Bias-Corrected Sample Skewness

Computes the sample skewness of a numeric vector using the bias-corrected (SAS/SPSS Type 2) formula. Skewness measures asymmetry of the distribution: zero is symmetric, positive values indicate a right-tail heavier than the left, negative values the reverse.

Usage

skewness(x, na_rm = TRUE)

Arguments

x: A numeric vector.
na_rm: Logical. If TRUE (the default), missing values are removed before computation. If FALSE, the result is NA when x contains any NA.

Value

A single numeric value: the bias-corrected sample skewness, or NA_real_ when fewer than three non-missing observations are available or when the sample standard deviation is zero.

Details

The reported value is $$\hat\gamma_1^{(2)} = \frac{n}{(n-1)(n-2)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^3,$$ where $s$ is the (divisor-$n-1$) sample standard deviation. This is sometimes called the “Type 2” or SAS/SPSS-default form; it is approximately unbiased under normality.

Why isn't this in base R? R Core has historically deferred higher-order moment statistics to contributed packages, in part because three popular formulas exist (biased Type 1, bias-corrected Type 2, and Minitab Type 3) and choosing a default would be opinionated. DMAR adopts Type 2, which is the form most often used in psychometric reporting and the one already used internally by descriptives.

Diagnostic interpretation. As a rough rule of thumb, $|\mathrm{skewness}| > 2$ is sometimes flagged as indicative of departures from normality large enough to threaten normal-theory inference (e.g., maximum likelihood estimation in factor analysis or structural equation modeling).

References

Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47(1), 183–189. doi:10.1111/1467-9884.00122

Author

Ken Kelley kkelley@nd.edu

Examples

# Symmetric data: skewness near zero.
set.seed(113)
skewness(rnorm(1000))
#> [1] -0.03196212

# Right-skewed data: positive value.
skewness(rexp(1000, rate = 1))
#> [1] 1.735017

# The classic 1:5 example: exactly symmetric (returns 0).
skewness(1:5)
#> [1] 0