# kurtosis: Kurtosis In e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien

## Description

Computes the kurtosis.

## Usage

 `1` ```kurtosis(x, na.rm = FALSE, type = 3) ```

## Arguments

 `x` a numeric vector containing the values whose kurtosis is to be computed. `na.rm` a logical value indicating whether `NA` values should be stripped before the computation proceeds. `type` an integer between 1 and 3 selecting one of the algorithms for computing kurtosis detailed below.

## Details

If `x` contains missings and these are not removed, the kurtosis is `NA`.

Otherwise, write x_i for the non-missing elements of `x`, n for their number, mu for their mean, s for their standard deviation, and m_r = ∑_i (x_i - mu)^r / n for the sample moments of order r.

Joanes and Gill (1998) discuss three methods for estimating kurtosis:

Type 1:

g_2 = m_4 / m_2^2 - 3. This is the typical definition used in many older textbooks.

Type 2:

G_2 = ((n+1) g_2 + 6) * (n-1) / ((n-2)(n-3)). Used in SAS and SPSS.

Type 3:

b_2 = m_4 / s^4 - 3 = (g_2 + 3) (1 - 1/n)^2 - 3. Used in MINITAB and BMDP.

Only G_2 (corresponding to `type = 2`) is unbiased under normality.

## Value

The estimated kurtosis of `x`.

## References

D. N. Joanes and C. A. Gill (1998), Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183–189.

## Examples

 ```1 2``` ```x <- rnorm(100) kurtosis(x) ```

### Example output

``` 0.5231317
```

e1071 documentation built on May 23, 2021, 5:07 p.m.