Strength of relationship vs strength of evidence for relationship

It is important to distinguish between the correlation coefficient, r, and the p-value for testing whether there is a relationship between X and Y.

Correlation coefficient
Describes the strength of the relationship between X and Y
The p-value for testing whether X and Y are related
Describes the strength of evidence for whether X and Y are related at all

It is important not to confuse these two values when interpreting the p-value for a test.

This is partly explained by an alternative formula for the test statistic,

The test statistic and the p-value therefore both depend on both r and the sample size, n. Increasing n and increasing r both result in a lower p-value.