Statistics Definitions > Jarque-Bera Test

## What is the Jarque-Bera Test?

The Jarque-Bera Test,a type of Lagrange multiplier test, is a test for normality. Normality is one of the assumptions for many statistical tests, like the t test or F test; the Jarque-Bera test is usually run before one of these tests to confirm normality. It is usually used for large data sets, because other normality tests are not reliable when n is large (for example, Shapiro-Wilk isn’t reliable with n more than 2,000).

Specifically, the test matches the skewness and kurtosis of data to see if it matches a normal distribution. The data could take many forms, including:

- Time Series Data.
- Errors in a regression model.
- Data in a Vector.

A normal distribution has a skew of zero (i.e. it’s perfectly symmetrical around the mean) and a kurtosis of three; kurtosis tells you how much data is in the tails and gives you an idea about how “peaked” the distribution is. It’s not necessary to know the mean or the standard deviation for the data in order to run the test.

## Running the Test

The formula for the Jarque-Bera test statistic (usually shortened to just **JB test statistic**) is:

JB = n [(√b1)

^{2}/ 6 + (b_{2}– 3)^{2}/ 24].

the

Where:

n is the sample size,

√b_{1} is the sample skewness coefficient,

b_{2} is the kurtosis coefficient.

The null hypothesis for the test is that the data is normally distributed; the alternate hypothesis is that the data does not come from a normal distribution.

## What the Results Mean

In general, a large J-B value indicates that errors are **not **normally distributed.

For example, in MATLAB, a result of 1 means that the null hypothesis has been rejected at the 5% significance level. In other words, the data does not come from a normal distribution. A value of 0 indicates the data is normally distributed.

Unfortunately, most statistical software does not support this test. In order to interpret results, you may need to do a little comparison (and so you should be intimately familiar with hypothesis testing). Checking p-values is always a good idea. For example, a tiny p-value and a large chi-square value from this test means that you can reject the null hypothesis that the data is normally distributed.

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“a result of 1 means that the null hypothesis has been rejected at the 5% significance level. In other words, the data does not come from a normal distribution. A value of 0 indicates the data is normally distributed.”

In most statistical tests I am familiar with, failure to reject the null cannot be used as evidence in support of the null. The conclusion is instead that the results are not statistically significant. Is the Jarque-Bera test doing a different kind of inference testing?

It’s doing the same thing. It’s not

proofof anything, but it doesindicatesomething (that the data is probably normal). I find when I teach statistics, the concept of “not statistically significant” and “rejecting the null” (as opposed to accepting the alternate) is hard to grasp. Instead, I choose to use more concrete terms (like “accepting the null hypothesis”). For elementary statistics, that’s good enough. For more advanced courses, I’ll get into the nit-picky stuff :)Ah, it does make sense that the probability that the null is true increases somewhat if this test fails to reject the null.

Do you have any recommendations on resources that go into more depth on explaining this further? Greatly appreciated.

Not that I can think of. If it were me, I’d limit my Google search to *.edu though :)