Skewness and Kurtosis
Skew computes the skewness,
Kurt the kurtosis of the values in x.
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, ...)
a numeric vector. An object which is not a vector is coerced (if possible) by
logical, indicating whether
NAvalues should be stripped before the computation proceeds. Defaults to
- integer out of 1, 2 or 3 (default). See Details.
- confidence level of the interval. If set to
NA(which is the default) no confidence interval will be calculated.
- The type of confidence interval required. The value should be any subset
of the values
"all"which would compute all five types of intervals, is not supported).
- 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
Rwould 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.
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(n-1)) / (n-2)
method = 3: b_1 = m_3 / s^3 = g_1 ((n-1)/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) * (n-1) / ((n-2)(n-3))
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(n-1)/((n-2)(n+1)(n+3)) ) ASE.kurt = sqrt( (n^2 - 1)/((n-3)(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
This implementation of the two functions is comparably fast, as the expensive sums are coded in C.
conf.levelis set to
NAthen the result will be then the result will be and if a
conf.levelis provided, a named numeric vector with 3 elements:
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, 183-189.
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)