# Correlation and Regression

> Edexcel International A-Level Mathematics · IAL S1
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s1-correlation-and-regression/

This guide covers all Edexcel IAL S1 correlation and regression content: scatter diagrams, PMCC calculation, y-on-x regression lines, valid predictions, and linear coding, fully aligned to the 2018 specification.

**Prerequisites:** [Descriptive statistics and measures of central tendency](https://www.owlsprep.com/study/edexcel-ial-math-s1-descriptive-statistics/); [Basic algebra and substitution into formulae](https://www.owlsprep.com/study/edexcel-ial-math-gcse-algebra-substitution/)

## Learning objectives

- Identify explanatory and response variables, interpret scatter diagrams for linear correlation
- Calculate and interpret the product moment correlation coefficient (PMCC) r
- Fit the y-on-x least squares regression line using provided formulae
- Make valid predictions for the response variable and explain dangers of extrapolation
- Apply linear coding to simplify correlation and regression calculations

## Scatter Diagrams, Variable Types and Correlation

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.

**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.

1. 1. Identify explanatory variable: Hours spent revising, as it is the independent variable expected to affect test score.
2. 2. 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.

*Calculator:* allowed

## Product Moment Correlation Coefficient (PMCC)

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.

**Product Moment Correlation Coefficient** — Calculated as $r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$, where $S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$, $S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n}$, $S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n}$, and $n$ is the number of data pairs. All formulae are provided in your exam booklet.

*Notation:* r

> **warning**
>
> PMCC only measures LINEAR correlation. A value of $r \approx 0$ does not mean there is no relationship at all, just no linear relationship (e.g. a quadratic relationship may exist).

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

1. 1. Calculate $S_{xx}$:
2. $$S_{xx} = 100 - \frac{24^2}{8} = 100 - 72 = 28$$
3. 2. Calculate $S_{yy}$:
4. $$S_{yy} = 250 - \frac{40^2}{8} = 250 - 200 = 50$$
5. 3. Calculate $S_{xy}$:
6. $$S_{xy} = 156 - \frac{24 \times 40}{8} = 156 - 120 = 36$$
7. 4. Calculate $r$:
8. $$r = \frac{36}{\sqrt{28 \times 50}} = \frac{36}{\sqrt{1400}} \approx 0.962$$
9. 5. 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'.

*Calculator:* allowed

## Fitting the y-on-x Least Squares Regression Line

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 $x$.

**y-on-x Regression Line** — Where the gradient $b = \frac{S_{xy}}{S_{xx}}$, and the y-intercept $a = \bar{y} - b\bar{x}$. $\bar{x}$ and $\bar{y}$ are the mean values of $x$ and $y$ respectively. All formulae are provided in your exam booklet.

*Notation:* y = a + bx

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

> **warning**
>
> Only use the regression line to predict values of $y$ for values of $x$ that lie within the range of the observed $x$ data. Predicting outside this range is called extrapolation, and is unreliable as you have no evidence the linear relationship continues outside the observed range.

**Worked example:** Using the same data from the previous example: $\sum x = 24$, $\sum y = 40$, $S_{xx} = 28$, $S_{xy} = 36$, $n=8$. 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.

1. 1. Calculate gradient $b$:
2. $$b = \frac{S_{xy}}{S_{xx}} = \frac{36}{28} \approx 1.29$$
3. 2. Calculate means:
4. $$\bar{x} = 24 / 8 = 3, \bar{y} = 40 / 8 = 5$$
5. 3. Calculate intercept $a$:
6. $$a = 5 - (1.29 \times 3) \approx 1.13$$
7. 4. Regression line equation: $y = 1.13 + 1.29x$ (3 significant figures)
8. 5. Predict for x=4:
9. $$y = 1.13 + (1.29 \times 4) = 6.29 \approx 6.3$$
10. 6. 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.

*Calculator:* allowed

## Linear Coding for Simplified Calculations

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.

> **info**
>
> PMCC $r$ is unaffected by linear coding of x or y, as it is a measure of relative linear relationship. The regression line coefficients $a$ and $b$ will change, but you can convert back to the original variables by substituting the coding equations into the coded regression line.

**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 $Y = 2 + 3X$. Find the regression line for the original x and y variables.

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

> **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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using x-on-y regression line instead of y-on-x
  - Why it fails: 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: Always fit and use the y-on-x regression line for predicting response variable y from explanatory variable x.
- **Wrong:** Interpreting a low PMCC as meaning no relationship between variables
  - Why it fails: PMCC only measures linear correlation; a low r may mean a strong non-linear relationship exists.
  - Correct: State only that there is no *linear* correlation, and refer to the scatter diagram to check for non-linear relationships.
- **Wrong:** Extrapolating beyond the range of observed x values to make predictions
  - Why it fails: There is no evidence that the linear relationship continues outside the range of collected data, so predictions are unreliable.
  - Correct: Only make predictions for x values within the observed range, and explicitly state if a prediction is unreliable due to extrapolation.
- **Wrong:** Confusing explanatory and response variables when plotting or doing regression
  - Why it fails: The regression line is dependent on which variable is explanatory, so swapping them will give an incorrect line.
  - Correct: Always identify the explanatory variable (x) first: it is the variable that is expected to cause changes in the response variable (y).
- **Wrong:** Trying to memorise all formulae for Sxx, Syy, Sxy, PMCC and regression lines
  - Why it fails: All required formulae are provided in the exam formula booklet, so memorising is unnecessary and leads to errors if you misremember.
  - Correct: Familiarise yourself with where each formula is located in the booklet so you can quickly access it during the exam.

## Cheatsheet

| Concept | Formula / Rule | Key Note |
| --- | --- | --- |
| PMCC r | $r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ | Range: $-1 \leq r \leq 1$; measures only linear correlation; unaffected by linear coding |
| Regression Line $y = a + bx$ | $b = \frac{S_{xy}}{S_{xx}}$, $a = \bar{y} - b\bar{x}$ | Passes through $(\bar{x}, \bar{y})$; 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 |

## 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.

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