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Measures of Skewness and Kurtosis

NIST/SEMATECH Section 1.3.5.11 Measures of Skewness and Kurtosis

What It Is

Skewness measures the asymmetry of a probability distribution about its mean, while kurtosis measures the heaviness of the distribution tails relative to a normal distribution. Together, they characterize the shape of a distribution beyond what location and scale can describe.

When to Use It

Use skewness and kurtosis to assess whether data conform to distributional assumptions required by many statistical methods. Large skewness suggests the data are not symmetric, which may invalidate mean-based analyses. Excess kurtosis indicates heavier or lighter tails than normal, affecting the reliability of confidence intervals and hypothesis tests that assume normality. These measures guide decisions about data transformations and the selection of appropriate statistical tests.

How to Interpret

A skewness near zero suggests the distribution is approximately symmetric. Values greater than 1 or less than -1 indicate substantial skewness. For kurtosis, values greater than zero (leptokurtic) indicate heavier tails than normal, while values less than zero (platykurtic) indicate lighter tails. A common rule of thumb is that skewness or kurtosis outside the range [-2, 2] warrants concern about normality. The D'Agostino-Pearson omnibus test combines skewness and kurtosis into a single normality test. High kurtosis suggests the data may contain outliers or come from a mixture distribution.

Assumptions and Limitations

Skewness and kurtosis are computed from sample moments and require independent observations from a continuous distribution. They are sensitive to outliers, and their sampling distributions converge slowly, so large samples (n > 50) are preferred for reliable inference. For small samples, graphical methods like probability plots are generally more informative.

Reference: NIST/SEMATECH e-Handbook, Section 1.3.5.11

Formulas

Sample Skewness

g1=1n∑i=1n(xi−xˉ)3[1n∑i=1n(xi−xˉ)2]3/2g_1 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3}{\left[\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\right]^{3/2}}

The ratio of the third central moment to the cube of the standard deviation. Note that s here uses N (not N-1) in the denominator. Positive values indicate right skew; negative values indicate left skew.

Sample Kurtosis (Excess)

g2=1n∑i=1n(xi−xˉ)4[1n∑i=1n(xi−xˉ)2]2−3g_2 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4}{\left[\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\right]^{2}} - 3

The ratio of the fourth central moment to the square of the variance, minus 3 so that the normal distribution has excess kurtosis zero. Positive values indicate heavy tails.

Adjusted Fisher-Pearson Skewness

G1=n(n−1)n−2⋅g1G_1 = \frac{\sqrt{n(n-1)}}{n-2} \cdot g_1

The adjusted skewness coefficient used by many software packages. It corrects the bias of the sample skewness for finite samples.

Standard Error of Skewness

SE(g1)≈6n\text{SE}(g_1) \approx \sqrt{\frac{6}{n}}

An approximate standard error for the sample skewness, used to test whether the skewness is significantly different from zero.