The idea, step by step
Describe three different features
Direction says whether the response generally increases or decreases as the explanatory variable increases. Form describes the shape: a straight-line tendency, a curve, or no clear pattern. Strength describes how closely points follow that form, not how steep the graph looks.
Use a sentence such as “a strong positive, approximately linear association”. A steep line is not automatically stronger than a gentle one. Rescaling the axes can make a graph steeper without changing the association.
A curved relationship can still be strong
If points closely follow a U-shaped curve, the association is strong but nonlinear. Pearson’s r measures linear association and can be near zero for such a pattern. Always inspect the scatter plot before choosing a numerical summary.
Outlying points and separate groups should be mentioned. Do not describe a pattern as exact unless the points really follow the relation without variation.
A common trap
A large slope means strong correlation.
Instead: Strength is about closeness to the fitted form. Slope depends on units and axis scale.
See the thinking
Negative and linear
Describe the pattern in the supplied distances and remaining fuel.
| Distance (km) | Fuel remaining (L) |
|---|---|
| 0 | 40 |
| 50 | 36 |
| 100 | 32 |
| 150 | 29 |
| 200 | 25 |
- As distance increases from 0 to 200 km, remaining fuel decreases.
- The points lie close to a straight descending pattern.
- The tight clustering makes the association strong.
Strong but not linear
Describe the pairs (-2,4), (-1,1), (0,0), (1,1), (2,4).
- The values fall then rise, so one positive/negative direction does not describe the whole data set.
- Every point follows y=x².
- This is an exact U-shaped nonlinear relationship.
Check your understanding
Points lie very close to a downward-sloping line. Describe direction, form and strength.
Show answer
Strong negative linear association.
Downward gives negative; the straight pattern gives linear; close clustering gives strong.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- Describe direction, form and strength for (1,10), (2,8), (3,6), (4,4).3 marks
Paired measurements x y 1 10 2 8 3 6 4 4 - Describe the supplied pairs (1,2), (2,5), (3,3), (4,7), (5,6).3 marks
- A scatter plot rises sharply but its points are widely scattered. Is “strong because it is steep” valid?3 marks
- What form is suggested by (0,1), (1,2), (2,4), (3,8)?3 marks
Paired measurements x y 0 1 1 2 2 4 3 8 - The U-shaped data (-2,4), (-1,1), (0,0), (1,1), (2,4) have r=0. Is there no relationship?3 marks
- Two graphs show the same data but use centimetres and metres on the vertical axes. Should their correlation strengths differ?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Describe direction, form and strength for (1,10), (2,8), (3,6), (4,4).3 marks
Paired measurements x y 1 10 2 8 3 6 4 4 Each one-unit rise in x reduces y by 2.
All points lie exactly on y=12-2x.
Perfect negative linear association. - Describe the supplied pairs (1,2), (2,5), (3,3), (4,7), (5,6).3 marks
The overall response increases as x increases.
It is not exact and no strong curvature is evident.
Positive, roughly linear association with visible scatter. - A scatter plot rises sharply but its points are widely scattered. Is “strong because it is steep” valid?3 marks
Slope and strength describe different features.
Assess closeness to the trend, not just steepness.
No. Wide scatter can indicate weak linear association despite a steep trend. - What form is suggested by (0,1), (1,2), (2,4), (3,8)?3 marks
Paired measurements x y 0 1 1 2 2 4 3 8 Successive y-values double.
The increase is not constant, so a straight line is not the natural form.
Positive nonlinear, exponential-looking association. - The U-shaped data (-2,4), (-1,1), (0,0), (1,1), (2,4) have r=0. Is there no relationship?3 marks
The increasing and decreasing sections balance the linear correlation.
Inspect the graph; r only summarises a straight-line tendency.
No. There is a strong nonlinear relationship y=x². - Two graphs show the same data but use centimetres and metres on the vertical axes. Should their correlation strengths differ?3 marks
Every vertical value is multiplied by the same positive constant.
The display’s apparent steepness can change while r stays the same.
No. A positive unit conversion does not change the association strength.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.
Official curriculum source ↗