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Explain how a correlation coefficient is tested

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.7.d

The idea

A sample correlation can be non-zero purely by chance, so a test is needed before concluding that two variables are related. The null is that the population correlation is zero, and the test statistic follows a t distribution with n - 2 degrees of freedom. The statistic rises with both the sample correlation and the sample size, which is why a weak correlation can be statistically significant in a large sample while a strong one is not significant in a small one. Significance answers whether a relationship exists, never whether it is large enough to act on.

Formulas

Test statistic for a correlation coefficient

t = r × √(n − 2) ÷ √(1 − r²)

r
the sample correlation
n
the number of paired observations

Tests whether a correlation differs from zero, with n − 2 degrees of freedom. A small correlation becomes significant on a large enough sample, which is why significance and importance are not the same finding.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is the null hypothesis in a standard test of a correlation coefficient?

  1. That the population correlation is zero — correct
  2. That the sample correlation is zeroWhy not: The sample correlation is observed, not hypothesised — you can simply look at it. The test is about the population value behind it.
  3. That the two variables are causally relatedWhy not: No correlation test addresses causation, and a null is normally the absence of an effect rather than its presence.

The test asks whether the observed sample correlation is large enough to rule out a population correlation of zero. It concerns the population, not the sample already in hand.

A harder one

A correlation of 0.08 is found to be statistically significant in a sample of 5,000. What does this tell an analyst?

  1. The relationship is strong enough to trade onWhy not: Significance says the relationship is probably not zero. At 0.08 it explains about 0.6% of the variation, which is unlikely to survive costs.
  2. A relationship almost certainly exists, but it is very weak — correct
  3. The sample must contain an errorWhy not: This is the expected result for a large sample. Detecting tiny effects reliably is what a sample of 5,000 is good at.

Large samples detect tiny effects reliably. Significance establishes existence; the coefficient itself says the relationship explains well under 1% of the variation.

The hardest tier

A correlation of 0.04 is significant in a sample of a million. What should an analyst conclude?

  1. That it exists and is too weak to use — correct
  2. That the relationship is strongWhy not: It explains about 0.16% of the variation. Significance is confidence that it is not zero, not a claim about size.
  3. That the test has failedWhy not: The test worked exactly as designed. A million observations can detect an effect this small, which is a feature rather than a fault.

A relationship almost certainly exists, and it is far too weak to act on. Significance answers whether a relationship exists, never whether it is large enough to matter.

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