What is statistics? – Standard Error of Skewness – Standard Error of Kurtosis

Standard Error of Skewness and Standard Error of Kurtosis
The Skewness and Kurtosis statistics are useful because tells us about some properties of the dispersion of a dataset and thus the shape as a Graph Figure of this dataset. When the size of a dataset is small, the sample skewness statistics or sample kurtosis statistics can be not representative of the true skewness or true kurtosis that exists in the reference population. Therefore, the Standard Error of Skewness and the Standard Error of Kurtosis can help. These standard errors can show the deviation that can exist between the values of Skewness or Kurtosis in multiple samples that will be taken randomly from the same underlying population distribution as the sample of analysis was taken.

Standard Error of Skewness: Definition
The Standard Error of Skewness shows the deviation that can exist between the values of Skewness in multiple samples that will be taken randomly from the the same underlying population distribution as the sample of analysis. A zero value shows that the deviation of values of skewness between multiple samples is zero and thus, the underlying distribution of the current sample also does not deviate from a symmetric distribution. Therefore, in that case, the current sample can be said that has a symmetric distribution, too. Note that, higher values show higher deviation of the underlying distribution of the sample from a symmetric distribution.

Skewness_Kurtosi_SES__SEK_standard error

Standard Error of Kurtosis: Definition
Same logic applies here, too. The Standard Error of Kurtosis shows the deviation that can exist between the values of Kurtosis in multiple samples that will be taken randomly from the the same underlying population distribution as the sample of analysis. A zero value shows that the deviation of values of Kurtosis between multiple samples is zero and thus, the underlying distribution of the current sample also does not deviate from a distribution with zero excess kurtosis or from a distribution that has a mesokurtic peak. Therefore, the current sample can be said that has also a distribution with a zero excess kurtosis. Note that, higher values show higher deviation of the underlying distribution of the sample from a symmetric distribution.

Skewness_Kurtosi_SES__SEK_standard error

Standard Error of Skewness: Statistical Definition
The statistical formula for Standard Error of Skewness (SES) for a normal distribution is the following one:

SES=\sqrt{\frac{6n(n-1)}{(n-2)(n+1)(n+3)}}

Note that n is the size of the sample. Because this formula has dependence only on the size of the sample, then SES can easily be calculated for any given size of sample. In the following table, you can see the values that SES takes for some specific sizes of sample. If the sample skewness is divided by SES, it can show how much the underlying distribution deviates from a symmetric distribution.

Standard Error of Kurtosis: Statistical Definition
The statistical formula for Standard Error of Kurtosis (SEK) for a normal distribution is the following one:

SEK=2(SES)\sqrt{\frac{(n^2-1)}{(n-3)(n+5)}}

Note that “n” is the size of the sample. Because this formula has dependence only on the size of the sample, -SES is also solely based on “n” the size of sample- then SEK can easily be calculated for any given size of sample. In the following table, you can see the values that SEK takes for some specific sizes of sample. If the sample (excess) Kurtosis is divided by SEK, it can show how much the underlying distribution deviates from a distribution with a mesokurtic peak or from a distribution with a zero Excess Kurtosis.

Skewness and Normal Distribution
There is no a universal accepted Statistical Formula to detect Skewness in all cases. However:
i) There are some Rule of Thumbs that mostly work that their reference is for population data that follow Normal Distribution. However, their thresholds are arbitrary set.
ii) The visual inspection of Histograms, Boxplots, and other related statistical graph figures is the best way to check for Skewness.

A) You can divide Skewness Statistic with Standard Error of Skewness: Skew(X)/SES

i) If the result of this division Skew(X)/SES<-2 is lower than minus two (-2) then this may suggest FOR population that is negatively Skewed
ii) If the result of this division is Skew(X)/SES>2 then this may suggest that population data are Positively Skewed
iii) If the result of this division is between -2<Skew(X)/SES<2 then it can suggest that population data are neither Positively or Negatively Skewed.
iv) A 95% Confidence Interval can be constructed by using these values: Skew(X)\pm2*SES

This Rule of thumb can be worded in a different way with the same meaning: When the Standard Error of Skewness (SES) is two times higher than the absolute value of the Skewness Statistic then population is probably, highly skewed: |Skew(X)|>2*SES

SES and SEK per sample size (n)
We can observe in the following table and in the Graph Figure that as the sample size increases, the Standard Error of Kurtosis and the Standard Error of Skewness take values closer to zero e.g. for n=10.000, we have: SES=.024, SEK=.048.

Table 1. It shows what values can take Standard Error of Skewness and Standard Error of Kurtosis when the sample size is from 5 to 10000.

n (sample size)Standard Error of Skewness (SES)Standard Error of Kurtosis (SEK)
50.913
2.000
100.687
1.334
150.580
1.121
200.512
0.992
300.427
0.833
400.374
0.733
500.337
0.663
1000.241
0.478
2000.172
0.342
10000.077
0.154
100000.024
0.048

Standard error of Skewness - Standard Error of Kurtosis - as n goes to 1000, SES and SEK go close to zero