Question

Get the sample skewness coefficient of a random variable X from a random sample of size n. Explain why it measures asymmetry of the distribution of X. Explain the relation between the sample skewness coefficient and the population skewness coefficient.

I know how to get a sample skewness coefficient but what can I say about the relationship between the sample and population skewness coefficient?

Answer 1

Population skewness is given by g₁:-

where

Sample skewness is given by G₁:-

The above equation is also the relationship between the population skewness coefficient and sample skewness coefficient .

Skewness indicates if the deviations from the mean are going to be positive or negative, i.e., how the distribution is different from the normal curve. If the deviations from the mean are going to be positive then it is called positively skewed or skewed right. Which means that the bulk of the data is at the left and the right tail is longer. If the deviations from the mean are going to be negative then it is called negatively skewed or skewed left. Which means that the bulk of the data is at the right and the left tail is longer.

The sample skewness doesn't necessarily apply to the whole population. So in order to conclude anything about the population skewness from the sample skewness, we must use significance test.

Test statistic for this test is as follows:-

Where

CONCLUSION:-

If this test statistic is less than -2, the population is most likely skewed negative

If this test statistic is between -2 and +2, can;t reach any conclusion about the population skewness.

If this test statistic is more than +2, the population is most likely skewed positive