Meaning of Skewness

Earlier chapters of the Maharashtra State Board Std XI Mathematics and Statistics (Commerce) course measured where a set of data centres (mean, median, mode) and how widely it spreads (range, quartile deviation, standard deviation). Neither of those tells us whether the data lie evenly on the two sides of the centre, or whether they pile up on one side and trail away on the other. That property — the lack of symmetry in a frequency distribution — is called skewness.

A distribution is symmetric when the values are balanced about the central value: the frequencies rise to a peak and fall away in mirror image on either side, so the left half is a reflection of the right half. When this balance is broken — when one tail of the distribution is drawn out longer than the other — the distribution is skewed. Skewness therefore has both a direction (which side the long tail lies on) and a degree (how far from symmetric the distribution is).

The idea matters in commerce because most real business data are not symmetric. Incomes in a firm, the value of orders received, the number of items sold per customer, the marks in an easy or a hard examination — all typically bunch near one end with a few extreme values stretching the other. Two distributions can share the very same mean and the very same standard deviation yet be shaped quite differently, one leaning left and the other leaning right; only a measure of skewness separates them.

Dispersion and skewness answer different questions. Dispersion asks how scattered the data are and is always a non-negative number. Skewness asks whether the scatter is one-sided and carries a sign: it is $0$ for a symmetric distribution, positive when the longer tail runs to the right (towards the larger values), and negative when the longer tail runs to the left (towards the smaller values). A measure of skewness must be a pure number, free of the units of the data, so that the skewness of, say, a distribution of rupees can be compared directly with the skewness of a distribution of kilograms.