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.