How Do You Test Whether a Correlation Coefficient Is Significantly Different From 0?

Use a two-tailed hypothesis test for correlation. Test $H_0:\rho=0$ against $H_1:\rho\neq 0$, obtain the p-value using a GDC, and compare it with the chosen significance level $\alpha$.

This is HL-only content under AHL 4.18: hypothesis testing for correlation. The test concerns the population correlation coefficient $\rho$, not the sample coefficient $r$.

The Method

1. State the hypotheses:

   $$H_0:\rho=0,\qquad H_1:\rho\neq 0$$

For example, if $p=0.032$ and $\alpha=0.05$, then $0.032<0.05$. Reject $H_0$ and conclude that there is significant evidence of a non-zero linear correlation in the population.

The numerical result must be interpreted in context. For example, if the variables are study time and examination score, write “there is significant evidence of a non-zero linear correlation between study time and examination score in the population.” This does not establish that additional study time causes higher scores. A lurking variable or the sampling method may explain the association. If the scatter plot shows curvature or influential outliers, Pearson’s test may be misleading instead.

Exam Technique

State hypotheses using $\rho$, explicitly compare the p-value with the significance level, and conclude in context. A common misconception is to write $H_0:r=0$; this is incorrect because $r$ is a sample statistic. Also write “insufficient evidence to reject $$,” not “accept ,” and never claim that significant correlation proves causation.