Regression and Correlation | IEB Grade 12 Maths

Q: What is a strong correlation?

The closer $|r|$ is to 1, the stronger the linear relationship. There is no fixed cut-off, so describe it in words and in context, and always give the direction: positive or negative.

Regression finds the straight line that best fits a scatter plot, and the correlation coefficient measures how strong that straight-line relationship is. Calculator skills are essential here, but so is explaining what the numbers mean in context.

Least squares regression line by calculator

The least squares regression line $\overset{y}{^} = a + b x$ is the line that minimises the sum of the squared vertical distances from the points to the line.

Use your calculator's STAT mode (A+BX): enter the $x$ - and $y$ -values, then read off $A$ and $B$ .

Example. Hours studied ( $x$ ) and test mark ( $y$ ) for 8 learners:

$x$ : 1, 2, 3, 4, 5, 6, 7, 8
$y$ : 42, 48, 51, 57, 60, 66, 68, 75

The calculator gives $a \approx 38, 07$ and $b \approx 4, 51$ :

$\overset{y}{^} = 38, 07 + 4, 51 x$

Interpreting $b$ : each extra hour of study is linked to about 4,51 more marks.

The regression line always passes through $(\overset{x}{ˉ}; \overset{y}{ˉ})$ .

The correlation coefficient

The correlation coefficient $r$ is between $- 1$ and $1$ .

$r$ close to $1$ : strong positive linear relationship
$r$ close to $- 1$ : strong negative linear relationship
$r$ close to $0$ : weak or no linear relationship

There are no fixed cut-offs in the syllabus. Describe the strength by how close $∣ r ∣$ is to 1 (very strong, strong, moderate, weak, no linear relationship) and support it with the scatter plot and the context.

Example. For the study data, $r \approx 0, 996$ : a very strong positive correlation.

Correlation does not prove that one variable causes the other.

Outliers on a scatter plot are points far from the general trend. One outlier can lower $r$ sharply. Outliers are identified by looking at the scatter plot.

Predictions and interpolation

Substitute an $x$ -value into the regression equation to predict $y$ .

Interpolation: predicting within the range of the data. Usually reliable.
Extrapolation: predicting outside the range. Less reliable, because the trend may not continue.

Example. For 5,5 hours: $\overset{y}{^} \approx 38, 07 + 4, 51 (5, 5) \approx 62, 9$ marks. This is interpolation.

For 12 hours, the equation predicts about 92,2 marks, but this is extrapolation: nobody in the data studied that long, and marks cannot rise forever.

The strength of $r$ also affects reliability: predictions from a weak correlation are less trustworthy.

Key terms

Scatter plot: A graph of paired data shown as points.
Regression line: The line of best fit, $\overset{y}{^} = a + b x$ .
Correlation coefficient: A number $r$ from $- 1$ to $1$ measuring the strength of a linear relationship.
Interpolation: Predicting within the range of the data.
Extrapolation: Predicting outside the range of the data.

Common mistakes

Swapping $a$ and $b$ when writing the equation from the calculator.
Entering the $x$ - and $y$ -values the wrong way round.
Saying a strong correlation proves cause and effect.
Trusting extrapolated predictions without comment.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Hours studied ( $x$ ) and test marks ( $y$ ) for 8 learners are:

$x$ : 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8
$y$ : 42 ; 48 ; 51 ; 57 ; 60 ; 66 ; 68 ; 75

Determine the equation of the least squares regression line.

Worked solution

Hidden - try the question first

Solution

$a \approx 38, 07$
$b \approx 4, 51$
$\overset{y}{^} = 38, 07 + 4, 51 x$

Question 2

4 marks total

routine

For the same data:

(a) Write down the correlation coefficient. (1)

(b) Describe the correlation. (1)

Worked solution

Hidden - try the question first

Solution

(a) $r \approx 0, 996$
(b) Very strong positive correlation
(c) For every extra hour studied
the mark is about 4,51 higher on average

Question 3

6 marks total

complex

Use the regression line for the study data.

(a) Predict the mark of a learner who studies for 5,5 hours. (2)

(b) Predict the mark of a learner who studies for 12 hours. (2)

Worked solution

Hidden - try the question first

Solution

(a) $r \approx 0, 996$
$\approx 62, 9$ marks
(b) $\overset{y}{^} = 38, 07 + 4, 51 (12)$
$\approx 92, 2$ marks
(c) The prediction for 5,5 hours
It is interpolation (inside the data range); 12 hours is extrapolation, where the trend may not continue

Question 4

8 marks total

problem-solving

A ninth learner studied for 2 hours and scored 85.

(a) Is this point an outlier? Explain by referring to the scatter plot. (2)

(b) With all nine points, the correlation coefficient is $r \approx 0, 49$ . Describe the effect of the new point on the correlation. (2)

(d) Give one other factor, apart from hours studied, that could explain the ninth learner's result. (2)

Worked solution

Hidden - try the question first

Solution

(a) Yes: the regression line predicts about 47 marks for 2 hours
85 is far above the trend of the other points
(b) $r$ drops from about 0,996 to about 0,49
The correlation changes from very strong to weak, because one point lies far from the line
(c) The original line fits the general trend closely ( $r$ close to 1)
One unusual point distorts the line and makes predictions for typical learners worse
(d) Any reasonable factor, for example prior knowledge of the topic
explained in context, for example the learner had already studied the work the previous year

FAQs

Do I calculate the regression line by hand in the IEB exam?

No. Use your calculator's regression mode. You must still write the equation correctly and interpret it.

What is a strong correlation?

The closer $∣ r ∣$ is to 1, the stronger the linear relationship. There is no fixed cut-off, so describe it in words and in context, and always give the direction: positive or negative.

Why is extrapolation risky?

The relationship may change outside the range of the data, so predictions there are less reliable.

Scatter plots, regression and correlation

Least squares regression line by calculator

The correlation coefficient

Predictions and interpolation

Key terms

Common mistakes

Practice questions

FAQs

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