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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:

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