Correlation and Regression

Q: Do I need to memorise formulae for PMCC or regression lines?

No, all formulae for , , , PMCC and regression line coefficients are provided in the Edexcel Mathematical Formulae & Statistical Tables booklet for your exam. Familiarise yourself with their location to save time during the exam.

Edexcel International A-Level Mathematics· 2018 Specification Issue 3, S1 §4.1-4.3· 25 min read

1. Scatter Diagrams, Variable Types and Correlation★★☆☆☆⏱ 5 min

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Correlation describes the strength and direction of linear relationship between two variables. When plotting data, the explanatory (independent) variable is plotted on the x-axis, and the response (dependent) variable, which is affected by the explanatory variable, is plotted on the y-axis.

📘 Definition

Explanatory and Response Variables

The explanatory variable (x) is controlled or measured independently; the response variable (y) is the outcome being measured, expected to change as x changes.

Example:

If measuring the effect of study time on test score, study time = x, test score = y.

Scatter diagrams visualise correlation: positive correlation means y increases as x increases, negative correlation means y decreases as x increases, and zero correlation means no linear relationship. Strength ranges from perfect (all points on a straight line) to no linear relationship.

📐 Worked Example

A researcher records the number of hours 7 students spent revising for a test, and their test scores out of 10. State which variable is explanatory, and describe the correlation you would expect to see.

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1. Identify explanatory variable: Hours spent revising, as it is the independent variable expected to affect test score.
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1. Describe expected correlation: Moderate to strong positive linear correlation, as longer revision time is expected to lead to higher test scores.

Exam tip:

Always confirm which variable is explanatory before plotting or calculations, as this determines how you set up regression lines later.

2. Product Moment Correlation Coefficient (PMCC)★★★☆☆⏱ 7 min

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The PMCC, denoted $r$ , is a numerical measure of the strength and direction of linear correlation between two variables, ranging from $r = - 1$ (perfect negative linear correlation) to $r = + 1$ (perfect positive linear correlation), with $r = 0$ meaning no linear correlation.

📘 Definition

Product Moment Correlation Coefficient

r

Calculated as $r = \frac{S _{x y}}{S _{xx} S _{y y}}$ , where $S_{xx} = \sum x^{2} - \frac{( \sum x ) ^{2}}{n}$ , $S_{y y} = \sum y^{2} - \frac{( \sum y ) ^{2}}{n}$ , $S_{x y} = \sum x y - \frac{( \sum x ) ( \sum y )}{n}$ , and $n$ is the number of data pairs. All formulae are provided in your exam booklet.

📐 Worked Example

For a set of 8 data pairs, you are given: $\sum x = 24$ , $\sum y = 40$ , $\sum x^{2} = 100$ , $\sum y^{2} = 250$ , $\sum x y = 156$ . Calculate the PMCC $r$ and interpret its value.

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1. Calculate $S_{xx}$ :
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$S_{xx} = 100 - \frac{24^2}{8} = 100 - 72 = 28$
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1. Calculate $S_{y y}$ :
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$S_{yy} = 250 - \frac{40^2}{8} = 250 - 200 = 50$
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1. Calculate $S_{x y}$ :
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$S_{xy} = 156 - \frac{24 \times 40}{8} = 156 - 120 = 36$
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1. Calculate $r$ :
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$r = \frac{36}{\sqrt{28 \times 50}} = \frac{36}{\sqrt{1400}} \approx 0.962$
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1. Interpretation: There is a very strong positive linear correlation between the two variables.

Exam tip:

Always give context when interpreting $r$ in exam questions, not just 'strong positive correlation' — link it to the variables given, e.g. 'strong positive correlation between revision time and test score'.

3. Fitting the y-on-x Least Squares Regression Line★★★☆☆⏱ 7 min

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The y-on-x least squares regression line is the straight line that minimises the sum of the squares of the vertical distances between each data point and the line. It is used to predict values of the response variable $y$ for given values of the explanatory variable $x$ .

📘 Definition

y-on-x Regression Line

y = a + bx

Where the gradient $b = \frac{S _{x y}}{S _{xx}}$ , and the y-intercept $a = \overset{y}{ˉ} - b \overset{x}{ˉ}$ . $\overset{x}{ˉ}$ and $\overset{y}{ˉ}$ are the mean values of $x$ and $y$ respectively. All formulae are provided in your exam booklet.

To draw the regression line on a scatter diagram, calculate two points on the line (e.g. when $x = 0$ , $y = a$ , and when $x = \overset{x}{ˉ}$ , $y = \overset{y}{ˉ}$ — the line always passes through the mean point $(\overset{x}{ˉ}, \overset{y}{ˉ})$ ) and draw a straight line through them.

📐 Worked Example

Using the same data from the previous example: $\sum x = 24$ , $\sum y = 40$ , $S_{xx} = 28$ , $S_{x y} = 36$ , $n = 8$ . Find the equation of the y-on-x regression line, and use it to predict the test score for a student who revised for 4 hours (x = revision time in hours, y = test score out of 10). The observed x values range from 1 hour to 6 hours.

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1. Calculate gradient $b$ :
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$b = \frac{S_{xy}}{S_{xx}} = \frac{36}{28} \approx 1.29$
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1. Calculate means:
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$\bar{x} = 24 / 8 = 3, \bar{y} = 40 / 8 = 5$
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1. Calculate intercept $a$ :
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$a = 5 - (1.29 \times 3) \approx 1.13$
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1. Regression line equation: $y = 1.13 + 1.29 x$ (3 significant figures)
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1. Predict for x=4:
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$y = 1.13 + (1.29 \times 4) = 6.29 \approx 6.3$
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1. Validity: x=4 is within the observed range 1 to 6, so the prediction is reliable.

Exam tip:

Always state whether a prediction is reliable or not, with justification linked to extrapolation.

4. Linear Coding for Simplified Calculations★★★★☆⏱ 6 min

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Linear coding (linear transformation) of variables can simplify correlation and regression calculations, especially when data values are large. A linear code has the form $X = \frac{x - c}{d}$ or $Y = \frac{y - e}{f}$ , where c, d, e, f are constants.

📐 Worked Example

A set of data uses coding $X = x - 10$ and $Y = \frac{y}{5}$ to simplify calculations. The regression line for the coded data is $Y = 2 + 3 X$ . Find the regression line for the original x and y variables.

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1. Substitute the coding expressions into the coded regression line:
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$\frac{y}{5} = 2 + 3(x - 10)$
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1. Multiply both sides by 5 to eliminate the denominator:
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$y = 10 + 15(x - 10)$
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1. Expand and simplify:
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$y = 10 + 15x - 150 = 15x - 140$
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1. Final regression line for original variables: $y = - 140 + 15 x$

Exam tip:

If you are given coded values to calculate PMCC, you do not need to convert back to original variables to get the correct r value.

5. Common Pitfalls

Wrong move:

Using x-on-y regression line instead of y-on-x

Why:

Only y-on-x lines are assessed in S1, and x-on-y lines are designed to predict x from y, not y from x, so predictions will be incorrect.

Correct move:

Always fit and use the y-on-x regression line for predicting response variable y from explanatory variable x.

Wrong move:

Interpreting a low PMCC as meaning no relationship between variables

Why:

PMCC only measures linear correlation; a low r may mean a strong non-linear relationship exists.

Correct move:

State only that there is no linear correlation, and refer to the scatter diagram to check for non-linear relationships.

Wrong move:

Extrapolating beyond the range of observed x values to make predictions

Why:

There is no evidence that the linear relationship continues outside the range of collected data, so predictions are unreliable.

Correct move:

Only make predictions for x values within the observed range, and explicitly state if a prediction is unreliable due to extrapolation.

Wrong move:

Confusing explanatory and response variables when plotting or doing regression

Why:

The regression line is dependent on which variable is explanatory, so swapping them will give an incorrect line.

Correct move:

Always identify the explanatory variable (x) first: it is the variable that is expected to cause changes in the response variable (y).

Wrong move:

Trying to memorise all formulae for Sxx, Syy, Sxy, PMCC and regression lines

Why:

All required formulae are provided in the exam formula booklet, so memorising is unnecessary and leads to errors if you misremember.

Correct move:

Familiarise yourself with where each formula is located in the booklet so you can quickly access it during the exam.

6. Quick Reference Cheatsheet

Concept	Formula / Rule	Key Note
PMCC r	$r = \frac{S _{x y}}{S _{xx} S _{y y}}$	Range: $- 1 \leq r \leq 1$ ; measures only linear correlation; unaffected by linear coding
Regression Line $y = a + b x$	$b = \frac{S _{x y}}{S _{xx}}$ , $a = \overset{y}{ˉ} - b \overset{x}{ˉ}$	Passes through $(\overset{x}{ˉ}, \overset{y}{ˉ})$ ; only use for y-on-x predictions
Prediction Validity	Valid only if x is within observed x range	Extrapolation outside observed range is unreliable
Linear Coding Effect	No change to r value	Regression coefficients change; substitute coding to convert back to original variables

7. Frequently Asked

Do I need to memorise formulae for PMCC or regression lines?

No, all formulae for $S_{xx}$ , $S_{y y}$ , $S_{x y}$ , PMCC and regression line coefficients are provided in the Edexcel Mathematical Formulae & Statistical Tables booklet for your exam. Familiarise yourself with their location to save time during the exam.

Can I use x-on-y regression lines in S1 exams?

No, only y-on-x least squares regression lines are assessed in S1. x-on-y lines are designed to predict x from y, so they will give incorrect predictions for the response variable y.

When is a regression line prediction unreliable?

Predictions are unreliable if you extrapolate beyond the range of observed explanatory variable values, or if the linear correlation between variables is very weak. Always justify the validity of predictions in exam answers.

Going deeper

What's Next

Now that you have mastered correlation and regression for Edexcel IAL S1, you are ready to move on to probability distributions, the next high-weightage topic in the S1 syllabus. Correlation and regression questions make up approximately 10-15% of your S1 exam, so be sure to practice past paper questions to familiarise yourself with common question styles, including coding and interpretation of results. Remember that all formulae are provided in your exam booklet, so focus on applying them correctly and justifying your answers, especially when commenting on correlation strength or prediction validity. This topic also lays the foundation for more advanced statistical analysis in S3, including significance testing for correlation, which you will encounter if you study A2 Further Mathematics.