Skewness and Kurtosis

Describes the shape of a distribution with two numbers. Skewness measures lopsidedness: positive means a long right tail, negative a long left tail. Kurtosis measures tail weight against the normal distribution, which sits at 0; larger values mean outliers are more likely.

A mean and a standard deviation do not pin down the shape of a distribution. Two data sets can share both and still look nothing alike, one leaning to the right and the other carrying heavy tails full of outliers. Skewness and kurtosis put numbers on that shape.

g_1 = \dfrac{1}{n}\sum \left(\dfrac{x_i - \bar{x}}{s}\right)^{3}

g_2 = \dfrac{1}{n}\sum \left(\dfrac{x_i - \bar{x}}{s}\right)^{4} - 3

$g_1$ is the skewness and $g_2$ the kurtosis
$x_i$ is each value
$\bar{x}$ is the mean
$s$ is the standard deviation
$n$ is the number of values

Each deviation is divided by the standard deviation before being cubed or raised to the fourth power. That makes both figures unit-free: measuring in centimetres or in metres gives the same answer.

Cubing keeps the sign, so a value far out to the right pushes skewness positive and one far to the left pushes it negative. A symmetric distribution cancels out to 0. The fourth power destroys the sign, so any value far from the centre, on either side, raises kurtosis. A normal distribution scores exactly 3 there, so 3 is subtracted to put the normal case at 0.

Example

Take the default data 2, 3, 3, 4, 4, 4, 5, 6, 9, 15.

The mean is 5.5 and the standard deviation about 3.6674. Skewness comes out at about 1.6543, strongly positive. Looking at the numbers, most cluster between 2 and 6 while 9 and 15 sit far out to the right. That right tail is what lifts the skewness.

Kurtosis is about 1.7200. Being above the normal distribution's 0 means heavier tails, so outliers are more likely here than under a normal curve.

When skewness is positive the mean sits above the median. Here the mean is 5.5 while the median is 4, the mean having been dragged rightwards by the tail.

Points to watch

These are the population formulas, dividing plainly by $n$ . Versions exist that correct for estimating a population from a sample, and spreadsheet functions generally use those, so the same data can produce different figures elsewhere.

Neither figure means much on small samples. Because deviations are cubed and raised to the fourth power, a single outlier can decide the answer almost by itself. Even with these 10 values, dropping the 15 changes the skewness dramatically. Treat them as a rough guide, and prefer at least a few dozen observations.

If every value is identical there is no spread to divide by, and neither figure exists.