**4.1 Sample Moments**

Let *x*_{1}, *x*_{2}, *x*_{3},
. . . . ., *x*_{n} be a sample of size *n* from a
distribution
whose theoretical mean is *µ* and variance
^{2}. This is known as the
sample population. The *sample mean*,
is then defined as

which is just the arithmetic average of the sample. In the limit
*n* -> ,
this can be shown to approach the theoretical mean,

Similarly, the *sample variance*, which we denote by
*s*^{2} is

which is the average of the squared deviations. In the limit *n* ->
,
this also approaches the theoretical variance
^{2}.

In the case of multivariate samples, for example, (*x*_{1},
*y*_{1}), (*x*_{2}, *y*_{2}),
. . ., the sample means and variances for each variable are calculated as
above. In an analogous manner, the *sample covariance* can be calculated
by

In the limit of infinite *n*, (28), not surprisingly, also approaches
the theoretical covariance (10).