Correlation, causation & coefficient of determination
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Question
Positive vs negative correlation?
Answer
**Positive**: as one variable rises, the other **rises** (line slopes up). **Negative**: as one rises, the other **falls** (line slopes down).
Question
What does the correlation coefficient r tell you, and what is its range?
Answer
It measures a linear correlation from **−1 to +1**: the **sign** = direction (+ together, − opposite) and the **size** (how close to ±1) = **strength**. $r ≈ 0$ = no linear relationship.
Question
How is the coefficient of determination R² related to r?
Answer
**$R^2 = r^2$.** It is the **fraction (or %) of the variation** in y explained by x. e.g. $r = 0.9 → R^2 = 0.81 →$ ~81% explained. Given R², $r = \pm\sqrt{R^2}$.
Question
Why does correlation NOT prove causation?
Answer
A **third (confounding) variable** or coincidence could cause both. Only a **controlled experiment** can establish that one variable causes a change in another.
Question
How do you DESCRIBE a relationship shown by a graph?
Answer
State the **direction** (positive/negative correlation) **and** quote a **figure/comparison** from the data — not just 'there is a relationship'.
Question
r = −0.9 vs r = +0.9 — which is stronger?
Answer
**Equally strong** — strength depends on $|r|$ (how close to 1), not the sign. They differ only in **direction**.
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