kurtosis: Calculate excess kurtosis
In rockchalk: Regression Estimation and Presentation

kurtosis

R Documentation

Calculate excess kurtosis

Description

Kurtosis is a summary of a distribution's shape, using the Normal distribution as a comparison. A distribution with high kurtosis is said to be leptokurtic. It has wider, "fatter" tails and a "sharper", more "peaked" center than a Normal distribution. In a standard Normal distribution, the kurtosis is 3. The term "excess kurtosis" refers to the difference kurtosis - 3. Many researchers use the term kurtosis to refer to "excess kurtosis" and this function follows suit. The user may set excess = FALSE, in which case the uncentered kurtosis is returned.

Usage

kurtosis(x, na.rm = TRUE, excess = TRUE, unbiased = TRUE)

Arguments

`x`	A numeric variable (vector)
`na.rm`	default TRUE. If na.rm = FALSE and there are missing values, the mean and variance are undefined and this function returns NA.
`excess`	default TRUE. If true, function returns excess kurtosis (kurtosis -3). If false, the return is simply kurtosis as defined above.
`unbiased`	default TRUE. Should the denominator of the variance estimate be divided by N-1, rather than N?

Details

If kurtosis is smaller than 3 (or excess kurtosis is negative), the tails are "thinner" than the normal distribution (there is lower chance of extreme deviations around the mean). If kurtosis is greater than 3 (excess kurtosis positive), then the tails are fatter (observations can be spread more widely than in the Normal distribution).

The kurtosis may be calculated with the small-sample bias-corrected estimate of the variance. Set unbiased = FALSE if this is not desired. It appears somewhat controversial whether this is necessary. According to the US NIST, http://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm, kurtosis is defined as

kurtosis = ( mean((x - mean(x))^4) )/ var(x)^2

where var(x) is calculated with the denominator N, rather than N-1.

A distribution is said to be leptokurtic if it is tightly bunched in the center (spiked) and there are long tails. The long tails reflect the probability of extreme values.