Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.


A distribution is said to be skewed to the right (left) if it shows a tailing at the right (left) end. The amount of skewing can be determined by the third moment of the distribution, which is usually called skewness:

This definition of the skewness of a sample is a biased estimator of the skewness of the population. In order to estimate the skewness of a population the following adjusted formula has to be used:

In order to test, whether the calculated sample skewness originates from a skewed distribution, the following test statistic may be calculated:

If the test statistic Zn exceeds the (1-/2) quantile of the standard normal distribution, the null hypothesis of a symmetric distribution has to be rejected at a level of significance of . The number of data points n should be greater than 100 for this test.

Hint: In some software packages the test statistic Zn is called "standardized skewness". As a rule of thumb you can assume a skewed distribution with 95% confidence if the standardized skewness is less than -2 or greater than +2.

Please note that the skewness is occasionally defined by a somewhat different formula, leading to different values.

Below you find two examples of skewed distributions. You may also start the following  interactive example  in order to see the effect of skewed distributions on the mean and the median.

Last Update: 2012-10-08