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

image9258

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.

21 Responses to Analysis of Skewness and Kurtosis

  1. soharb says:

    I think there is some thing wrong with this formula
    for example for this series
    26.83946269
    26.95131935
    8.371060164
    10.40495872
    18.38858378
    20.12905135
    24.2843167
    1.76670796
    20.19191695
    41.06557085
    16.09877032
    13.34390071
    0.426210193
    28.31166689
    11.89051087
    109.3641761
    25.50859431
    61.26802436
    32.5178008
    66.58119511
    41.27546773
    14.67351611
    2.048435245
    28.01590722
    44.93746991

    the JARQUE(R1)=38.28239095
    but if we use an array formula like this:
    =COUNT(A2:A26)*(((((SUM((A2:A26-AVERAGE(A2:A26))^3)/COUNT(A2:A26))/((SUM((A2:A26-AVERAGE(A2:A26))^2)/COUNT(A2:A26))^1.5))^2)/6)+((((SUM((A2:A26-AVERAGE(A2:A26))^4)/COUNT(A2:A26))/((SUM((A2:A26-AVERAGE(A2:A26))^2)/COUNT(A2:A26))^2)-3))^2)/24)
    + CTRL + SHIFT + ENTER
    the answer will be: 26.69155055
    not to mention, I completely sure about this formula to be the Jarque–Bera test coefficient.

    • Charles says:

      I am using the following Excel formula =COUNT(A2:A26)*(SKEW(A2:A26)^2/6+KURT(A2:A26)^2/24)
      Charles

      • soharb says:

        Then there is some thing wrong (bug) in excel formula, since I calculated the SKEW, KURT and JB with “EViews 9.5” and my array formula turn up to be the correct answer!

        • Charles says:

          What value did you get for SKEW and KURT_
          Charles

          • soharb says:

            EViews 9.5:
            SKEW= 1.769081
            KURT= 3.620125
            JB= 26.69155

            Excel regular formula:
            =SKEW(A2:A26) = 1.884063081
            =SKEW.P(A2:A26) =1.769080723
            =KURT(A2:A26) = 4.748928357
            Note: there is no KURT.P!!!

            Excel array formula:
            for SKEW
            =((SUM((A2:A26-AVERAGE(A2:A26))^3)/COUNT(A2:A26))/((SUM((A2:A26-AVERAGE(A2:A26))^2)/COUNT(A2:A26))^1.5))
            + CTRL + SHIFT + ENTER
            =1.769080723

            for KURT
            =((SUM((A2:A26-AVERAGE(A2:A26))^4)/COUNT(A2:A26))/((SUM((A2:A26-AVERAGE(A2:A26))^2)/COUNT(A2:A26))^2)-3)
            + CTRL + SHIFT + ENTER
            =3.620124598

          • Charles says:

            Soharb,
            Thanks for sending me this information. It looks like if we use the population values of skewness and kurtosis then we get the result that you have seen from EViews.
            In particular, the Real Statistics Resource Pack has functions SKEWP and KURTP. If these functions are used then the formula =COUNT(A2:A26)*(SKEWP(A2:A26)^2/6+KURT(A2:A26)^2/24) yields the result 26.69155.
            Thanks for bringing this up. I will revise the JARQUE and JBTEST functions in the next release of the software.
            Charles

  2. Jpso says:

    Hi and congrats for the great initiative.

    When you refer to Kurtosis, you mean the Excess kurtosis (i.e. kurt-3) or the outright kurtosis? For example when I perform the “D’Agostino-Pearson Test” as described in the relevant section (i.e. using outright kurtosis) I get results suggesting rejection of the null hypothesis, even if I use Kurt=3, Skew=0, which is the ND standards stats.

    Thank you.

  3. david oluyole ajekigbe says:

    thank you very much for this information. i have gained a lot from it. it will be appreciated if you can please attend to the question of zohreh of february 28, 2016 @ 9.31pm . i also will like to name of the person for reference. thank you .
    david

    • Charles says:

      David,

      As I wrote in response to that comment

      “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.”

      Thus, it is difficult to attribute this rule of thumb to one person, since this goes back to the beginning of statistics, or at least the use of the value 1.96. You will find this value of 1.96 in any elementary book on statistics.

      Charles

  4. 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

  5. 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…

  6. Rajesh says:

    Data distribution free how to apply 2 way anova

  7. 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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