Computes the skewness.
a numeric vector containing the values whose skewness is to be computed.
a logical value indicating whether
an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below.
x contains missings and these are not removed, the skewness
Otherwise, write x_i for the non-missing elements of
n for their number, mu for their mean, s for
their standard deviation, and
m_r = ∑_i (x_i - mu)^r / n
for the sample moments of order r.
Joanes and Gill (1998) discuss three methods for estimating skewness:
g_1 = m_3 / m_2^(3/2). This is the typical definition used in many older textbooks.
G_1 = g_1 * sqrt(n(n-1)) / (n-2). Used in SAS and SPSS.
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.
The estimated skewness of
D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183–189.