KURTOSIS

Purpose:

Calculates the kurtosis of a series.

Syntax:

KURTOSIS(series, bias, excess, vardef)

series

A series. The z value.

bias

Optional. An integer, the bias calculation. Valid methods are:

0:	compute unbiased estimate (default)
1:	compute biased estimate

excess

Optional. A real. Excess normalization subtracted from the base calculation. Defaults to 3.0.

vardef

Optional. An integer, the variance normalization method.

0:	1 / (N-1)
1:	1 / N

where N is the length of the input series. Defaults to 1/(N-1) for the unbiased computation and 1/N for the biased computation.

Returns:

A scalar, the kurtosis of the series.

Example:

W1: {3, 4, 5, 2, 3, 4, 5, 6, 4, 7}

W2: {kurtosis(w1)}

W3: {kurtosis(w1, 1)}

W2 == {-0.151800}, the unbiased or population kurtosis.

W3 == {-0.631321}, the biased kurtosis.

Remarks:

The fourth standardized moment or raw kurtosis is defined as:

$image\kurtosis01.gif$

where μ₄ is the fourth moment about the mean and σ is the standard deviation.

The excess kurtosis is defined as:

$image\kurtosis02.gif$

The value 3 is subtracted from the raw value to set the kurtosis of a normal distribution to 0.

The biased or sample kurtosis is computed as:

$image\kurtosis04.gif$

where the arithmetic mean is defined as:

$image\std02.gif$

The unbiased or population kurtosis is computed as:

$image\kurtosis03.gif$

By default, KURTOSIS computes the unbiased or population kurtosis as used in SAS, SPSS, and Excel. To calculate the biased kurtosis, set the bias flag to 1.

Set excess to 0.0 to compute the raw kurtosis.

Kurtosis characterizes the relative peakedness or flatness of a distribution compared with the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

A distribution with a high peak is called leptokurtic, a flat-topped curve is called platykurtic, and the normal distribution is called mesokurtic.