Skewness

Skewness is a measure of the lack of symmetry, or the lopsidedness, a distribution has. In other words, one tail of the distribution is longer than another. A positively skewed distribution has a longer right tail than left, corresponding to a smaller number of occurrences at the high end of the distribution. This might be the case when you have a test that is very difficult: Few people get scores that are very high, and many more get scores that are relatively low. A negatively skewed distribution has a shorter right tail than left, corresponding to a larger number of occurrences at the high end of the distribution. This would be the case for an easy test (lots of high scores and relatively few low scores).

Although skewness (and kurtosis) are used mostly as descriptive terms (such as “That distribution is negatively skewed”), mathematical indicators can indicate how skewed or kurtotic a distribution is. For example, skewness is computed by subtracting the value of the median from the mean:

where

Sk is Pearson measure of skewness,

X¯ is the mean,

M is the median, and

s is the standard deviation.

For example, if the mean of a distribution is 100 and the median is 95, the skewness value is 100 – 95 = 5, and the distribution is positively skewed. If the mean of a distribution is 85 and the median is 90, the skewness value is 85 – 90 = −5, and the distribution is negatively skewed. The formula takes the standard deviation of the distribution into account so that skewness indicators can be compared with one another.