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3.2 Empirical Distributions 35

Although the variance has the disadvantage of not sharing the dimension
of the original data, it is extensively used in may applications instead of the
standard deviation.
Furthermore, both skewness and kurtosis can be used to describe the
shape of a frequency distribution. Skewness is a measure of asymmetry of
the tails of a distribution. The most popular way to compute the asymmetry
of a distribution is Pearson·s mode skewness:

skewness = (mean-mode) / standard deviation

A negative skew indicates that the distribution is spread out more to the left
of the mean value, assuming increasing values on the axis to the right. The
sample mean is smaller than the mode. Distributions with positive skew-
ness have large tails that extend to the right. The skewness of the symmetric
normal distribution is zero. Although Pearson·s measure is a useful measure,
the following formula by Fisher for calculating the skewness is often used,
including the corresponding MATLAB function.

The second important measure for the shape of a distribution is the kurtosis.
Again, numerous formulas to compute the kurtosis are available. MATLAB
uses the following formula:

The kurtosis is a measure of whether the data are peaked or flat relative to
a normal distribution. A high kurtosis indicates that the distribution has a
distinct peak near the mean, whereas a distribution characterized by a low
kurtosis shows a fl at top near the mean and heavy tails. Higher peakedness
of a distribution is due to rare extreme deviations, whereas a low kurtosis is
caused by frequent moderate deviations. A normal distribution has a kurto-
sis of three. Therefore some defi nitions for kurtosis subtract three from the
above term in order to set the kurtosis of the normal distribution to zero.

After having defined the most important parameters to describe an em-
pirical distribution, the measures of central tendency and dispersion are il-