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:

Population Skewness Formula

Population skewness formula

Sample Skewness Formula

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):

Skewness formula - moments Skewness formula - third moment and standard deviation

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:

Deviation from the mean for i-th observation

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:

Third moment formula

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.

Second moment formula Variance formula Standard deviation formula

Skewness Formula

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

Skewness formula (using standard deviation) Skewness formula (applying the standard deviation formula) Skewness formula (directly from deviations from the mean)

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):

Sample skewness formula (using sample standard deviation)

Where:

n = sample size

s = sample standard deviation:

Sample standard deviation formula

Therefore:

Sample skewness formula (applying sample standard deviation formula) Sample skewness formula (directly from deviations from the mean)

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:

Sample skewness formula for large sample size

And therefore approximately equals population skewness formula:

Population skewness formula

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