4.1 Sample Moments

Let x₁, x₂, x₃, . . . . ., 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

(25)

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

(26)

Similarly, the sample variance, which we denote by s² is

(27)

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, (x₁, y₁), (x₂, y₂), . . ., the sample means and variances for each variable are calculated as above. In an analogous manner, the sample covariance can be calculated by

(28)

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