StatReason

Skewness and Kurtosis Calculator

Describe a dataset's asymmetry and tail shape numerically, complementing a visual check with the histogram.

At least 4 values are required (kurtosis needs a minimum sample size to be defined).

What this answers

Skewness answers "does this data lean toward one side, with a longer tail in one direction?" A positive skewness means a longer tail toward higher values; negative means a longer tail toward lower values; near zero suggests rough symmetry. Excess kurtosis answers a different question: "are this distribution's tails heavier or lighter than a normal distribution's?" Positive excess kurtosis means heavier tails and a sharper peak than normal; negative means lighter tails and a flatter peak.

How it is calculated

Both statistics use the bias-corrected sample formulas (the same conventions R's default methods and most statistical software use), based on standardized deviations from the mean raised to the third power for skewness and the fourth power for kurtosis. Excess kurtosis subtracts 3 from raw kurtosis so that a normal distribution's excess kurtosis is exactly 0, making departures from normal directly readable from the sign and size of the number.

Worked example

For the values 1 through 5, a perfectly symmetric set: skewness comes out to exactly 0, confirming no lean toward either tail. Compare that against a set like 1, 2, 2, 3, and 20, where the single extreme value 20 pulls skewness clearly positive, quantifying what a histogram of the same data would show visually as a long right tail.

Assumption audit

Calculated from your data: the exact skewness and excess kurtosis values, using every value you entered.
Evidence to review: both statistics are descriptive evidence about the shape of your specific sample, not a formal test. Pair them with the Histogram Maker or the Data Normality Checker for a fuller picture before concluding anything about normality.
You must verify: that your sample is large enough for these statistics to be stable; with a small n, both skewness and kurtosis can be strongly influenced by a single unusual value.

What this result does not mean

Neither skewness near 0 nor excess kurtosis near 0 proves your data is normally distributed; they only fail to detect two particular kinds of departure from normal. Conversely, a nonzero value describes your specific sample and does not by itself prove the underlying population is non-normal, especially with a small sample.

Limitations

Both statistics can be unstable and sensitive to individual extreme values in a small sample. Use them alongside a visual check, not as a standalone verdict on your data's shape.