Direction/form/strength of association | Year 12

01 / LEARN

Understand the idea.

Describe three different features

Direction describes whether the response tends to rise or fall as the explanatory variable increases. Form describes the shape, such as linear or curved. Strength describes how tightly observations follow that shape; assess these three features separately.

A steep pattern can have large scatter, while a gentle pattern can be very tight. Changing centimetres to metres changes the numerical slope and appearance of the axes, but does not change how consistently the measurements are related.

A curved relationship can still be strong

A U-shaped pattern can be strong even though it has no single increasing or decreasing direction over the whole range. Pearson's $r$ measures linear association, so a value near zero may conceal a clear curve. Inspect the scatter plot first.

Describe groups and unusual points as well as the main tendency. Use “approximately linear” when the dots scatter around a straight pattern. An exact relationship among the observed pairs does not establish that every future observation must follow it.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Negative and linear

Describe the pattern in the supplied distances and remaining fuel.

Scatter plot with Distance in kilometres on the horizontal axis and Fuel in litres on the vertical axis. Supplied points are (0,40), (50,36), (100,32), (150,29) and (200,25). No fitted line is added.

Paired measurements
Distance (km)	Fuel remaining (L)
0	40
50	36
100	32
150	29
200	25

Worked solution

1
Read the overall direction.
$\begin{aligned} Distance: & 0 \to 200 km \\ Fuel: & 40 \to 25 L \end{aligned}$
Larger distances correspond to less remaining fuel, so the association is negative.
2
Compare changes over equal distance intervals.
$Δ F = - 4, - 4, - 3, - 4 L$
The fuel changes are similar for successive 50 km increases, supporting an approximately straight pattern.
3
Judge the scatter around that pattern.
The points lie close to a descending line, giving a strong negative, approximately linear association.

Answer

Strong negative, approximately linear association.

EXAMPLE 02

Strong but not linear

Describe the pairs (-2,4), (-1,1), (0,0), (1,1), (2,4).

A scatter plot showing exactly the five points (-2,4), (-1,1), (0,0), (1,1) and (2,4). The points are not joined.

Worked solution

1
Check a possible curved rule against the supplied pairs.
$\begin{aligned} {(- 2)}^{2} & = 4, & {(- 1)}^{2} & = 1, \\ 0^{2} & = 0, & 1^{2} & = 1, & 2^{2} & = 4 \end{aligned}$
Every supplied response equals the square of its paired explanatory value.
2
Describe the form and direction.
$y = x^{2}$
The responses fall to zero and then rise, forming a U rather than a straight line or one overall positive or negative direction.
3
State the strength for these observations.
All five observed points lie exactly on the curve, so the observed association is strong and nonlinear.

Answer

Strong nonlinear U-shaped association.

03 / YOUR TURN

Try it yourself.

Check your understanding

Points lie very close to a downward-sloping line. Describe direction, form and strength.

Show answer & reasoning

Strong negative linear association.

The downward direction is negative, the straight form is linear, and the close clustering makes the association strong. All three features are needed in the description.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

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

Paired measurements
$x$	$y$
1	10
2	8
3	6
4	4

Paired measurements
$x$	$y$
0	1
1	2
2	4
3	8

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 $y = 12 - 2 x$ .
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 $y = x^{2}$ .
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.

Paired measurements
$x$	$y$
1	10
2	8
3	6
4	4

Paired measurements
$x$	$y$
0	1
1	2
2	4
3	8

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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 ↗

In this lesson

Understand the idea.

Describe three different features

A curved relationship can still be strong

See the method in action.

Negative and linear

Read the overall direction.

Compare changes over equal distance intervals.

Judge the scatter around that pattern.

Strong but not linear

Check a possible curved rule against the supplied pairs.

Describe the form and direction.

State the strength for these observations.

Try it yourself.

Check your understanding

Keep practising