Skewness Calculator - Distribution Shape

Skewness tells you which direction a distribution leans. Positive skew, meaning the right tail is longer, tends to pull the mean above the median. Income data, reaction times and prices nearly always look like this. Negative skew pushes the mean below the median, which is what you see in age-of-death in developed countries and in scores on an easy exam.

The formula used here is the Fisher-Pearson standardized moment:

g1 = [sum((xi - x-bar)^3) / n] / sigma^3

where sigma is the population standard deviation. As a rough guide: |g1| below 0.5 is approximately symmetric. Between 0.5 and 1 is moderately skewed. Above 1 is highly skewed. These thresholds appear in Bulmer’s 1965 Principles of Statistics and are widely cited, though not universal.

Note that Excel SKEW() and R’s e1071 type-2 return a sample-adjusted version, G1, which relates to the g1 above by a factor of sqrt(n(n-1))/(n-2). At n=5 that factor is 1.49, so Excel will report a skew half again as large as this page does for the same numbers. At n=10 it is 1.19, and by n=30 it has fallen to 1.05 and stopped mattering. Neither is wrong; they estimate slightly different things, and the population formula used here is the one most textbooks introduce first. If you are reconciling against a spreadsheet, that factor is the whole discrepancy.

This calculator also returns excess kurtosis: g2 = [sum((xi - x-bar)^4) / n] / sigma^4 - 3. A normal distribution has g2 = 0. Positive kurtosis (leptokurtic) means fatter tails and a sharper peak than normal, which financial returns show reliably. Negative kurtosis (platykurtic) means thinner tails and a flatter distribution.

Neither skewness nor kurtosis alone tells you the distribution is normal. That needs a Shapiro-Wilk test or similar. But together they catch most obvious departures from normality in a quick scan.

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