Subscript 1 comma 1 Baseline 2nd Column sigma Subscript 1 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column sigma Subscript 1 comma p Baseline 2nd Row 1st Column sigma Subscript 2 comma 1 Baseline 2nd Column sigma Subscript 2 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column sigma Subscript 2 comma p Baseline 3rd Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 4th Row 1st Column sigma Subscript p comma 1 Baseline 2nd Column sigma Subscript p comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column sigma Subscript p comma p Baseline EndMatrix period"/>
Just like the sample covariance case defined in (3.1), the diagonal elements are the population variances of the 's, and the off‐diagonal elements are the population covariances of all possible pairs of s, i.e. for .
The notation for the covariance matrix is widely used and seems natural because is the uppercase version of .
Example 3.3 Consider the following data matrix introduced in Example 3.1:
Each receipt yields a pair of measurements, total dollar sales, and number of movies sold. Since there are three receipts, we have a total of three observations on each variable. We find the sample variances and covariance as follows:
Therefore,
3.5 Correlation Matrices
A correlation matrix is a table showing correlation coefficients between variables. Correlation is a statistical technique that can show whether and how strongly pairs of variables are related. The sample correlation between the th and th variables is defined as
(3.6)
where
Substituting and into (3.6) and canceling terms, we obtain
(3.7)
for and . We note that the sample correlation is symmetric since for all and .
The
