# 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

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.

### 21 Responses to Analysis of Skewness and Kurtosis

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2. 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

3. Rajesh says:

Data distribution free how to apply 2 way anova

• Charles says:

Sorry, but I don’t understand your question.
Charles

4. 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…

5. 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

• Denny Yu says:

Thanks for replying.
I’m really looking forward to it.

6. 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

7. 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.

• Charles says:

Jpso,
I am using excess kurtosis (as does Excel).
Charles

8. 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