Skewness

Computes the skewness.

skewness(x, na.rm = FALSE, type = 3)

Arguments

x	a numeric vector containing the values whose skewness is to be computed.
na.rm	a logical value indicating whether NA values should be stripped before the computation proceeds.
type	an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below.

Details

If x contains missings and these are not removed, the skewness is NA.

Otherwise, write \(x_i\) for the non-missing elements of x, \(n\) for their number, \(\mu\) for their mean, \(s\) for their standard deviation, and \(m_r = \sum_i (x_i - \mu)^r / n\) for the sample moments of order \(r\).

Joanes and Gill (1998) discuss three methods for estimating skewness:

Type 1:

\(g_1 = m_3 / m_2^{3/2}\). This is the typical definition used in many older textbooks.

Type 2:

\(G_1 = g_1 \sqrt{n(n-1)} / (n-2)\). Used in SAS and SPSS.

Type 3:

\(b_1 = m_3 / s^3 = g_1 ((n-1)/n)^{3/2}\). Used in MINITAB and BMDP.

All three skewness measures are unbiased under normality.

Value

The estimated skewness of x.

References

D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183--189.

Examples

x <- rnorm(100)
skewness(x)
#> [1] -0.2006999