# 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 Descriptive Statistics Excel Calculator.

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

## 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):

## 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:

## 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:

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

## 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:

## Kurtosis Formula

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

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

## 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:

Where:

n = sample size

s = sample standard deviation

s2 = sample variance:

Therefore sample kurtosis equals:

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:

And therefore approximately equals 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):

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:

And therefore approximately equals population excess kurtosis formula:

You can easily calculate kurtosis, skewness, and other measures in Excel using the Descriptive Statistics Excel Calculator.

## Related pages

form 13f secvix optionquote vxxxxv etnthe greeks in financeassumptions of black scholesblack scholes calculatorcboe trading hourshistorical volatility formulablacksholes modelspot markets versus futures marketsetf etcmoment coefficient of kurtosisstraddle options examplewhat is the cboe volatility indexprice weighted index calculationinverse vxxcovered call payoffcumulative function in excelannualized volatility formulashort put option payoffhow to calculate cdf in excelcalculate the variance in excelblack sholes equationeuro stoxx 50 wikiformula sharpe ratioema buutcalculate variance excelscholes option pricing modelexponential moving average definitiondelta formula mathblack scholes model in excelvelocityshares dailydefine skewnessmsft financecost of equity formula waccvar value at risk formulalong short equity hedge fundsdefine sharpe ratioyahoo msftwacc market valueforex volatility chartwhat is straddle optiondisadvantages of arithmetic meandemerits of meanweighted average method calculatorkurtosis graphestimated standard deviation calculatoraveraging percentages calculatorwarren buffett sec filingssharpes ratiologarithm calculator step by stepcfe cboeapple stock closing price historyexcel formulas minusstandard deviation calculatorsvix futures contract specificationsaapl earnings historyipath etncost of ending inventory formulawhat does kirt meanleveraged volatility etfdownloading stock prices from yahoo finances&p futures bloombergdark notepadinverse sp500 etfskewness of a distributionskewness and kurtosis formulaarithmetic mean formulasnotepad backgroundwhat is svxyweighted average cost accountingput option black scholesrsi calculationpercentile coefficient of kurtosiscalculating standard deviation calculatormethods of calculating var