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)
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? |
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.
A scalar value or NA
Paul Johnson <pauljohn@ku.edu>
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