Descriptive statistics Updated November 2026

Skewness and kurtosis

Skewness measures whether a distribution leans left or right: positive skew has a long right tail and a mean above the median, negative skew the opposite. Kurtosis measures how heavy the tails are compared with a normal distribution, whose excess kurtosis is 0.

Syntax

Pearson skewness = 3 * (mean - median) / sd

Part	What it is
`mean - median`	positive for right skew, negative for left skew
`sd`	the standard deviation, which scales the measure

Examples

Data: 1, 2, 3, 10
mean = 4, median = 2.5, sample sd = 4.082
skewness = 3 * (4 - 2.5) / 4.082

Result: 1.10 (right skewed)

Excel: =SKEW(A1:A4) gives the moment-based version, 1.76 for this data

Symmetric data: 1, 2, 3, 4, 5
mean = 3, median = 3

Result: skewness = 0

Excess kurtosis of a sample in Excel
=KURT(A1:A100)

Result: about 0 for normal data, positive for heavy tails, negative for light tails

Common mistakes

Reading kurtosis as peakedness; it is driven by the tails and outliers, not by how pointed the center looks.
Comparing Excel KURT (excess kurtosis, normal = 0) with a textbook kurtosis where normal = 3; subtract 3 from the textbook value to compare.
Judging skew from a histogram with too few bins; compute the statistic or compare mean and median.

Practice it

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Questions

What is an acceptable skewness value?

A common rule of thumb treats skewness between -0.5 and 0.5 as roughly symmetric, 0.5 to 1 (or -1 to -0.5) as moderately skewed, and beyond 1 in either direction as highly skewed.

Does right skew mean the mean is bigger than the median?

Usually yes. The long right tail pulls the mean up while the median stays put. Income is the classic example.

Mean, median and modeHow to calculate the mean, median and mode of a data set by hand and in Excel, when each average is the right one, and the mistakes that give a wrong answer.
Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.
Standard deviationThe standard deviation formula, sample versus population versions, a worked example with exact numbers and the Excel functions STDEV.S and STDEV.P.