Computes the skewness.
1 |
x |
a numeric vector containing the values whose skewness is to be computed. |
na.rm |
a logical value indicating whether |
type |
an integer between 1 and 3 selecting one of the algorithms for computing skewness detailed below. |
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 = ∑_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 x
.
D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183–189.
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