# Skewness Formula

This page provides and explains the formula for skewness and sample skewness. Skewness is one of the summary statistics; it is used for describing or estimating symmetry of a distribution (relative frequency of positive and negative extreme values). Skewness is very important in portfolio management, risk management, and option pricing and trading. You can easily calculate skewness 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 Skewness Formula

Detailed derivation and explanation of the formulas follows.

## Skewness Definition

Skewness is the ratio of (1) the third moment and (2) the second moment raised to the power of 3/2 (= the ratio of the third moment and standard deviation cubed):

## Deviations from the Mean

For calculating skewness, 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:

## Third Moment Formula

The third moment about the mean is the sum of each value’s deviation from the mean cubed, which (the whole sum) is then divided by the number of values:

## 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. This is the same formula as the one you probably know as variance (σ2). Variance is standard deviation (σ) squared.

## Skewness Formula

The direct skewness formula (ratio of the third moment and standard deviation cubed) therefore is:

## Sample Skewness Formula

The formulas above are for population skewness (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 skewness from a sample.

When calculating sample skewness, you need to make a small adjustment to the skewness formula (the function of the adjustment is to correct a bias inherent in small samples):

Where:

n = sample size

s = sample standard deviation:

Therefore:

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

And therefore approximately equals population skewness formula:

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

## Related pages

vix inverse etf 2xexcel 95 percentilemonospace font listhow to calculate stdev in excelskewness interpretationstandard deviation from mean calculatorstock volatility calculatorhow to calculate market value of debt for waccvxx barclayskurtosis examplefinance msftetf swiss franchistorical implied volatility charthedge fund definition examplewhats a good sharpe ratiost dev calccboe futuresnatural log in excellogarithm calculator step by stepschole greek definitionreverse rsi calculatorblack scholes equation explainedwhat is hedge fund with examplevariance calcpopulation covariance formulayahoo finance eurvolatile stocks meanings&p etf shortstandard deviation calculator frequencyportfolio management advantages and disadvantagesblack scholes option pricing model examplecalculating the intrinsic value of a stockvix trading hourshow to calculate 95th percentile in excelpro shares ultra shortvariance statssec 13f securitiesfx option valuationhow to calculate variance using excelcalculate sharpe ratio excelexcel computation formulahow to write ln in excelportfolio sharpe ratioetf for vixstdev excelmorgan stanley historical stock pricesquaring negativesformula for skewnessblack scholes option pricing model formulashort uvxyvix historical charthow to calculate rsi of stockput spread payoffimplied volatility meaningannualization calculatortheta call optionwhat is meant by weighted average cost of capital wacchow to calculate mean and standard deviation on excelcalculate sigma in excelmeasure of skewness formulaarithmetic mean in excelstatistics kurtosiscalculate standard deviation excelimplied volatility meaningsec hedge fund filingsmomentum indicators technical analysisstandard deviation formula financekurtosicalculation of waccoptions straddle calculatorhow to calculate variation in excelkurtosis is a measure ofkurtosis calculationsample variance formula in excelstandard deviation excel formulacall option payoff graphcalculate standard deviation with excelparametric var formulaexcel calculations formulas