Kurtosis Calculator
How heavy your distribution's tails are — the fourth moment, reported as excess kurtosis so that a normal distribution reads zero.
Kurtosis is the average fourth power of the z-scores.
How the kurtosis calculator works
Kurtosis is the average fourth power of the z-scores. Because the fourth power grows so fast, it is dominated by the extremes: kurtosis measures tail weight and the propensity for outliers, not "peakedness", despite the common description.
A normal distribution has kurtosis 3, so excess kurtosis subtracts it. Positive excess means fatter tails than normal — more extreme events than a normal model would predict, which is the central fact of financial risk.
Formula: excess kurtosis = (1/n) Σ((xᵢ − x̄)/σ)⁴ − 3
Worked examples
| Inputs | Excess kurtosis (sample) | Note |
|---|---|---|
| With one far value | 7.331113 | strongly positive excess |
| A uniform-ish spread | -1.2 | negative excess — light tails |
| Two clumps | -2.8 | the most light-tailed shape there is |
FAQFrequently asked questions
What does kurtosis measure?
Tail weight — how prone the distribution is to extreme values. It is often described as peakedness, which is misleading.
What is excess kurtosis?
Kurtosis minus 3, so that a normal distribution reads exactly zero and the sign tells you which way you differ from normal.
What does positive excess mean?
Fatter tails than normal: extreme observations happen more often than a normal model would predict. Daily stock returns are the standard example.
Why does that matter?
Because risk models built on normal assumptions systematically underestimate the chance of large losses. The 1987 crash was a 20-sigma event under a normal model.
What is the minimum possible?
−2, achieved by data split evenly between two values. There is no upper limit.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.