16.2 Statistics Methods
The CÆSAR Statistics Class is used to keep track of the statistics
associated with a sample population or distribution. Currently, the class
stores only conglomerate information about the population and does not store
the entire distribution. Therefore, certain types of statistical information,
such as the median and the mode, are not available. The following values
are calculated by the class.
Given a set of N variables denoted by
x_{i}, the arithmetic mean,
or average, of the distribution is given by
The geometric mean of the distribution (calculated only if all x_{i} are
positive) is given by
or equivalently by
The harmonic mean of the distribution is given by:
The standard deviation of the distribution is given by
s = ,

(16.5) 
or equivalently by
s = .

(16.6) 
Note that the factor N  1, rather than N, is required in the denominator
to account for the fact that the parameter
x has been determined from
the distribution and not independently. This formula for the standard
deviation is sometimes called the sample standard deviation. The limits of
the sample mean and the sample standard deviation give the true values:

= 
x , 
(16.7) 

= 
s . 
(16.8) 
The Statistics Class also calculates minimum and maximum values for the
distribution. The Valid_State procedure verifies that all of the means
lie within the extremum bounds, and that the following mathematical
relationship holds:
Michael L. Hall