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1 Calculation of Covariance Matrix from Data Matrix
Suppose we have a data matrix with rows corresponding to
subjects and columns corresponding to variables. We can calculate a
mean for each variable and replace the data matrix with a matrix of
deviations from the mean. That is, each element is
replaced by
where is the mean of the
variable. Let us call the new matrix . The
covariance matrix is then simply calculated as
where is the number of subjects.
For example, suppose we have the following data:




1 
2 
2 
4 
2 
8 
1 
2 
3 
6 
0 
0 
4 
4 
1 
2 
5 
10 
2 
4 
So the matrix of deviations from the mean is
and therefore the covariance matrix of the observations is
The diagonal elements of this matrix are the variances of the variables, and
the offdiagonal elements are the covariances between the variables. The
standard deviation is the square root of the variance (see
Chapter 2).
The correlation is
In general, a correlation matrix may be calculated from a covariance
matrix by pre and postmultiplying the covariance matrix by a
diagonal matrix in which each diagonal element is
, i.e., the reciprocal of the standard deviation for
that variable. Thus, in our two variable example, we have:
Next: 2 Transformations of Data
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Jeff Lessem
20020321