The mathematical notation in this document was created using MathType's anti-aliasing feature. Notice the edges of the equations are smooth and overall much higher-quality than the jagged-edged equations created by converting Microsoft Word documents to HTML.

Mean, Variance, and Standard Deviation

Let X₁ , X₂,…, X_n be n observations of a random variable X. We wish to measure the average of X₁, X₂ ,…, X_n in some sense. One of the most commonly-used statistics is the mean, m_X, defined by the formula

Next, we wish to obtain some measure of the variability of the data. The statistics most often used are the variance  and the standard deviation . We have

It is easy to show that the variance is simply the mean squared deviation from the mean.

Covariance and Correlation

Next, let (X₁ , Y₁), (X₂ , Y₂),…, (X_n , Y_n) be n pairs of values of two random variables X and Y. We wish to measure the degree to which X and Y vary together, as opposed to being independent. The first statistic we will calculate is the covariance s_XY  given by

Actually, a much better measure of correlation can be obtained from the formula

The quantity r_XY  is known as the coefficient of correlation of X and Y.

The Covariance Matrix

Covariances and variances are sometimes arranged in a matrix known as the covariance matrix. In our case, the covariance matrix will be a 2«2 matrix, C. We calculate its eigenvalues in the usual way by finding the roots of the characteristic polynomial:

Notes About Viewing this Document

This document is tuned for viewing in Netscape 3.0 or later or in Microsoft Internet Explorer 4.0 or later. If you view it using other browsers, you may experience the following problems: