Analysis of Skewness and Kurtosis

Since the skewness and kurtosis of the normal distribution are zero, values for these two parameters should be close to zero for data to follow a normal distribution.

  • A rough measure of the standard error of the skewness is \sqrt{6/n} where n is the sample size.
  • A rough measure of the standard error of the kurtosis is \sqrt{24/n} where n is the sample size.

If the absolute value of the skewness for the data is more than twice the standard error this indicates that the data are not symmetric, and therefore not normal. Similarly if the absolute value of the kurtosis for the data is more than twice the standard error this is also an indication that the data are not normal.

Example 1: Use the skewness and kurtosis statistics to gain more evidence as to whether the data in Example 1 of Graphical Tests for Normality and Symmetry is normally distributed.

As we can see from Figure 4 of Graphical Tests for Normality and Symmetry (cells D13 and D14), the skewness for the data in Example 1 is .23 and the kurtosis is -1.53. The standard error for the skewness is .55 (cell D16) the standard error for the kurtosis is 1.10 (cell D17). Both statistics are within two standard errors, which suggest that the data is likely to be relatively normally distributed.

Observation: Related to the above properties is the Jarque-Barre (JB) test for normality which tests the null hypothesis that data from a sample of size n with skewness skew and kurtosis kurt

image9257

For Example 1

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Since CHISQ.DIST.RT(2.13, 2) = .345 > .05, based on the JB test, we conclude there isn’t sufficient evidence to rule out the data coming from a normal population.

Real Statistics Functions: The Real Statistics Resource Pack contains the following functions.

JARQUE(R1) = the Jarque-Barre test statistic JB for the data in the range R1

JBTEST(R1) = p-value of the Jarque-Barre test on the data in R1

Any empty cells or cells containing non-numeric data are ignored. For Example 1, we see that JARQUE(A4:A23) = 2.13 and JBTEST(A4:A23) = .345.

Observation: See D’Agostino-Pearson Test for another more accurate test for normality which is based on the skewness and kurtosis of sample data.

11 Responses to Analysis of Skewness and Kurtosis

  1. Denny Yu says:

    Thank you very much! The Real Statistics Functions are really of great help.
    However, I came across a problem that JBTEST, as well as DPTEST, doesn’t allow ranges expressed in array form. For example, the expression: =jbtest(IF(INDIRECT(“G”&6):INDIRECT(“G”&10)0,INDIRECT(“AE”&6):INDIRECT(“AE”&10))) cannot be recognized by Excel and the result is #VALUE!. By comparing with another expression: =jbtest(INDIRECT(“AE”&6):INDIRECT(“AE”&10)) in Evaluating Fomula, I found that JBTEST can only read data with form of “Am:Bn”, not expressed in a set of data like “0.1, 0.2, …”. Is there any solution to it? I have to deal with ranges within which there are certain values that should not be included in the test.
    Thank you again!

    • Charles says:

      Denny,
      The current implementation of these functions supports only arrays which are ranges. I have just changed this so that they should support any arrays. I will include these changes in the next release of the software. I hope to issue this release in the next few days.
      Charles

  2. Zohreh says:

    Salaam
    May you please cite the reference for “If the absolute value of the skewness for the data is more than twice the standard error this indicates that the data are not symmetric, and therefore not normal”. I need it. Thanks.

    • Charles says:

      We often use alpha = .05 as the significance level for statistical tests. The critical value for a two tailed test of normal distribution with alpha = .05 is NORMSINV(1-.05/2) = 1.96, which is approximately 2 standard deviations (i.e. standard errors) from the mean. This is source of the rule of thumb that you are referring to.

      The Jarque-Barre and D’Agostino-Pearson tests for normality are more rigorous versions of this rule of thumb.

      Charles

      • Zohreh says:

        Thanks for replying. I’ve heard that one way to check normality is to divide skewness by standard error, if the results falls between the range +-1.96, then normality will be satisfies. Using this formula my data was proved to be not normal. I used another formula to which you referred “If the absolute value of the skewness for the data is more than twice the standard error this indicates that the data are not symmetric, and therefore not normal”, then my data revealed to be normal. As I want to use the latter procedure in my study I need to cite the name of the person whose opinion I will use. By reference I meant based on whose opinion “If the absolute value of the skewness for the data is … Will you please provide the name of the person?
        Many thanks…

  3. Rajesh says:

    Data distribution free how to apply 2 way anova

  4. Colin says:

    Sir

    What are the rough measure of the standard error of the skewness and kurtosis ? I cannot see the pictures.

    • Charles says:

      Colin,

      A rough measure of the standard error of the skewness is \sqrt{6/n} where n is the sample size.
      A rough measure of the standard error of the kurtosis is \sqrt{24/n} where n is the sample size.

      Charles

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