Measures of Shape
Skewness and Kurtosis
Skew
computes the skewness, Kurt
the kurtosis of the values in x.
 Keywords
 math
Usage
Skew(x, na.rm = FALSE, method = 3, conf.level = NA, ci.type = "bca", R = 1000, ...)
Kurt(x, na.rm = FALSE, method = 3, conf.level = NA, ci.type = "bca", R = 1000, ...)
Arguments
 x

a numeric vector. An object which is not a vector is coerced (if possible) by
as.vector
.  na.rm

logical, indicating whether
NA
values should be stripped before the computation proceeds. Defaults toFALSE
.  method
 integer out of 1, 2 or 3 (default). See Details.
 conf.level
 confidence level of the interval. If set to
NA
(which is the default) no confidence interval will be calculated.  ci.type
 The type of confidence interval required. The value should be any subset
of the values
"classic"
,"norm"
,"basic"
,"stud"
,"perc"
or"bca"
("all"
which would compute all five types of intervals, is not supported).  R
 The number of bootstrap replicates. Usually this will be a single positive integer. For importance resampling,
some resamples may use one set of weights and others use a different set of weights. In this case
R
would be a vector of integers where each component gives the number of resamples from each of the rows of weights.  ...
 the dots are passed to the function
boot
, when confidence intervalls are calculated.
Details
If na.rm
is TRUE
then missing values are removed before computation proceeds.
The methods for calculating the skewness can either be:
method = 1: g_1 = m_3 / m_2^(3/2)
method = 2: G_1 = g_1 * sqrt(n(n1)) / (n2)
method = 3: b_1 = m_3 / s^3 = g_1 ((n1)/n)^(3/2)
and the ones for the kurtosis:
method = 1: g_2 = m_4 / m_2^2  3
method = 2: G_2 = ((n+1) g_2 + 6) * (n1) / ((n2)(n3))
method = 3: b_2 = m_4 / s^4  3 = (g_2 + 3) (1  1/n)^2  3
method = 1 is the typical definition used in many older textbooks. method = 2 is used in SAS and SPSS. method = 3 is used in MINITAB and BMDP.
Cramer et al. (1997) mention the asymptotic standard error of the skewness, resp. kurtosis:
ASE.skew = sqrt( 6n(n1)/((n2)(n+1)(n+3)) ) ASE.kurt = sqrt( (n^2  1)/((n3)(n+5)) )to be used for calculating the confidence intervals. This is implemented here with
ci.type="classic"
. However, Joanes and Gill (1998) advise against this approach, pointing out that the normal assumptions would virtually always be violated.
They suggest using the bootstrap method. That's why the default method for the confidence interval type is set to "bca"
. This implementation of the two functions is comparably fast, as the expensive sums are coded in C.
Value

If
conf.level
is set to NA
then the result will be
then the result will be
and
if a conf.level
is provided, a named numeric vector with 3 elements:References
Cramer, D. (1997): Basic Statistics for Social Research Routledge.
Joanes, D. N., Gill, C. A. (1998): Comparing measures of sample skewness and Kurt. The Statistician, 47, 183189.
See Also
Examples
Skew(d.pizza$price, na.rm=TRUE)
Kurt(d.pizza$price, na.rm=TRUE)
# use sapply to calculate skewness for a data.frame
sapply(d.pizza[,c("temperature","price","delivery_min")], Skew, na.rm=TRUE)
# or apply to do that columnwise with a matrix
apply(as.matrix(d.pizza[,c("temperature","price","delivery_min")]), 2, Skew, na.rm=TRUE)