Skewness and Kurtosis Calculator
Measure how lopsided (skewness) and how heavy-tailed (kurtosis) your data are, with the same formulas Excel, SPSS and most textbooks use.
Formulas used
G₁ = n / ((n−1)(n−2)) · Σ((x − x̄)/s)³
G₂ = n(n+1) / ((n−1)(n−2)(n−3)) · Σ((x − x̄)/s)⁴ − 3(n−1)² / ((n−2)(n−3))
These are the sample statistics reported by Excel (SKEW, KURT), SPSS and SAS, so your numbers will match software output. Standard errors use the exact small-sample formulas: SE(G₁) = √(6n(n−1) / ((n−2)(n+1)(n+3))).
Skewness and kurtosis summarise what a histogram shows. With fewer than about 30 values they are very noisy, so look at the plot and the box plot too, rather than relying on the numbers alone.
Frequently asked questions
What does skewness tell you?
Skewness measures asymmetry. Positive (right) skew means a long tail of large values, so the mean is usually above the median, as with incomes or waiting times. Negative (left) skew means a long tail of small values, as with scores on an easy test. Zero means symmetric.
What is a normal range for skewness and kurtosis?
A common rule of thumb (Bulmer, 1979) calls |skewness| < 0.5 roughly symmetric, 0.5–1 moderately skewed and > 1 highly skewed. For normality checks many texts accept skewness and excess kurtosis between −2 and +2 (George & Mallery); another approach divides each by its standard error and treats |z| > 1.96 as significant.
What is the difference between kurtosis and excess kurtosis?
A normal distribution has kurtosis 3. Excess kurtosis subtracts 3 so that normal = 0. Excel’s KURT, SPSS and this page’s headline value report excess kurtosis; the population kurtosis b₂ shown below is the raw version.
Why do Excel and other tools give slightly different skewness?
There are several formulas. Excel SKEW, SPSS and SAS use the adjusted Fisher–Pearson coefficient G₁ (the headline here). Excel SKEW.P and many textbooks’ population formula give g₁. R’s e1071 package defaults to yet another variant. They converge as n grows.
Does high kurtosis mean a sharper peak?
Not really. Kurtosis is driven almost entirely by the tails, meaning how often extreme values occur, not by the shape of the peak (Westfall, 2014). Read high kurtosis as “more outliers than a normal distribution would produce.”