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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,…,
Xn be n observations of a random variable X.
We wish to measure the average of X1, X2
,…, Xn in some sense. One of the most commonly-used
statistics is the mean, mX, 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),…, (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 sXY given
by
Actually, a much better measure of correlation can be obtained
from the formula
The quantity rXY 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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