Correlation & Causation

TipLearning Objectives
  • Distinguish correlation from causation.
  • Identify lurking variables and confounding factors.
  • Understand correlation coefficient meaning (qualitatively).
  • Recognize common real-world examples of misleading correlations.

Key Ideas

Correlation measures association between variables.
Causation means one variable directly affects another.

Correlation does not imply causation.

Correlation coefficient \(r\) (not required numerically on SAT):
- \(r>0\) → positive association
- \(r<0\) → negative association
- \(|r|\) near 1 → strong
- \(|r|\) near 0 → weak

Common Problem Types

Identifying Whether Causation Is Plausible

Ask: Does A directly cause B?

Example:
Ice cream sales ↑ and drowning ↑.
No causation — lurking variable = temperature.

Correlation Without Causation

Variables rise together but do not cause each other.

Example:
Years of education vs. shoe size (in adults).
No causal link.

Recognizing Lurking Variables

Hidden factor affects both variables.

Example:
More firefighters at larger fires → correlation, not causation.
Lurking variable = fire size.

Weak vs. Strong Correlation (Qualitative)

SAT may ask which scatterplot shows stronger association.

Example:
Tighter clustering = stronger.

Misleading Graph Interpretations

Graphs may stretch axes or omit scales.

Strategies

  • Ask: Could another factor explain both variables?
  • Do NOT assume cause/effect unless explicitly tested.
  • Check graph scales for misleading presentations.
  • Think logically about real-world likelihood.

Worked Examples

Example 1

More umbrellas sold when rainfall increases.
Correlation and likely causation (rain causes umbrella purchases).

Example 2

Coffee consumption vs. productivity.
Correlation exists; causation uncertain — need controlled study.

WarningCommon Mistakes
  • Assuming correlation means causation.
  • Ignoring lurking/confounding variables.
  • Misreading manipulated graphs.
  • Claiming causation without experimental evidence.

Practice Problems

  1. Does higher SAT score cause students to buy more textbooks?
  2. More people celebrate holidays in December; electricity use rises. Causation?
  3. A scatterplot shows strong negative correlation. Describe it.
  4. More police officers present correlates with more crime reports. Lurking variable?
  1. No — correlation only, many other factors.
  2. Not necessarily; both may be caused by cold weather & shorter days.
  3. As x increases, y tends to decrease strongly.
  4. Crime severity/size brings more officers; underlying cause = crime, not officers.

Summary

  • Correlation does NOT imply causation.
  • Look for lurking variables.
  • Interpret correlation qualitatively on SAT.
  • Always ask: Could something else be causing both?
  • Strong patterns still don’t prove causation.
  • Beware misleading graphs.