Absolute and Relative Measures of Skewness

Skewness can be measured in two ways.

Absolute measures of skewness express skewness in the same units as the original data:

An absolute measure has a real limitation: it cannot be used to compare the skewness of two distributions that are measured in different units (say, marks out of 100 versus wages in rupees), or even two distributions in the same unit but with very different overall scale (a spread of 5 marks means something very different from a spread of 500 marks). A skewness of '8 marks' tells you nothing about whether that is a large or small departure from symmetry relative to how spread out the data is.

Relative measures of skewness (coefficients of skewness) solve this by dividing the absolute measure by a suitable measure of dispersion, which makes the coefficient a pure number — free of the original units — so distributions of different scales or different units can be compared directly. Because Gujarat Std 11 Commerce Statistics questions and answers on skewness almost always ask you to compute and interpret a coefficient rather than an absolute measure, this chapter focuses on the two relative measures prescribed in the Std 11 syllabus:

1. Karl Pearson's Coefficient of Skewness — based on the mean, mode (or median) and the standard deviation.
2. Bowley's Coefficient of Skewness — based only on the three quartiles ($Q_1$, Median, $Q_3$).

Interpreting the sign and size of a coefficient. Whichever formula is used, the sign of the resulting coefficient always tells you the direction of skewness, and its numeric size tells you the degree: