Kurtosis Calculator
Find the kurtosis and excess kurtosis of your data — how heavy the tails are compared with a normal distribution — from a list of values.
How it works
kurtosis = m₄ ÷ m₂² · excess = kurtosis − 3 · m_k = (1/n)·Σ(x−x̄)ᵏ
Kurtosis measures how much of a distribution’s variance comes from rare extreme values. It divides the fourth central moment by the variance squared; a normal curve has a raw kurtosis of exactly 3, so subtracting 3 gives the excess kurtosis. Positive excess (leptokurtic) means heavier tails and a sharper peak, while negative excess (platykurtic) means lighter tails and a flatter middle.
Worked example
For 2, 4, 4, 5, 5, 5, 6, 6, 8, 20 the single large value of 20 fattens the right tail, pushing m₄ up sharply. The raw kurtosis lands well above 3, so the excess kurtosis is strongly positive — a leptokurtic, heavy-tailed shape driven by that outlier.
Sources: Kurtosis — Wikipedia
Frequently asked questions
What is the difference between kurtosis and excess kurtosis?
Raw kurtosis is m₄ ÷ m₂², and a normal distribution scores exactly 3. Excess kurtosis subtracts that 3 so the normal reference sits at 0, making positive and negative departures easy to read. Many software packages report the excess value by default.
Does high kurtosis mean a tall peak?
Not mainly. Modern statistics ties kurtosis to tail weight — the influence of outliers — rather than peakedness. A high-kurtosis distribution has more extreme values than a normal curve; the sharper peak is a side effect, not the definition.
Can kurtosis be negative?
Raw kurtosis is always positive, but excess kurtosis can be negative. A uniform distribution, for example, has excess kurtosis around −1.2 because its tails are lighter than a normal curve’s.
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