come from a normal distribution.

Typically, a p-value is used to determine the significance of the test. A small p-value (typically ( p < 0.05 )) i ndicates that we can reject the null hypothesis.

To use this in practice, most statistical software or programming languages with statistical libraries (like Python's `scipy`) provide built-in functions to compute the Jarque-Bera Test.

Step by Step Process for Jarque-Bera Test

1. understand your data.

First and foremost, ensure that you have a clear understanding of your dataset. Are there any obvious outliers or data points that need addressing?

2. Compute Sample Mean and Standard Deviation

Calculate the sample mean and standard deviation for your dataset.

3. Calculate Skewness

4. Calculate Kurtosis

5. Compute the Jarque-Bera Test Statistic

6. Determine the Significance

Compare the test statistic against the critical value from the chi-squared distribution with two degrees of freedom. If the test statistic is significantly different, reject the null hypothesis that the data comes from a normal distribution.

7. Draw Conclusions

Based on the test result and p-value, conclude whether the dataset is likely to have come from a normal distribution or not.

Calculating Jarque-Bera Test with Sample Data

Let's say you have a sample data set: `data = [2.1, 2.4, 2.3, 2.9, 2.8, 3.0, 3.2, 3.1, 2.9, 2.7]`.

5. Determine the Significance

For a significance level of 0.05 and 2 degrees of freedom, the critical chi-squared value is 5.991. Our computed is much less than this value, so we fail to reject the null hypothesis.

6. Conclusion

 There is not enough evidence to conclude that the data does not come from a normal distribution.

Remember, while manual computation provides insight into the workings of the Jarque-Bera Test, for practical purposes and especially with large datasets, using a statistical software package or a programming language with appropriate libraries will be more efficient.

Interpreting the Jarque-Bera test Results

Once the Jarque-Bera test is performed, the results must be interpreted to determine whether the dataset follows a normal distribution or not.

The results of the Jarque-Bera test are typically reported in the form of a p-value. The p-value is a measure of the probability of observing a test statistic as extreme as the one calculated, assuming that the null hypothesis is true (i.e., the data is normally distributed). 

A p-value less than the significance level (usually set at 0.05) indicates that the null hypothesis should be rejected and the data does not follow a normal distribution.

A p-value greater than the significance level indicates that the null hypothesis cannot be rejected, and there is no evidence to suggest that the data does not follow a normal distribution.

What do the test results indicate about the dataset's normality?

The data are not normally distributed if the p-value is less than the significance level, and there may be signs of skewness, kurtosis, or both. Alternative statistical techniques that don't rely on the normality assumption may need to be looked into in this situation.

It is typically believed that the data are normally distributed if the p-value is higher than the significance level. It is crucial to remember that the test might not have enough power to find outliers from normality, especially for small sample sizes. 

In order to confirm normality, it is crucial to combine the Jarque-Bera test with visual analysis of the data, such as a histogram or a normal probability plot.

Strengths and Limitations Of Jarque-Bera Test

Like any statistical test, the Jarque-Bera test has both strengths and limitations. Understanding these can help ensure appropriate use of the test and accurate interpretation of its results.

Strengths of the Jarque-Bera Test

One strength of the Jarque-Bera test is that it can test for normality of a dataset without assuming a specific mean or variance. This makes it a useful tool for assessing normality in a wide variety of datasets.

Another strength is that the test can detect non-normality caused by either skewness or kurtosis. This is important because other normality tests may only detect one type of non-normality.

The Jarque-Bera test is also relatively easy to perform using statistical software, making it accessible to a wide range of researchers and analysts.

Limitations of the Jarque-Bera Test

One limitation of the Jarque-Bera test is that it may not have sufficient power to detect deviations from normality, particularly for small sample sizes. This means that the test may not always accurately detect non-normality in a dataset, even when it exists.

Another limitation is that the test assumes independence between observations, and may not be appropriate for datasets with autocorrelation or other forms of dependence.

Finally, it is important to note that a failure to reject the null hypothesis (i.e., the data is normally distributed) using the Jarque-Bera test does not necessarily guarantee that the data is actually normally distributed. It is always important to supplement the test with visual inspection of the data to confirm normality.

Practical Applications

The Jarque-Bera test is a useful tool for assessing normality in statistical analysis, and it can be used in a variety of practical applications.

How is the Jarque-Bera test used in statistical analysis?

The Jarque-Bera test can be used in statistical analysis to determine whether a dataset is normally distributed. This is important because many statistical techniques, such as linear regression , assume that the data is normally distributed. If the data is not normally distributed, the results of the analysis may be invalid.

How can the test be used to assess normality in datasets?

To use the Jarque-Bera test to assess normality in a dataset, researchers typically perform the following steps:

Collect the dataset of interest.

• Use statistical software to calculate the Jarque-Bera test statistic and p-value.

Interpret the results. If the p-value is less than the chosen significance level (usually 0.05), the null hypothesis (that the data is normally distributed) is rejected, indicating non-normality.

• Jarque-Bera statistic: 0.7475697750125799
• p-value: 0.6881249201767494
• The dataset is normally distributed.

In this code, we first generate a sample dataset of 1000 observations from a standard normal distribution. We then use the jarque_bera function from the SciPy library to calculate the Jarque-Bera test statistic and p-value for the dataset.

Finally, we interpret the results by checking whether the p-value is less than the significance level of 0.05. If the p-value is less than 0.05, we conclude that the dataset is not normally distributed. Otherwise, we conclude that the dataset is normally distributed.

How can the test be used in hypothesis testing?

The Jarque-Bera test can also be used in hypothesis testing to determine whether a sample comes from a normal distribution.

In hypothesis testing, the null hypothesis is that the sample comes from a normal distribution, and the alternative hypothesis is that the sample does not come from a normal distribution.

To use the Jarque-Bera test in hypothesis testing, researchers typically perform the following steps:

• Collect a sample of interest.
• Set a significance level (usually 0.05).
• Interpret the results. If the p-value is less than the chosen significance level, the null hypothesis is rejected, indicating that the sample does not come from a normal distribution.
• Jarque-Bera statistic (data1): 1.56011816872522
• p-value (data1): 0.4583789274783522
• Jarque-Bera statistic (data2): 5.474623451308658
• p-value (data2): 0.06474416322853181
• The null hypothesis (data1 comes from a normal distribution) cannot be rejected.
• The null hypothesis (data2 comes from a normal distribution) cannot be rejected.

In this code, we first generate two sample datasets: data1 and data2. We then use the jarque_bera function from the SciPy library to calculate the Jarque-Bera test statistic and p-value for each dataset. 

We print out the results and then perform hypothesis testing by checking whether the p-values are less than the significance level of 0.05. If the p-value is greater than 0.05, we cannot reject the null hypothesis that the data comes from a normal distribution. Otherwise, we reject the null hypothesis.

Note that in this example, data1 is generated from a normal distribution, while data2 is generated from a uniform distribution.

As expected, the Jarque-Bera test indicates that data1 comes from a normal distribution (since the p-value is greater than 0.05), while data2 does not come from a normal distribution (since the p-value is less than 0.05).

The Jarque-Bera test is a valuable tool in statistical analysis for testing the normality of datasets. By assessing the skewness and kurtosis of the dataset, the test provides a statistical measure of the deviation from normality. 

It is important to keep in mind that the test is only as reliable as the assumptions it relies on, and care should be taken to ensure that those assumptions are met before applying the test. 

Despite its limitations, the Jarque-Bera test remains a popular and useful tool for assessing normality and is widely used in a variety of fields, from finance and economics to the natural sciences.

While the test is not without its limitations, it remains an important tool in the statistical toolkit of any data scientist or analyst. By understanding the strengths and weaknesses of the Jarque-Bera test, researchers can make informed decisions about its applicability to their specific research questions and draw more accurate conclusions from their data.

Frequently Asked Questions (FAQs) on Jrque-Bera Test

1. what is the jarque-bera test.

The Jarque-Bera test is a statistical procedure to test if a given dataset has the skewness and kurtosis matching a normal distribution.

2. Why is Testing for Normality Important?

Many statistical techniques assume that data is normally distributed . Testing for normality ensures that the assumptions underlying these methods are valid.

3. How Does the Jarque-Bera Test Work?

The test statistic is based on the difference between the sample skewness and kurtosis, and those of a normal distribution. A significant result indicates the data is not normally distributed.

4. What are Skewness and Kurtosis?

Skewness measures the asymmetry of the data distribution, while kurtosis measures the "tailedness" or the sharpness of the peak of the distribution.

5. How Do I Interpret the Results of the Jarque-Bera Test?

A low p-value (typically ≤ 0.05) rejects the null hypothesis, suggesting the data is not normally distributed. A higher p-value suggests the opposite.

6. Is the Jarque-Bera Test Suitable for Small Sample Sizes?

The Jarque-Bera test is more reliable for larger sample sizes. For small samples, it might not have enough power to detect deviations from normality.

7. How Does Jarque-Bera Compare to the Shapiro-Wilk Test?

Both tests check for normality. While the Shapiro-Wilk test is more appropriate for smaller datasets, the Jarque-Bera test is suitable for larger datasets.

8. Can I Use the Jarque-Bera Test for Time Series Data?

Yes, the Jarque-Bera test can be applied to residuals of time series models to check if they're normally distributed, which is a common assumption in many time series techniques.

9. Are There Limitations to the Jarque-Bera Test?

Like any test, it has its assumptions and conditions under which it's most effective. It's sensitive to sample size and might not detect subtle deviations from normality.

10. When Shouldn't I Use the Jarque-Bera Test?

 If you have a very small sample size or if you suspect that deviations from normality are subtle, other tests or methods might be more appropriate.

11. Does the Jarque-Bera Test Only Work with Continuous Data?

 The test is designed for continuous data as it relies on measures of skewness and kurtosis which are most meaningful for continuous distributions.

12. Is a Visual Inspection Enough to Determine Normality?

 While visual methods like Q-Q plots provide a good preliminary check, statistical tests like Jarque-Bera provide more objective measures of deviation from normality.

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Jarque-Bera test

Description

h = jbtest( x ) returns a test decision for the null hypothesis that the data in vector x comes from a normal distribution with an unknown mean and variance, using the Jarque-Bera test . The alternative hypothesis is that it does not come from such a distribution. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise.

h = jbtest( x , alpha ) returns a test decision for the null hypothesis at the significance level specified by alpha .

h = jbtest( x , alpha , mctol ) returns a test decision based on a p -value computed using a Monte Carlo simulation with a maximum Monte Carlo standard error less than or equal to mctol .

[ h , p ] = jbtest( ___ ) also returns the p -value p of the hypothesis test, using any of the input arguments from the previous syntaxes.

[ h , p , jbstat , critval ] = jbtest( ___ ) also returns the test statistic jbstat and the critical value critval for the test.

collapse all

Test for a Normal Distribution

Load the data set.

Test the null hypothesis that car mileage, in miles per gallon ( MPG ), follows a normal distribution across different makes of cars.

The returned value of h = 1 indicates that jbtest rejects the null hypothesis at the default 5% significance level.

Test the Hypothesis at a Different Significance Level

Test the null hypothesis that car mileage in miles per gallon ( MPG ) follows a normal distribution across different makes of cars at the 1% significance level.

The returned value of h = 1 , and the returned p -value less than α = 0.01 indicate that jbtest rejects the null hypothesis.

Test for a Normal Distribution Using Monte Carlo Simulation

Test the null hypothesis that car mileage, in miles per gallon ( MPG ), follows a normal distribution across different makes of cars. Use a Monte Carlo simulation to obtain an exact p -value.

The returned value of h = 1 indicates that jbtest rejects the null hypothesis at the default 5% significance level. Additionally, the test statistic, jbstat , is larger than the critical value, critval , which indicates rejection of the null hypothesis.

Input Arguments

X — sample data vector.

Sample data for the hypothesis test, specified as a vector. jbtest treats NaN values in x as missing values and ignores them.

Data Types: single | double

alpha — Significance level 0.05 (default) | scalar value in the range (0,1)

Significance level of the hypothesis test, specified as a scalar value in the range (0,1). If alpha is in the range [0.001,0.50], and if the sample size is less than or equal to 2000, jbtest looks up the critical value for the test in a table of precomputed values. To conduct the test at a significance level outside of these specifications, use mctol .

Example: 0.01

mctol — Maximum Monte Carlo standard error nonnegative scalar value

Maximum Monte Carlo standard error for the p -value, p , specified as a nonnegative scalar value. If you specify a value for mctol , jbtest computes a Monte Carlo approximation for p directly, rather than interpolating into a table of precomputed values. jbtest chooses the number of Monte Carlo replications large enough to make the Monte Carlo standard error for p less than mctol .

If you specify a value for mctol , you must also specify a value for alpha . You can specify alpha as [] to use the default value of 0.05.

Example: 0.0001

Output Arguments

H — hypothesis test result 1 | 0.

Hypothesis test result, returned as 1 or 0 .

If h = 1 , this indicates the rejection of the null hypothesis at the alpha significance level.

If h = 0 , this indicates a failure to reject the null hypothesis at the alpha significance level.

p — p -value scalar value in the range (0,1)

p -value of the test, returned as a scalar value in the range (0,1). p is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis. Small values of p cast doubt on the validity of the null hypothesis.

jbtest warns when p is not found within the tabulated range of [0.001,0.50], and returns either the smallest or largest tabulated value. In this case, you can use mctol to compute a more accurate p -value.

jbstat — Test statistic nonnegative scalar value

Test statistic for the Jarque-Bera test, returned as a nonnegative scalar value.

critval — Critical value nonnegative scalar value

Critical value for the Jarque-Bera test at the alpha significance level, returned as a nonnegative scalar value. If alpha is in the range [0.001,0.50], and if the sample size is less than or equal to 2000, jbtest looks up the critical value for the test in a table of precomputed values. If you use mctol , jbtest determines the critical value of the test using a Monte Carlo simulation. The null hypothesis is rejected when jbstat > critval .

Jarque-Bera Test

The Jarque-Bera test is a two-sided goodness-of-fit test suitable when a fully specified null distribution is unknown and its parameters must be estimated.

The test is specifically designed for alternatives in the Pearson system of distributions. The test statistic is

J B = n 6 ( s 2 + ( k − 3 ) 2 4 )   ,

where n is the sample size, s is the sample skewness, and k is the sample kurtosis. For large sample sizes, the test statistic has a chi-square distribution with two degrees of freedom.

Monte Carlo Standard Error

The Monte Carlo standard error is the error due to simulating the p -value.

The Monte Carlo standard error is calculated as

S E = ( p ^ ) ( 1 − p ^ ) mcreps ,

where p ^ is the estimated p -value of the hypothesis test, and mcreps is the number of Monte Carlo replications performed. jbtest chooses the number of Monte Carlo replications, mcreps , large enough to make the Monte Carlo standard error for p ^ less than the value specified for mctol .

Jarque-Bera tests often use the chi-square distribution to estimate critical values for large samples, deferring to the Lilliefors test (see lillietest ) for small samples. jbtest , by contrast, uses a table of critical values computed using Monte Carlo simulation for sample sizes less than 2000 and significance levels from 0.001 to 0.50. Critical values for a test are computed by interpolating into the table, using the analytic chi-square approximation only when extrapolating for larger sample sizes.

[1] Jarque, C. M., and A. K. Bera. “A Test for Normality of Observations and Regression Residuals.” International Statistical Review . Vol. 55, No. 2, 1987, pp. 163–172.

[2] Deb, P., and M. Sefton. “The Distribution of a Lagrange Multiplier Test of Normality.” Economics Letters . Vol. 51, 1996, pp. 123–130. This paper proposed a Monte Carlo simulation for determining the distribution of the test statistic. The results of this function are based on an independent Monte Carlo simulation, not the results in this paper.

Version History

Introduced before R2006a

adtest | kstest | lillietest

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Jarque-Bera

The Jarque-Bera test [ JAR1 ] is a two-sided goodness-of-fit test for Normality suitable when a fully-specified null distribution is unknown and its parameters must be estimated. It is based on the sample skewness and kurtosis and was developed for use in connection with regression analysis. The test statistic is

where n is the sample size, s is the sample skewness, and k is the sample kurtosis. For (very) large sample sizes, the test statistic has a chi-square distribution with two degrees of freedom, but more generally its distribution is obtained via Monte Carlo simulation. The test is, by definition, particularly well suited to evaluating the departure of the sample skewness and kurtosis from that expected under the assumption of a Normal distribution .

[JAR1] Jarque C M, Bera A K (1987) A test for normality of observations and regression residuals. Intl Statistical Review,  55(2), 163–172

Jarque-Bera Test

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References and Further Reading

Jarque CM, Bera AK (1980) Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Econ Lett 6(3):255–259

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Jarque CM, Bera AK (1987) A test for normality of observations and regression residuals. Int Stat Rev 55(2):163–172

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Ord JK (1972) Families of frequency distributions, Griffin’s statistical monographs and courses 30. Griffin, London

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Carlos M. Jarque

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Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

Miodrag Lovric

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Jarque, C.M. (2011). Jarque-Bera Test. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_319

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Published : 02 December 2014

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scipy.stats.jarque_bera #

Perform the Jarque-Bera goodness of fit test on sample data.

The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution.

Note that this test only works for a large enough number of data samples (>2000) as the test statistic asymptotically has a Chi-squared distribution with 2 degrees of freedom.

Observations of a random variable.

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None , the input will be raveled before computing the statistic.

Defines how to handle input NaNs.

propagate : if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

omit : NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

raise : if a NaN is present, a ValueError will be raised.

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

An object with the following attributes:

The test statistic.

The p-value for the hypothesis test.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix . Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False .

Jarque, C. and Bera, A. (1980) “Efficient tests for normality, homoscedasticity and serial independence of regression residuals”, 6 Econometric Letters 255-259.

Shapiro, S. S., & Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4), 591-611.

B. Phipson and G. K. Smyth. “Permutation P-values Should Never Be Zero: Calculating Exact P-values When Permutations Are Randomly Drawn.” Statistical Applications in Genetics and Molecular Biology 9.1 (2010).

Panagiotakos, D. B. (2008). The value of p-value in biomedical research. The open cardiovascular medicine journal, 2, 97.

Suppose we wish to infer from measurements whether the weights of adult human males in a medical study are not normally distributed [2] . The weights (lbs) are recorded in the array x below.

The Jarque-Bera test begins by computing a statistic based on the sample skewness and kurtosis.

Because the normal distribution has zero skewness and zero (“excess” or “Fisher”) kurtosis, the value of this statistic tends to be low for samples drawn from a normal distribution.

The test is performed by comparing the observed value of the statistic against the null distribution: the distribution of statistic values derived under the null hypothesis that the weights were drawn from a normal distribution. For the Jarque-Bera test, the null distribution for very large samples is the chi-squared distribution with two degrees of freedom.

The comparison is quantified by the p-value: the proportion of values in the null distribution greater than or equal to the observed value of the statistic.

If the p-value is “small” - that is, if there is a low probability of sampling data from a normally distributed population that produces such an extreme value of the statistic - this may be taken as evidence against the null hypothesis in favor of the alternative: the weights were not drawn from a normal distribution. Note that:

The inverse is not true; that is, the test is not used to provide evidence for the null hypothesis.

The threshold for values that will be considered “small” is a choice that should be made before the data is analyzed [3] with consideration of the risks of both false positives (incorrectly rejecting the null hypothesis) and false negatives (failure to reject a false null hypothesis).

Note that the chi-squared distribution provides an asymptotic approximation of the null distribution; it is only accurate for samples with many observations. For small samples like ours, scipy.stats.monte_carlo_test may provide a more accurate, albeit stochastic, approximation of the exact p-value.

Furthermore, despite their stochastic nature, p-values computed in this way can be used to exactly control the rate of false rejections of the null hypothesis [4] .

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Goodness of fit test- jarque-bera test in r.

Posted on August 16, 2021 by finnstats in R bloggers | 0 Comments

Goodness of fit test, The Jarque-Bera test is a goodness-of-fit test that measures if sample data has skewness and kurtosis that are similar to a normal distribution.

The Jarque-Bera test statistic is always positive, and if it is not close to zero, it shows that the sample data do not have a normal distribution.

Goodness of Fit Test

The test statistic Jarque-Bera Test is defined as:

JB  =[(n-k+1) / 6] * [S 2 + (0.25*(C-3) 2 )]

Under the null hypothesis of normality, Jarque-Bera Test(JB) ~ X 2 (2)

where n denotes the number of observations in the sample, k denotes the number of regressors (k=1 if not used in a regression), S denotes sample skewness, and C denotes sample kurtosis.

This tutorial describes how to execute a Jarque-Bera test in R.

Kruskal Wallis test in R-One-way ANOVA Alternative »

Jarque-Bera test in R

First, need to call tseries library in R.

library(“tseries”)

Let’s generate some random data and make use of the set.seed function for reproducibility.

Case Study 1:-

set.seed(123)

The above function generates normally distributed random variables and we can expect the result is not significant. Let’s verify the same based on the Jarque-Bera test

LSTM Network in R » Recurrent Neural network »

Analyze data based on Jarque-Bera test

The above function generates the following outputs.

This indicates that the test statistic is 0.16908, with a p-value of 0.9189. We would not be able to reject the null hypothesis that the data is normally distributed in this scenario.

Principal component analysis (PCA) in R »

Case Study 2:-

Now instead of rnorm make use of runif function and check the Jarque-Bera test in R.

Execute Jarque-Bera test in R

It generates the following outputs.

This indicates that the test statistic is 6.1759, with a p-value of 0.0456. We would reject the null hypothesis that the data is normally distributed in this circumstance.

We have enough evidence to conclude that the data in this scenario is not normally distributed.

KNN Algorithm Machine Learning » Classification & Regression »

The post Goodness of Fit Test- Jarque-Bera Test in R appeared first on finnstats .

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Jarque-Bera test for goodness-of-fit to a normal distribution

• H = jbtest(X) H = jbtest(X,alpha) [H,P,JBSTAT,CV] = jbtest(X,alpha)

H = jbtest(X) performs the Jarque-Bera test on the input data vector X and returns H , the result of the hypothesis test. The result is H=1 if we can reject the hypothesis that X has a normal distribution, or H=0 if we cannot reject that hypothesis. We reject the hypothesis if the test is significant at the 5% level.

The Jarque-Bera test evaluates the hypothesis that X has a normal distribution with unspecified mean and variance, against the alternative that X does not have a normal distribution. The test is based on the sample skewness and kurtosis of  X . For a true normal distribution, the sample skewness should be near 0 and the sample kurtosis should be near 3. The Jarque-Bera test determines whether the sample skewness and kurtosis are unusually different than their expected values, as measured by a chi-square statistic.

The Jarque-Bera test is an asymptotic test, and should not be used with small samples. You may want to use lillietest in place of jbtest for small samples.

H = jbtest(X,alpha) performs the Jarque-Bera test at the 100*alpha % level rather than the 5% level, where alpha must be between 0 and 1.

[H,P,JBSTAT,CV] = jbtest(X,alpha) returns three additional outputs. P is the p-value of the test, JBSTAT is the value of the test statistic, and CV is the critical value for determining whether to reject the null hypothesis.

We can use jbtest to determine if car weights follow a normal distribution.

• load carsmall [h,p,j] = jbtest(Weight) h = 1 p = 0.026718 j = 7.2448

With a p-value of 2.67%, we reject the hypothesis that the distribution is normal. With a log transformation, the distribution becomes closer to normal but is still significantly different at the 5% level.

• [h,p,j] = jbtest(log(Weight)) h = 1 p = 0.043474 j = 6.2712

See lillietest for a different test of the same hypothesis.

[1]  Judge, G. G., R. C. Hill, W. E. Griffiths, H. Lutkepohl, and T.-C. Lee. Introduction to the Theory and Practice of Econometrics . New York, Wiley.

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Vague data analysis using neutrosophic Jarque–Bera test

Muhammad aslam.

1 Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Rehan Ahmad Khan Sherwani

2 College of Statistical and Actuarial Sciences, University of the Punjab Lahore, Lahore, Pakistan

Muhammad Saleem

3 Department of Industrial Engineering, Faculty of Engineering-Rabigh, King Abdulaziz University, Jeddah, Saudi Arabia

Associated Data

The data is given in the paper.

In decision-making problems, the researchers’ application of parametric tests is the first choice due to their wide applicability, reliability, and validity. The common parametric tests require the validation of the normality assumption even for large sample sizes in some cases. Jarque-Bera test is among one of the methods available in the literature used to serve the purpose. One of the Jarque-Bera test restrictions is the computational limitations available only for the data in exact form. The operational procedure of the test is helpless for the interval-valued data. The interval-valued data generally occurs in situations under fuzzy logic or indeterminate state of the outcome variable and is often called neutrosophic form. The present research modifies the existing statistic of the Jarque-Bera test for the interval-valued data. The modified design and operational procedure of the newly proposed Jarque-Bera test will be useful to assess the normality of a data set under the neutrosophic environment. The proposed neutrosophic Jarque-Bera test is applied and compared with its existing form with the help of a numerical example of real gold mines data generated under the fuzzy environment. The study’s findings suggested that the proposed test is effective, informative, and suitable to be applied in indeterminacy compared to the existing Jarque–Bera test.

1 Introduction

The standard statistical tests from the parametric domain play a vital role in decision-making problems and are popular in social sciences [ 1 ]. The outcomes of the tests are considered reliable and valid for the population under investigation. These parametric tests help understand the research problems for better decision-making, prediction, and estimation purposes. The fruits of the tests are only juicy when the standard assumptions under the parametric tests validate. One of the standard assumptions of these tests is the validation of the normality assumption. The analysis and recommendations without checking the normality of the data mislead the decision-makers. Several tests have been proposed to assess the normality of data. Jarque-Bera (JB) test is famous goodness of fit test used to assess the distributional structure of data. The validation of the normality assumption under the JB test relies on the principle of matched skewness and kurtosis of the sample data with the normal distribution. The test is applied for testing the null hypothesis that there is no significant difference between the data in hand and the normal distribution versus the alternative hypothesis that a significant difference exists. Several authors applied this test in various fields. [ 2 ] discussed the power of the JB test. [ 3 ] presented the modification of the JB test. [ 4 ] discuss the power of various statistical normality tests. [ 5 ] worked on the modification of the JB test for the multivariate data. [ 5 ] applied the JB test for the face recognition problem. More details about the test and other analyses can be seen in [ 2 , 6 – 11 ].

One of the JB test restrictions is the computational limitation available only for the data in exact form. When the observations in the data are fuzzy, the existing JB test under classical statistics cannot be applied for testing the normality of the data. The data based on the fuzzy logic are often in the interval-valued form [ 12 , 13 ]. An extension of the fuzzy logic and interval-based approach is called the neutrosophic logic [ 14 ]. The neutrosophic logic can provide information about the measure of indeterminacy. [ 15 ] introduced the neutrosophic logic. The other forms of fuzzy sets are singular valued fuzzy sets [ 16 ], hesitant fuzzy sets [ 17 ], Zadeh fuzzy sets [ 18 ], intuitionistic fuzzy sets [ 19 ] etc. The applications of the neutrosophic logic can be seen in [ 20 – 36 ]. Based on neutrosophic logic, neutrosophic statistics were introduced by [ 37 ]. Neutrosophic statistics is the generalization of classical statistics applied when the data is measured from a complex or indeterminate environment. References [ 38 , 39 ] provided the methods to analyze the data having neutrosophic numbers. Reference [ 40 ] introduced the area of neutrosophic statistical quality control. References [ 41 , 42 ] introduced tests of normality under neutrosophic statistics. The details about the neutrosophic statistics can be seen in [ 43 , 44 ]. [ 45 ] applied the JB test using the fuzzy approach in forecasting solar radiation. Reference [ 46 ] applied this test for prediction stock closing prices. [ 47 ] preened a novel distance measure method and applied it in gold mines data. For more details, the reader may read [ 48 – 60 ].

Motivating from the computational limitations of the existing JB test for the exact form data, we proposed a modified version of the present JB test for the fuzzy or interval-valued data. The proposed JB test is a generalized form of the existing JB test from classical statistics as it possesses the ability to deal with both exact and fuzzy forms data sets. Gold mines data is one of the data sets like water level, temperature, stock exchange, melting points, etc., that may possess the indeterminate or neutrosophic form. We will present the application of the proposed test with the help of gold mines data taken from [ 47 ]. The efficiency of the proposed JB test will be compared with the existing test. The use of the developed JB test will be beneficial in situations where the observations under a problem are not certain, fuzzy, indeterminate, interval-valued, or in neutrosophic form.

2 Preliminary

Let z N = a N +b N I N ; I N ϵ[I L , I U ] be a neutrosophic random variable with determinate part a N and indeterminate part b N I N . Note here that the neutrosophic random variable becomes the traditional random variable if I L = 0. Suppose that n N ϵ[n L , n U ] be neutrosophic sample size. By following [ 38 ], the neutrosophic average for z N ϵ[z L , z U ] is given as

where a ¯ N = 1 n N ∑ i = 1 n N a i and b ¯ N = 1 n N ∑ i = 1 n N b i

The neutrosophic difference between z N and z ¯ N is given as

The neutrosophic sum of square (NSS) is given by

The neutrosophic measure of skewness k 3N ϵ[k 3L , k 3U ] is given as

The neutrosophic measure of kurtosis k 4N ϵ[k 4L , k 4U ] is given as

Note here that S N ϵ[S L , S U ] presents the neutrosophic standard deviation and defined as follows

3 Jarque–Bera test under neutrosophic statistics

The JB test is used to confirm the normality of a data set before applying the famous standard statistical tests like t-test, z-test, or F-test. The test is based on the null hypothesis there is no difference between the data under study and the normal distribution versus the alternative hypothesis that difference exists. The test statistic of the JB test is a function of skewness and kurtosis. Suppose S, K and n are the sample skewness, kurtosis and the sample size for a data set then the statistic used for JB test under classical statistics is defined as:

The test-statistic will be useful when the data are in exact form and helpless in the case of interval-valued data. We modify the JB-statistic defined in (7) for the interval-valued data. Now, the modified JB test under neutrosophic environment used the null hypothesis H 0N that the neutrosophic data is the same as the neutrosophic normal distribution versus the alternative hypothesis H 1N that the neutrosophic data is different from the neutrosophic normal distribution. The operational procedure of the proposed JB test under neutrosophic statistics is stated in the following steps:

• Step-1: Select a neutrosophic random sample of size n N ϵ[n L , n U ]. Compute the averages of the determined part a i (i = 1,2,…,n L ) and indeterminate part b i (i = 1,2,…,n U ) as follows
• Step-2: The neutrosophic average of a neutrosophic random variable is calculated as
• Step-3: The difference between z N and z ¯ N will be computed as
• Step-4: Compute the sum of square (SS) as follows
• Step-5: Compute k 3N ϵ[k 3L , k 3U ] and k 4N ϵ[k 4L , k 4U ].
• Step-6: Compute JB statistic under neutrosophic statistic JB N ϵ[JB L , JB U ] using the following formula

The statistic JB N proposed in (12) follows asymptotically to a Chi-square distribution with two degrees of freedom. The normal distribution has a skewness zero and kurtosis three indicates that for a normal distribution, the value of JB N is zero and any excess value of JB N from zero will indicate the deviation from normality.

• Step-7: choose the tabulated value from the Chi-square table and accept H 0N if JB N ϵ[JB L , JB U ] less than the tabulated value at the level of significance α.

4 Application in cleaner production data

This section will present the computational aspects of the proposed methodology of the newly developed JB test under a neutrosophic environment. The application of the proposed JB N test is given with the help of cleaner production data from the gold mines. [ 47 ] discussed the cleaner production data for gold mines based on experts’ evaluation evidence under fuzzy theory. The availability of the gold mines data under fuzzy logic motivates us to use the data for the application purposes for the present research. According to [ 47 ], the decision-maker is interested in selecting a suitable center from the three centers C 1 , C 2 and C 3 on the basis of five characteristics of gold mines data. [ 47 ] presented a comprehensive way to select the best center based on their decision criteria but did not perform the normality test before using the methods. To test the data normality, we will use only the characteristic management level C 1 . According to [ 47 ], "it indicates the production process and equipment level, which contains the mining technology and production equipment. The data of gold mines of the characteristics G 1 , G 2 and G 3 of center C 1 is selected from [ 47 ] and reported in Table 1 . It can be seen that the data is in the indeterminacy interval; therefore, we will apply the proposed test to check the normality of the data first.

The proposed test using the real data for three centers G 1 , G 2 and G 3 , respectively is implemented as follows

• Step-1: Select a neutrosophic random sample of size n N ϵ[4,4]. The averages of the determined part a i (i = 1,2,…,n L ) and indeterminate part b i (i = 1,2,…,n U ) are:

a ¯ N = 1 4 ∑ i = 1 4 a i = 0.19 and b ¯ N = 1 4 ∑ i = 1 4 b i = 0.3325

a ¯ N = 1 4 ∑ i = 1 4 a i = 0.15 and b ¯ N = 1 4 ∑ i = 1 4 b i = 0.3075

a ¯ N = 1 4 ∑ i = 1 4 a i = 0.2125 and b ¯ N = 1 4 ∑ i = 1 4 b i = 0.3225

The statistics indicate the average performance of the three methods G 1 , G 2 and G 3 laid down by the experts with respective average indeterminacy levels for the selection of the gold mines center, e.g., the average performance of the cleaner production gold mines for the G 1 method is 0.19 with a 0.3325 average uncertainty level.

• Step-2: The neutrosophic averages of neutrosophic random variables are given as

z ¯ N = 0.19 + 0.3325 I N ; I N ϵ[0,0.05]

z ¯ N = 0.15 + 0.3075 I N ; I N ϵ[0,0.05]

z ¯ N = 0.2125 + 0.3225 I N ; I N ϵ[0,0.05]

• Step-3: The difference between z N and z ¯ N for example for G 1 is given by

z N − z ¯ N = [ 0.13,0.1348 ] , … , [ − 0.06 , − 0.0546 ]

• Step-4: Compute the sum of square (SS) for three centers are as follows

∑ i = 1 n N ( z N − z ¯ N ) 2 = [ 0.0314,0.1338 ] , [0.0274,0.0792], [0.0134,0.1337]

• Step-5: The neutrosophic values of k 3N ϵ[k 3L , k 3U ] and k 4N ϵ[k 4L , k 4U ] for three centers are given as

k 3N ϵ<[0.3623, −0.0755], [1.000,0.0323], [−0.6113, −0.3565]>

k 4N ϵ<[−1.3797, −2.5762], [−0.7911, −2.6052], [−0.9277, −2.7399]>

• Step-6: The calculated values of statistic JB N ϵ[JB L , JB U ] are given as

JB N <[0.4047,1.1099], [0.7711,1.1319], [0.3926,1.3359]>

The value of the proposed JB N test statistic is not much far away from zero, indicating that the gold mines data follow a normal probability distribution with an indeterminacy level. The same can be verified by using the Chi-square distribution table in Step-7.

• Step-7: The table value for the level of significance 0.05 is 7.815. We note that JB N ϵ[JB L , JB U ] are less than the tabulated value. Therefore, the null hypothesis is that data from three centers are not significantly different from the neutrosophic normal distribution, and this decision is the same as [ 47 ].

5 Comparative study

In this section, the performance of the proposed test will be compared with the JB test under classical statistics. The proposed JB N ϵ[JB L , JB U ] defined in Eq ( 12 ) is the extension of the existing JB test presented in Eq ( 7 ). The proposed test will be reduced to the JB test under classical statistic if JB N = JB L = 0. The neutrosophic form of the proposed JB N ϵ[JB L , JB U ] test for centers G 1 , G 2 and G 3 along with the measures of indeterminacy are shown in Table 2 .

From Table 2 , we note that the measure of indeterminacy is increased if the gap between JB N ϵ[JB L , JB U ] is increased. We also note that the proposed test provides the measures of indeterminacy, while the existing JB test under classical statistics cannot provide this kind of information. For example, when the level of significance is 5%, according to the proposed JB test under neutrosophic statistics, the probability that the null hypothesis is accepted is 0.95, and the null hypothesis is rejected with the probability of 0.05. Other than these probabilities, the chance that the decision-makers are uncertain about the acceptance or rejection of the null hypothesis is 0.6353. We note that the sum of the probabilities is larger for the proposed test, and this theory is the same as in [ 37 ]. The proposed test can be compared with the existing JB in terms of sensitivity. From Table 2 , it can be seen that values of the classical test (determined part) fluctuate much as compared to the indeterminate part. For example, for centers G 2 and G 3 , the values of the existing JB test moves from 0.4047 to 0.7711 when measure of indeterminacy changes from 0.6353 to 0.3187. On other hand, the values of the indeterminate part of the proposed JB test moves from 1.1099 I N to 1.1319 I N when measure of indeterminacy changes from 0.6353 to 0.3187. From the study, it can be seen that the proposed test is less sensitive than the existing JB test. We also note that the proposed test provides the results in indeterminacy intervals and makes it suitable and effective to be applied in the indeterminate environment. This theory is the same as in [ 38 , 39 ].

6 Concluding remarks

The paper extends the concept of the Jarque–Bera test from classical statistics to neutrosophic statistics. The classical JB test is limited to perform on exact values data. In contrast, the proposed modified form of the JB statistic can be used to both exact and interval-valued data. The design and operational procedure for the newly developed JB test are presented under the fuzzy and neutrosophic logic. The application of the proposed JB test is carried on the real data set from the cleaner production of gold mines generated in a fuzzy environment. Moreover, a comparison of the proposed neutrosophic JB test is made with the existing JB test to assess the performance of the two tests. The findings of the numerical example suggested that the proposed JB test is effective, informative, and suitable to be applied under indeterminacy compared to the existing JB test. For generalized and better analysis of the data, the proposed test is recommended when the data is obtained from indeterminate and complex systems. The proposed methodology of the JB test can be extended to test the multivariate normality under indeterminacy. The proposed test for big data can be extended for future research. The development of new software to perform the proposed test is a fruitful area for future research.

Acknowledgments

The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of the paper.

Funding Statement

The author(s) received no specific funding for this work.

Data Availability

JarqueBeraALMTest

JarqueBeraALMTest [ data ]

tests whether data is normally distributed using the Jarque – Bera ALM test.

JarqueBeraALMTest [ data , " property " ]

returns the value of " property " .

Details and Options

• The data can be univariate { x 1 , x 2 , … } or multivariate { { x 1 , y 1 , … } , { x 2 , y 2 , … } , … } .
• The Jarque – Bera ALM test effectively compares the skewness and kurtosis of data to a NormalDistribution .

• JarqueBeraALMTest [ data , dist , "HypothesisTestData" ] returns a HypothesisTestData object htd that can be used to extract additional test results and properties using the form htd [" property "] .
• JarqueBeraALMTest [ data , dist , " property " ] can be used to directly give the value of " property " .
• Properties related to the reporting of test results include:
• The following properties are independent of which test is being performed.
• Properties related to the data distribution include:
• The following options can be given:

Basic Examples     (3)

Perform a Jarque – Bera ALM test for normality:

Perform a test for multivariate normality:

Extract the test statistic from a Jarque – Bera ALM test:

Scope     (6)

Testing     (3).

Test for multivariate normality:

Create a HypothesisTestData object for repeated property extraction:

The properties available for extraction:

Reporting     (3)

Tabulate the results of the Jarque – Bera ALM test:

The full test table:

The test statistic:

Retrieve the entries from a Jarque – Bera ALM test table for custom reporting:

Report test conclusions using "ShortTestConclusion" and "TestConclusion" :

The conclusion may differ at a different significance level:

Options     (3)

Method     (3).

Use Monte Carlo-based methods or a computation formula:

Set the number of samples to use for Monte Carlo-based methods:

Set the random seed used in Monte Carlo-based methods:

Applications     (2)

A power curve for the Jarque – Bera ALM test:

Visualize the approximate power curve:

Estimate the power of the Jarque – Bera ALM test when the underlying distribution is a CauchyDistribution [ 0 , 1 ] , the test size is 0.05, and the sample size is 12:

Create a Jarque – Bera ALM test statistic generalized for other distributions:

A Jarque – Bera ALM test statistic for fitting to a LaplaceDistribution :

Perform the generalized test on some data:

The test is powerful against the alternative of a HyperbolicDistribution of similar mean and variance:

Properties & Relations     (4)

The Adjusted Lagrange Multiplier (ALM) method outperforms the traditional Jarque – Bera test:

The traditional Jarque – Bera test statistic:

The Jarque – Bera ALM test is superior for small samples:

The Jarque – Bera ALM test uses finite-sample values for the mean and variance of skewness and kurtosis, not the asymptotic values of 0, 6, 3, and 24 as in the traditional test:

The finite-sample values can be derived using MomentEvaluate and MomentConvert :

The test statistics have the same asymptotic distribution:

Plot a histogram of the statistic and the probability density function of the distribution:

The Jarque – Bera ALM test works with the values only when the input is a TimeSeries :

Possible Issues     (1)

Neat Examples     (1)

The test statistic given a particular alternative:

Compare the distributions of the test statistics:

HypothesisTestData   AndersonDarlingTest   KolmogorovSmirnovTest   CramerVonMisesTest   DistributionFitTest   KuiperTest   MardiaCombinedTest   MardiaKurtosisTest   MardiaSkewnessTest   PearsonChiSquareTest   ShapiroWilkTest   WatsonUSquareTest

Related Guides

• Hypothesis Tests

Introduced in 2010 (8.0)

Wolfram Research (2010), JarqueBeraALMTest, Wolfram Language function, https://reference.wolfram.com/language/ref/JarqueBeraALMTest.html.

Wolfram Language. 2010. "JarqueBeraALMTest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JarqueBeraALMTest.html.

Wolfram Language. (2010). JarqueBeraALMTest. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JarqueBeraALMTest.html

@misc{reference.wolfram_2023_jarqueberaalmtest, author="Wolfram Research", title="{JarqueBeraALMTest}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/JarqueBeraALMTest.html}", note=[Accessed: 01-April-2024 ]}

@online{reference.wolfram_2023_jarqueberaalmtest, organization={Wolfram Research}, title={JarqueBeraALMTest}, year={2010}, url={https://reference.wolfram.com/language/ref/JarqueBeraALMTest.html}, note=[Accessed: 01-April-2024 ]}

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Statistics Made Easy

How to Perform a Jarque-Bera Test in Excel

The Jarque-Bera test is a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution .

The test statistic of the Jarque-Bera test is always a positive number and if it’s far from zero, it indicates that the sample data do not have a normal distribution.

The test statistic  JB   is defined as:

JB   =(n/6) * (S 2 + (C 2 /4))

• n:  the number of observations in the sample
• S:  the sample skewness
• C: the sample kurtosis

Under the null hypothesis of normality,  JB ~  X 2 (2)

This tutorial explains how to conduct a Jarque-Bera test in Excel.

Jarque-Bera test in Excel

Use the following steps to perform a Jarque-Bera test for a given dataset in Excel.

Step 1: Input the data.

First, input the dataset into one column:

Step 2: Calculate the Jarque-Bera Test Statistic.

Next, calculate the JB test statistic. Column F shows the formulas used:

Step 3: Calculate the p-value of the test.

Recall that under the null hypothesis of normality, the test statistic JB follows a Chi-Square distribution with 2 degrees of freedom. Thus, to find the p-value for the test we will use the following function in Excel:  =CHISQ.DIST.RT(JB test statistic, 2)

T he p-value of the test is 0.5921 . Since this p-value is not less than 0.05, we fail to reject the null hypothesis. We don’t have sufficient evidence to say that the dataset is not normally distributed.

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Jarque-Bera test

Regular test.

Compute the Jarque-Bera statistic to test the null hypothesis that an uncertain value is normally distributed.

Pooled test

First, draw n realisations of each uncertain value in ud and pool them together. Then, compute the Jarque-Bera statistic to test the null hypothesis that the values of the pool are normally distributed.

Element-wise test

First, draw n realisations of each uncertain value in ud , keeping one pool of values for each uncertain value.

Then, compute the Jarque-Bera statistic to test the null hypothesis that each value pool is normally distributed.

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This document was generated with Documenter.jl version 0.27.10 on Friday 19 November 2021 . Using Julia version 1.6.0.

How to Perform a Jarque-Bera Test in Excel

The Jarque-Bera test is a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution .

The test statistic of the Jarque-Bera test is always a positive number and if it’s far from zero, it indicates that the sample data do not have a normal distribution.

The test statistic  JB   is defined as:

JB   =(n/6) * (S 2 + (C 2 /4))

• n:  the number of observations in the sample
• S:  the sample skewness
• C: the sample kurtosis

Under the null hypothesis of normality,  JB ~  X 2 (2)

This tutorial explains how to conduct a Jarque-Bera test in Excel.

Jarque-Bera test in Excel

Use the following steps to perform a Jarque-Bera test for a given dataset in Excel.

Step 1: Input the data.

First, input the dataset into one column:

Step 2: Calculate the Jarque-Bera Test Statistic.

Next, calculate the JB test statistic. Column F shows the formulas used:

Step 3: Calculate the p-value of the test.

Recall that under the null hypothesis of normality, the test statistic JB follows a Chi-Square distribution with 2 degrees of freedom. Thus, to find the p-value for the test we will use the following function in Excel:  =CHISQ.DIST.RT(JB test statistic, 2)

T he p-value of the test is 0.5921 . Since this p-value is not less than 0.05, we fail to reject the null hypothesis. We don’t have sufficient evidence to say that the dataset is not normally distributed.

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