VCE General Mathematics (Year 12) · Data Analysis · 14m read

Scatterplots and Correlation

Scatterplots are the gateway to regression. Before you fit a line, you have to describe what you see — direction, form, strength — and back it up with Pearson's r. Just as important is knowing what r cannot tell you: a strong correlation never proves that one variable causes the other.

Learning objective: Describe the association in a scatterplot and interpret Pearson's correlation coefficient r, distinguishing association from causation.

By the end of this lesson you can:

I describe direction, form and strength from a scatterplot.
I interpret Pearson's r using the VCAA strength guidelines.
I explain why correlation does not imply causation.

Correlation measures how well two dancers move together. r near ±1: perfectly in step. Near 0: dancing to different songs.

Describing a scatterplot

PUT THE EXPLANATORY VARIABLE ON THE x-AXIS.

DESCRIBE THREE FEATURES (plus outliers):
• DIRECTION — positive (up to the right) or negative (down to the right).
• FORM — linear or non-linear (curved).
• STRENGTH — how closely points follow the form: strong, moderate, weak, none.
• OUTLIERS — points far from the pattern.

PEARSON CORRELATION COEFFICIENT (r):
• Measures the strength and direction of a LINEAR association.
• −1 ≤ r ≤ 1.
• Only valid when the data are numerical, the form is linear and there are no clear outliers.

VCAA STRENGTH GUIDELINES:
• |r| ≥ 0.75 → strong.
• 0.5 ≤ |r| < 0.75 → moderate.
• 0.3 ≤ |r| < 0.5 → weak.
• |r| < 0.3 → no association.

CORRELATION IS NOT CAUSATION:
• A strong r shows variables move together, not that one causes the other.
• Possible explanations: a CONFOUNDING (lurking) variable affecting both, coincidence, or reverse direction.
• Only a controlled experiment can establish causation.

Quick reference

| r value | Strength |
|---|---|
| ±0.75 to ±1 | Strong |
| ±0.5 to ±0.75 | Moderate |
| ±0.3 to ±0.5 | Weak |
| −0.3 to 0.3 | No association |
| Sign | Direction |

Key ideas at a glance

Describe DFS: Direction, form, strength.
r only for linear: Check form.
Outliers affect r: Note them.
No causation: Lurking variables.

Worked example (Medium)

For 20 suburbs, r = −0.82 between average house price and distance from the CBD. Describe the association.

Direction. r is negative → as distance increases, price tends to decrease.
Strength. |r| = 0.82 ≥ 0.75 → strong.
Form. r is only meaningful for linear form — assume the scatterplot is linear.

Answer: There is a strong, negative, linear association between distance from the CBD and average house price.

Worked example (Easy)

Ice-cream sales and drowning numbers have r = 0.9. Does eating ice-cream cause drowning?

Look for a confounder. Hot weather increases both ice-cream sales and swimming.
Conclude. The association is explained by temperature, not causation.

Answer: No — temperature is a likely confounding variable; correlation does not imply causation.

Common mistakes

Calling r = 0.6 strong VCAA guidelines classify 0.5–0.75 as moderate. Fix: Memorise the cut-offs 0.3, 0.5, 0.75.
Using r for a curved pattern r measures LINEAR association only. Fix: Check the form first; if curved, consider a transformation.
Claiming causation Observational data cannot prove cause. Fix: Say "is associated with" and suggest a possible confounding variable.

Anscombe's quartet

In 1973 statistician Francis Anscombe created four data sets with nearly identical r values and regression lines — but completely different scatterplots.

Lesson: always plot your data before trusting a single statistic.

Practice questions

Using VCAA guidelines, r = 0.42 indicates:
1. No association
2. Weak positive
3. Moderate positive
4. Strong positive
(1 marks)
Suburbs with more cafés have higher average incomes (r = 0.78). Suggest why opening more cafés may not raise incomes.
(2 marks)

Summary

Describe direction, form, strength and outliers.
r measures linear association only; −1 ≤ r ≤ 1.
Strength cut-offs: 0.3 weak, 0.5 moderate, 0.75 strong.
Correlation does not imply causation.