R: Skewness and Kurtosis

Measures of Shape {DescTools} | R Documentation

## Skewness and Kurtosis

### Description

`Skew` computes the skewness, `Kurt` the kurtosis of the values in x.

### Usage

    Skew(x, weights = NULL, na.rm = FALSE, method = 3, conf.level = NA,
         ci.type = "bca", R = 1000, ...)

    Kurt(x, weights = NULL, 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`.
`weights` | a numerical vector of weights the same length as `x` giving the weights to use for elements of `x`.
`na.rm` | logical, indicating whether `NA` values should be stripped before the computation proceeds. Defaults to `FALSE`.
`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(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 `"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

`` | a single numeric value

and if a `conf.level` is provided, a named numeric vector with 3 elements:

`skew, kurt` | the specific estimate, either skewness or kurtosis
`lwr.ci` | lower bound of the confidence interval
`upr.ci` | upper bound of the confidence interval

### Author(s)

Andri Signorell <andri@signorell.net>, David Meyer <david.meyer@r-project.org> (method = 3)

### 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, 183-189.

### See Also

`mean`, `sd`, similar code in `library(e1071)`

### 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)

[Package DescTools version 0.99.35 Index]