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 where
*n*is the sample size. - A rough measure of the standard error of the kurtosis is 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*

For Example 1

based on using the functions SKEW and KURT to calculate the sample skewness and kurtosis values. 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.

The JB test can also be calculated using the SKEWP (or SKEW.P) and KURTP functions to obtain the population values of skewness and kurtosis. In this case, we obtain

Since CHISQ.DIST.RT(1.93, 2) = .382 > .05, once again 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, *pop*) = the Jarque-Barre test statistic *JB* for the data in the range R1

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

If *pop* = TRUE (default), the population version of the test is used; otherwise the sample version of the test is used. Any empty cells or cells containing non-numeric data are ignored.

For Example 1, we see that JARQUE(A4:A23) = 1.93 and JBTEST(A4:A23) = .382. Similarly, JARQUE(A4:A23, FALSE) = 2.13 and JBTEST(A4:A23, FALSE) = .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.

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Sir

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

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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Sorry, but I don’t understand your question.

Charles

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.

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

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…

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!

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

Thanks for replying.

I’m really looking forward to it.

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

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

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.

Jpso,

I am using excess kurtosis (as does Excel).

Charles

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.

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

Charles

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!

What value did you get for SKEW and KURT_

Charles

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

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