Page 104 - Standard Handbook Of Petroleum & Natural Gas Engineering

Applied Statistics 93

For a discrete random variable. the variance is

and if f(X) = l/n
'
&J or S? - - t(X, -F)'(biased)
-
n i=l
or

The standard deviation is the square root of the variance and is denoted by Q
(population) or S (sample).
The skew, the third moment about the mean, is a measure of symmetry of
distribution and can be denoted by y (population) or g (sample). It is given for
a continuous random variable by

and for a discrete random variable by

A completely symmetrical distribution will have a skew = 0, and in a non-
symmetric distribution the sign of the skew will depend upon the location of
the tail of the distribution.
The kurtosis, the fourth moment, is a measure of peakedness or flatness of a
distribution. It has no common notation (k is used here) and is given for a
continuous random variable by

k = Jm (X - ~)~f(X)dx
-m
and for a discrete random variable by