Kurtosis Calculator

Measure the tail weight of a distribution relative to a normal distribution.

About Kurtosis Calculator

A normal distribution has kurtosis of exactly 3, so excess kurtosis measures deviation from it. Positive excess kurtosis means extreme values occur more often than a normal model would predict. Financial returns have strong positive excess kurtosis, which is why normal-distribution risk models systematically underestimate the frequency of crashes.

Formula

Kurtosis

Kurtosis = fourth central moment ÷ σ⁴; excess kurtosis subtracts 3

excess kurtosis: Zero for a normal distribution, positive for heavier tails

Worked example

With the default values below, Kurtosis Calculator returns Excess Kurtosis of 3.72000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Fourth central momentminimum 0: 4,200
Standard deviationminimum 0: 5
Sample sizeminimum 4: 200

Results

Excess Kurtosis: 3.72000
Kurtosis: 6.72000
Classification: Leptokurtic — heavier tails than normal
Standard Error of Kurtosis: 0.34641
Tail Risk Implication: Extreme events are substantially more likely than a normal model predicts

Step by step

1Values used
2Kurtosis
3Excess Kurtosis= 3.72000
4Kurtosis= 6.72000
5Classification= Leptokurtic — heavier tails than normal
6Standard Error of Kurtosis= 0.34641
7Tail Risk Implication= Extreme events are substantially more likely than a normal model predicts

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is Kurtosis calculated?

Kurtosis = fourth central moment ÷ σ⁴; excess kurtosis subtracts 3 A normal distribution has kurtosis of exactly 3, so excess kurtosis measures deviation from it. Positive excess kurtosis means extreme values occur more often than a normal model would predict.

Why does Kurtosis matter?

Financial returns have strong positive excess kurtosis, which is why normal-distribution risk models systematically underestimate the frequency of crashes.

What values do I need to enter?

This calculator takes 3 inputs: Fourth central moment, Standard deviation, Sample size. The pre-filled defaults are a realistic starting point — replace them with figures from your own environment for a result you can act on.