kurtosis function

Calculate excess kurtosis

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.

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?

Returns

A scalar value or NA

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

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.

Author(s)

Paul Johnson pauljohn@ku.edu