Sample size, p-values and many correlations at once

In short

• For a fixed r, more data raises the chance of rejecting in two ways: $\sqrt{n-2}$ grows, and the critical value shrinks as df rise.
• So the smallest |r| that is significant falls as n grows; a larger sample gives the test more power.
• In very large datasets almost every correlation is significant, so significance alone says little about how strong or useful a relationship is.
• A p-value is the smallest significance level at which $H_0$ can be rejected: reject when p < $\alpha$.
• With k variables there are $k(k-1)/2$ distinct correlations, and each is tested separately.