Univariate Statistics - MATLAB Recipes for Earth Sciences - page 61

h e standard deviation is the average deviation of each data point from the
mean. h e standard deviation of an empirical distribution is ot en used as
an estimate of the population standard deviation ˃. h e formula for the
population standard deviation uses N instead of N -1 as the denominator.
h e sample standard deviation s is computed with N -1 instead of N since it
uses the sample mean instead of the unknown population mean. h e sample
mean, however, is computed from the data x i , which reduces the number of
degrees of freedom by one. h e degrees of freedom are the number of values
in a distribution that are free to be varied. Dividing the average deviation of
the data from the mean by N would therefore underestimate the population
standard deviation ˃.
h e variance is the third important measure of dispersion. h e variance is
simply the square of the standard deviation.
Although the variance has the disadvantage of not having the same
dimensions as the original data, it is extensively used in many applications
instead of the standard deviation.
In addition, both skewness and kurtosis can be used to describe the
shape of a frequency distribution (Fig. 3.3). Skewness is a measure of the
asymmetry of the tails of a distribution. h e most popular way to compute
the asymmetry of a distribution is by Pearson's mode skewness:
skewness = ( mean - mode ) / standard deviation
A negative skew indicates that the distribution is spread out more to the let
of the mean value, assuming values increasing towards the right along the
axis. h e sample mean is in this case smaller than the mode. Distributions
with positive skewness have large tails that extend towards the right. h e
skewness of the symmetric normal distribution is zero. Although Pearson's
measure is a useful one, the following formula by Fisher for calculating the
skewness is ot en used instead, including in the relevant MATLAB function.
h e second important measure for the shape of a distribution is the kurtosis .
Again, numerous formulas to compute the kurtosis are available. MATLAB