where r is the sample correlation coefficient, and n is the size of the sample (the number of data pairs). The probability of t may be looked up in Table 3 (in "Statistics Tables") using n − 2 degrees of freedom.
The probability of obtaining a t of –5.807 with 8 df (drop the sign when looking up the value of t) is lower than the lowest listed probability of 0.0005. If the correlation between months of exercise‐machine ownership and hours of exercise per week were actually 0, you would expect an r of –0.899 or lower in fewer than one out of a thousand random samples.
To evaluate a correlation coefficient, first determine its significance. If the probability that the coefficient resulted from chance is not acceptably low, the analysis should end there; neither the coefficient's sign nor its magnitude may reflect anything other than sampling error. If the coefficient is statistically significant, the sign should give an indication of the direction of the relationship, and the magnitude indicates its strength. Remember, however, that all statistics become significant with a high enough n.
Even if it's statistically significant, whether a correlation of a given magnitude is substantively or practically significant depends greatly on the phenomenon being studied. Generally, correlations tend to be higher in the physical sciences, where relationships between variables often obey uniform laws, and lower in the social sciences, where relationships may be harder to predict. A correlation of 0.4 between a pair of sociological variables may be more meaningful than a correlation of 0.7 between two variables in physics.
Bear in mind also that the correlation coefficient measures only straight‐line relationships. Not all relationships between variables trace a straight line. Figure 5 shows a curvilinear relationship such that values of y increase along with values of x up to a point, then decrease with higher values of x. The correlation coefficient for this plot is 0, the same as for the plot in Figure 3. This plot, however, shows a relationship that Figure 3 does not.
Correlation does not imply causation. The fact that Variable A and Variable B are correlated does not necessarily mean that A caused B or that B caused A (though either may be true). If you were to examine a database of demographic information, for example, you would find that the number of churches in a city is correlated with the number of violent crimes in the city. The reason is not that church attendance causes crime, but that these two variables both increase as a function of a third variable: population.
Figure 5. Data that can cause trouble with correlation analysis.