Lesson 2 of 6 · All lessons in this module
  1. Parametric vs non-parametric tests, and correlation hypotheses
  2. The t-test for a Pearson correlation
  3. Sample size, p-values and many correlations at once
  4. The Spearman rank correlation testVideo lesson
  5. Contingency tables and the chi-square test of independenceVideo lesson
  6. Interpreting the result: standardized residuals and mosaics

Lesson 2 of 6 · 13 min

The t-test for a Pearson correlation

A sample correlation is never exactly zero, so convert it into a t-statistic with n − 2 degrees of freedom and ask whether it is too large to be sampling noise.

In short

  • Test statistic: t=rn−2/1−r2 t = r\sqrt{n-2}/\sqrt{1-r^2} , t-distributed with n − 2 degrees of freedom.
  • It is a parametric test: both variables are assumed to be normally distributed.
  • The sign of t is the sign of r; t grows as |r| rises and as n rises.
  • Two-sided test: reject if t is beyond ±tα/2 \pm t_{\alpha/2} . One-sided test: put all of α \alpha in the tail named by Ha H_a .
  • Rejecting H0:ρ=0 H_0: \rho = 0 means the evidence is sufficient to say the correlation differs from zero; failing to reject does not prove it is zero.

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