Kurtosis Formula

This page provides and explains the formula for kurtosis, excess kurtosis, sample kurtosis, and sample excess kurtosis. Kurtosis is one of the summary statistics; it is used for describing or estimating a distribution’s peakedness and frequency of extreme values. You can easily calculate kurtosis in Excel using the Summary Statistics Calculator & Charts Spreadsheet.

If you don’t want to go through the lengthy derivation and explanation below, the formulas are here:

Population Kurtosis Formula

Population kurtosis formula

Sample Kurtosis Formula

Sample kurtosis formula

Detailed derivation and explanation of the formulas follows.

Kurtosis Definition

Kurtosis is the ratio of (1) the fourth moment and (2) the second moment squared (= the ratio of the fourth moment and variance squared):

Kurtosis formula - moments

Kurtosis formula - fourth moment and variance

Deviations from the Mean

For calculating kurtosis, you first need to calculate each observation’s deviation from the mean (the difference between each value and arithmetic average of all values). The deviation from the mean for ith observation equals:

Deviation from the mean for i-th observation

Second Moment Formula

The second moment about the mean is the sum of each value’s squared deviation from the mean, divided by the number of values:

Second moment formula

It is the same formula as the one you probably know as variance (σ2):

Population variance formula

Fourth Moment Formula

The fourth moment about the mean is the sum of each value’s deviation from the mean raised to the power of 4, which (the whole sum) is then divided by the number of values:

Fourth moment formula

Kurtosis Formula

The direct kurtosis formula (ratio of the fourth moment and the second moment squared) therefore is:

Kurtosis formula (intermediate step)

The n’s in the denominators cancel out and this is the final nice version of population kurtosis formula:

Kurtosis formula

Excess Kurtosis Formula

Very often kurtosis is quoted in the form of excess kurtosis (kurtosis relative to normal distribution kurtosis). Excess kurtosis is simply kurtosis less 3. The excess kurtosis formula therefore is:

Excess kurtosis formula

Sample Kurtosis Formula

The kurtosis and excess kurtosis formulas above are for population kurtosis (when your data set includes the whole population). Very often, you don’t have data for the whole population and you need to estimate population kurtosis from a sample.

When calculating sample kurtosis, you need to make a small adjustment to the kurtosis formula:

Sample kurtosis formula

Where:

n = sample size

s = sample standard deviation

s2 = sample variance:

Sample variance formula

Therefore sample kurtosis equals:

Sample kurtosis formula after including variance

Sample kurtosis formula

For a very large sample (very high n), the differences between and among n+1, n, n-1, n-2, and n-3 are becoming negligible, and the sample kurtosis formula approximately equals:

Sample kurtosis formula for very large sample

And therefore approximately equals population kurtosis formula:

Population kurtosis formula

Sample Excess Kurtosis Formula

Sample excess kurtosis formula differs from sample kurtosis formula only by adding a little at the end (adjusting the minus 3 for a sample):

Sample excess kurtosis formula

For a very large sample, the differences between and among n+1, n, n-1, n-2, and n-3 are becoming negligible, and the sample excess kurtosis formula approximately equals:

Sample excess kurtosis formula for very large sample

And therefore approximately equals population excess kurtosis formula:

Population excess kurtosis formula

You can easily calculate kurtosis, skewness, and other measures in Excel using the Summary Statistics Calculator & Charts Spreadsheet.