Regression & Correlation — Grade 12 Maths (CAPS)

Drawing a scatter plot, finding the least squares regression line ŷ = a + bx on the calculator, using it to predict, and interpreting the correlation coefficient r.

1Scatter plots

A scatter plot shows whether two quantities move together. The pattern can be linear, quadratic, exponential or show no relationship at all.

Hours studied against test mark, with the least squares regression line drawn through the points.

2The least squares regression line

ŷ = a + bx

a = the y-intercept · b = the gradient
(ŷ, read ‘y-hat’, is the PREDICTED value of y)

In the exam you get a and b from the calculator's STAT mode, not by hand. Enter the x-values in one column and the y-values in the other; the calculator labels the intercept A and the gradient B, so its A is your a and its B is your b.

Worked ExampleFor the data shown, the calculator gives A = 13,00 and B = 8,86. Write down the regression line and predict the mark for 7 hours.

1
ŷ = 13,00 + 8,86x.
a = A = 13,00 and b = B = 8,86.
2
For x = 7: ŷ = 13,00 + 8,86(7).
3
= 13,00 + 62,02 = 75,02 ≈ 75%.
Round sensibly — a test mark is not given to two decimals.

The gradient b = 8,86 means that each extra hour of study is associated with roughly 8,86 more percentage points. Interpreting the gradient in context is a standard question.

⚠️

Predicting far outside the data range is unreliable. This data covers 1 to 6 hours; predicting the mark for 20 hours would give over 190%, which is nonsense. Say so if a question pushes you there.

3The correlation coefficient

−1 ≤ r ≤ 1

r close to 1 ⇒ strong positive
r close to −1 ⇒ strong negative
r close to 0 ⇒ very weak or none

A rough scale: |r| > 0,9 very strong · 0,7 to 0,9 strong · 0,5 to 0,7 moderate · below 0,5 weak.

Worked ExampleThe calculator gives r = 0,998 for this data. Interpret it.

1
r = 0,998 is very close to 1.
2
There is a very strong positive linear correlation between hours studied and the mark.
Say strength AND direction AND that it is linear — all three earn marks.

💡

Correlation is not causation. A strong r shows the two move together; it does not prove one causes the other. Examiners like asking this.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Using the scatter plot below, describe the type of relationship shown.
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A strong positive linear relationship
2
The calculator gives A = 13,00 and B = 8,86. Write down the equation of the least squares regression line.
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ŷ = 13,00 + 8,86x
3
Use the regression line to predict the mark of a learner who studies for 7 hours.
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13,00 + 8,86(7) = 75,02 ≈ 75%
4
Interpret the gradient of the regression line in this context.
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Each extra hour of study is associated with about 8,86 more percentage points.
5
The correlation coefficient is r = 0,998. Interpret it.
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A very strong positive linear correlation
6
What is the possible range of values for r?
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−1 ≤ r ≤ 1
7
Would it be reasonable to use this line to predict the mark for 20 hours of study? Explain.
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No — 20 hours is far outside the data range (1 to 6 hours) and the prediction would exceed 100%.
8
Does a strong correlation prove that studying causes higher marks?
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No — correlation does not prove causation.