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Inter-relationships between variables

Correlation

3.1 Pearson’s correlation coefficient = r

n ∑ xy – ∑ x ∑ y
 r =
2
2
(n ∑ x –( ∑ x) )(n ∑ y –( ∑ y) )
2
2
 This measure has the property of always lying in the range –1 to +1, where:

– r = +1 denotes perfect positive linear correlation

– r = –1 denotes perfect negative linear correlation

– r = 0 denotes no linear correlation.

 The strength of a correlation can be judged by its proximity to +1 or –1: the
nearer it is (and the further away from zero), the stronger is the linear
correlation.

 A common error is to believe that negative values of r cannot be strong. They
can be just as strong as positive values except that y is decreasing as
x increases.

2
3.2 The coefficient of determination = r

2
 The coefficient of determination, r , gives the proportion of changes in y that can
be explained by changes in x, assuming a linear relationship between x and y.
2
 E.g. If r = +0.7, then r = 0.49 and we could state that 49% of the observed
changes in y can be explained by the changes in x but that 51% of the changes
must be due to other factors.

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