Mean, Variance, and Standard Deviation
Statistics Examples
Mean, Variance, and Standard Deviation
Let [pic] be n observations of a random variable X. We wish to measure the average of [pic] in some sense. One of the most commonly used statistics is the mean, [pic], defined by the formula
[pic]
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 [pic]. We have
[pic]
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) ,…, (Xn , Yn) 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 [pic] given by
[pic]
Actually, a much better measure of correlation can be obtained from the formula
[pic]
The quantity [pic] 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 a covariance matrix. In our case, the covariance matrix will be a [pic] matrix:
[pic]
The eigenvalues of the covariance matrix are sometimes of interest. These are obtained in the usual way by solving the characteristic equation:
[pic]
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