Kurtosis Coefficient Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
The excess kurtosis coefficient is calculated as g₂ = m₄ ÷ m₂² − 3, the ratio of the fourth to the squared second central moment, minus 3. For the distribution 0, 0, 0, 0, 5 (very peaked, with a heavy tail), g₂ equals exactly 0.25: slightly more "peaked" than a normal distribution.
Explanation
The kurtosis coefficient complements the z-score: where a z-score locates a single value relative to a distribution's mean, kurtosis reveals the shape of the distribution's central "peak" and the weight of its tails (extreme values, far from the mean), across the whole sample. The "−3" in the formula isn't arbitrary: it sets the reference to a normal distribution, whose theoretical excess kurtosis equals exactly 0. A positive excess kurtosis (a so-called leptokurtic distribution) signals a distribution more peaked at the center than a normal one, with heavier tails — more values very close to the mean, but also more values very far from it, at the expense of intermediate values. A negative excess kurtosis (a so-called platykurtic distribution) signals the opposite: a flatter profile, closer to a uniform distribution like (1, 2, 3, 4, 5), whose excess kurtosis here equals exactly −1.3. This contrast between the two examples illustrates the measure well: the five uniformly spread values (1 to 5) produce a flat shape with no marked peak or tail, while the distribution concentrated at 0 with a single value isolated at 5 creates both a sharp peak (four identical values) and a heavy tail (the isolated value) — exactly the profile a positive kurtosis is meant to detect. In practice, a very high excess kurtosis often signals the presence of outliers worth examining before applying a statistical test that assumes normally distributed data, such as a Pearson correlation.
Example: the distribution 0, 0, 0, 0, 5
Inputs
Values: 0, 0, 0, 0, 5.
Calculation
Mean = (0+0+0+0+5) ÷ 5 = 1. m₂ = mean of (xᵢ−1)² = (1+1+1+1+16) ÷ 5 = 4. m₄ = mean of (xᵢ−1)⁴ = (1+1+1+1+256) ÷ 5 = 52. g₂ = 52 ÷ 4² − 3 = 52 ÷ 16 − 3 = 3.25 − 3 = 0.25.
Result
This distribution has an excess kurtosis coefficient of 0.25, slightly above that of a normal distribution (0), consistent with its peaked shape and heavy tail.
Frequently asked questions
What does an excess kurtosis of zero mean?
An excess kurtosis of 0 corresponds exactly to that of a theoretical normal distribution, the reference used by this formula (the "−3" in the formula exists precisely to recenter the scale on this reference). This isn't proof, however, that the data actually follows a normal distribution: other symmetric, non-normal distributions can also show a kurtosis close to 0, just as a skewness coefficient close to 0 isn't sufficient proof on its own either.
What is the difference between skewness and kurtosis?
Skewness (the 3rd moment) measures whether a distribution leans to the left or right; kurtosis (the 4th moment) instead measures the shape of the central peak and the weight of extreme values, independent of any left-right imbalance. A perfectly symmetric distribution can very well have a kurtosis quite different from 0, as shown by the two examples on this page.
Why does this calculator require at least 4 values?
Because kurtosis relies on the fourth central moment (m₄), a quantity that loses all statistical meaning below 4 values — unlike skewness, which remains computable, though unreliable, from just 3 values. This minimum of 4 values is in any case still far too small for a reliable estimate in practice: kurtosis is a sensitive measure, generally not very meaningful on very small samples.