Example 3.6
Measures of Association: Covariance
and Correlation
EXPENSES.XLS

A survey questions members of 100 households
about their spending habits.

The data in this file represent the salary, expense for
cultural activities, expense for sports-related
activities, and the expense for dining-out for each
household over the past year.

Do these variables appear to be related linearly?
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Covariance and Correlation

When we need to summarize the relationship
between two variables we can use the measures
covariance and correlation. We summarize the type
of behavior observed in a scatterplot.

Each measures the strength (and direction) of a
linear relationship between two numerical variables.

The relationship is “strong” if the points in a
scatterplot cluster tightly around some straight line. If
this line rises form left to right then the relationship is
“positive”. If it falls from left to right then the
relationship is “negative”.
3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.7 | 3.8 | 3.9 | 3.10 | 3.11
Determining Linear
Relationships

Scatterplots of each variable versus each other would
provide the answer to the question but six
scatterplots would be required, one for each pair.

To get a quick indication of possible linear
relationships we can use Stat-Proto obtain a table of
correlations and/or covariances.
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Table of Correlations and
Covariances

To get the table, place the cursor anywhere in the
data set and use the StatPro/Summary
Stats/Correlations, Covariances menu item and
proceed in the obvious way.
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Relationships

The only relationships that stand out are the positive
relationships between salary and cultural expenses
and between salary and dining expenses.

The negative relationships are between cultural and
sports-related expenses.

To confirm these graphically we show scatterplots of
Salary versus Culture and Culture versus Sports
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Scatterplot Indicating Positive
Relationship
3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.7 | 3.8 | 3.9 | 3.10 | 3.11
Scatterplot Indicating Negative
Relationship
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Correlation and Covariance
Properties

In general, the following properties are evident from
the Table of correlations and covariances.
– The correlation between a variable and itself is 1.
– The correlation between X and Y is the same as the
correlation between Y and X. Therefore, it is sufficient to list
the correlations below (or above) the diagonal in the table.
(The same is true for the covariances).
– The covariance between a variable and itself is the variance
of the variable. We indicate this in the heading of the
covariance table.
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Correlation and Covariance
Properties -- continued
– It is difficult to interpret the magnitudes of covariances.
These depend on the fact that the data are measured in
dollars rather than, say, thousands of dollars. It is such
easier to interpret the magnitudes of the correlations
because they are scaled to be between -1 and +1.
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