Questions: Correlation Coefficient

Question 1

A researcher computes r = 0.02 between two variables and concludes there is essentially no relationship. What is the problem with this conclusion?

• A. There is no problem — r = 0.02 is so close to zero that no relationship is confirmed
• B. The sample size might be too small to compute r reliably
• C. r near zero only rules out a linear relationship; a strong nonlinear relationship may still exist
• D. r should always be computed as an absolute value, so 0.02 might be negative

Answer: (C)

r = Cov(X,Y)/(σ_X·σ_Y) measures only *linear* association. A perfect quadratic relationship (e.g., Y = X²) or a sinusoidal relationship produces r ≈ 0 even though the variables are strongly dependent. Before concluding 'no relationship,' always examine a scatterplot. r near zero is a reliable indicator of no *linear* pattern, not of no relationship at all.

Question 2

A study finds r = 0.85 between ice cream sales and drowning rates across summer months. This correlation proves that eating ice cream increases the risk of drowning.

• T. True
• F. False

Answer: False

Correlation does not imply causation. Both ice cream sales and drowning rates rise in summer because of a common cause: hot weather brings more people to pools and beaches. This is a classic example of a confounding variable (season/temperature) producing high correlation between two variables that have no direct causal link. High |r| indicates a strong linear association, not a causal mechanism.

Question 3

Two variables have Pearson r = 0. Does this guarantee they are statistically independent? Explain.

Model answer: No. r = 0 only guarantees no linear association. Two variables can be strongly dependent in a nonlinear way (e.g., a U-shaped or circular relationship) while still having r = 0. Independence implies r = 0, but r = 0 does not imply independence.

Independence is a stronger condition than zero correlation. For example, if X is drawn uniformly from [-1, 1] and Y = X², then Cov(X,Y) = 0 (by symmetry), so r = 0 — yet knowing X tells you exactly what Y is. Statistical independence requires that the joint distribution factorizes: P(X,Y) = P(X)·P(Y) for all values, which is a much stricter requirement than uncorrelatedness.