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 \(r\) (often interpreted qualitatively):
- \(r>0\) → 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)

Problems 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 a higher test 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.
  • Always ask: Could something else be causing both?
  • Strong patterns still don’t prove causation.
  • Beware misleading graphs.