Bias-Corrected Sample Excess Kurtosis

Computes the sample excess kurtosis of a numeric vector using the bias-corrected (SAS/SPSS Type 2) formula. Excess kurtosis measures tailedness relative to the normal distribution: zero matches a normal, positive values indicate heavier tails (“leptokurtic”), negative values indicate lighter tails (“platykurtic”).

Usage

kurtosis(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 excess kurtosis, or NA_real_ when fewer than four non-missing observations are available or when the sample standard deviation is zero.

Details

The reported value is $$\hat\gamma_2^{(2)} = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)},$$ where $s$ is the (divisor-$n-1$) sample standard deviation. Subtracting the asymptotic correction $3(n-1)^2/((n-2)(n-3))$ centers the statistic at 0 for a normal distribution; that is, excess kurtosis is reported (rather than “raw” kurtosis, which centers at 3).

Why isn't this in base R? See the same Details in skewness: R Core defers higher-order moment statistics to contributed packages, partly because multiple formulas (biased Type 1, bias-corrected Type 2, Minitab Type 3) coexist. DMAR adopts Type 2, the form most common in psychometric reporting and used internally by descriptives.

Diagnostic interpretation. As a rough rule of thumb, $|\mathrm{kurtosis}| > 7$ 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

# Normal data: excess kurtosis near zero.
set.seed(113)
kurtosis(rnorm(1000))
#> [1] -0.1595807

# Heavy-tailed data: positive excess kurtosis.
kurtosis(rt(1000, df = 4))
#> [1] 5.076089

# The classic 1:5 example: bias-corrected excess kurtosis = -1.2.
kurtosis(1:5)
#> [1] -1.2