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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn7357Paper 3 · Pure and StatisticsData presentation and interpretationSummary statistics and the PMCC

Data presentation and interpretation · Subtopic 5 of 6

Summary Statistics and the PMCC

The summary statistics $S_{xx}$ , $S_{yy}$ and $S_{xy}$ measure spread and how $x$ and $y$ vary together. The product moment correlation coefficient $r=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ lies between $-1$ and $1$ and measures how close the points are to a straight line.

Edexcel YMA01 WST01Edexcel YMA01 WST03Edexcel 9MA0 Paper 3Edexcel 9FM0 OptionAQA 7357 Paper 3≈ 30 minRevision 6 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Calculate $S_{xx}$ , $S_{yy}$ and $S_{xy}$ from summary totals.
Calculate the PMCC $r$ .
Interpret $r$ in context, including its sign and strength.
Know that coding does not change $r$ , and that correlation is not causation.

Pause and think

For a set of points, multiply each $x$ -deviation from $\bar x$ by its $y$ -deviation from $\bar y$ . If most products are positive, what does the scatter look like?

Section 2 of 14

Core idea

Start hereHow straight is the cloud?

Plot height against shoe size for a class: the dots form a rising cloud. Plot height against birthday: a shapeless blob. One number, between $-1$ and $1$ , measures how close the cloud is to a straight line.

The big idea

$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ , $-1\le r\le1$ . Near $\pm1$ : strong linear correlation; near $0$ : none. Correlation is not causation.

1. Scatter diagrams

Scatter diagrams as the correlation changes

r (approx.) = -0.95

r

close to

\pm1

the points almost form a line. Near

0

they are a shapeless cloud.

2. The formula

S_{xx}=\sum x^2-\frac{(\sum x)^2}n\quad S_{yy}=\sum y^2-\frac{(\sum y)^2}n\quad S_{xy}=\sum xy-\frac{\sum x\sum y}n

$S_{xy}$: $\sum(x-\bar x)(y-\bar y)$ : how $x$ and $y$ vary together.
$n$: number of points.

Why

S_{xy}

measures correlation

$(x-\bar x)(y-\bar y)>0$
Point top-right or bottom-left of the centre $(\bar x,\bar y)$ .
$(x-\bar x)(y-\bar y)<0$
Point top-left or bottom-right.
$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$
Dividing removes units and scale, forcing $-1\le r\le1$ .

Worked example $S_{xx}=42$ , $S_{yy}=110$ , $S_{xy}=63$

Find and interpret $r$ .

Goal: $r$ .
Strategy: Formula.

$\sqrt{42\times110}=\sqrt{4620}=67.97$
Denominator.
$r=\frac{63}{67.97}=0.927$
Divide.

Answer

0.927

: strong positive linear correlation.

✓ CheckBetween

-1

and

1

✓.

Common misconceptionCorrelation is not causation Ice cream sales and drownings correlate because both rise in hot weather.

Edge caseCoding Adding constants or multiplying by positive constants doesn't change

r

Pause and think

What does $r=0$ mean?

Summary

$r=S_{xy}/\sqrt{S_{xx}S_{yy}}$ .
$-1\le r\le1$ .

PMCC: Product moment correlation coefficient.

Revision hours and test score: strong positive correlation

Points close to a rising straight line give a product moment correlation coefficient near

+1

(here about

0.98

). It measures linear association only, and never proves that one thing causes the other.

HistoryGalton and Pearson. Francis Galton introduced the idea of correlation in 1888 while studying heights of parents and children. His colleague Karl Pearson gave the precise formula in 1896, which is why

r

is often called "Pearson's correlation coefficient".

Fun factStorks and babies. Across European regions, the number of storks correlates with the number of births. Storks don't deliver babies: rural areas simply have both more storks and higher birth rates.

Section 3 of 14

Key formulae

Summary statistics

S_{xy}=\sum xy-\frac{\sum x\sum y}{n}

PMCC

r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

Range

-1\le r\le1

TrickSanity check the sign.

S_{xx}

and

S_{yy}

can never be negative;

S_{xy}

can. If you get a negative

S_{xx}

, you've divided the wrong thing by

n

. And

r

's sign always matches

S_{xy}

's.

Section 4 of 14

Worked examples

Example 1 · S values0 / 2 steps

n=8

\sum x=48

\sum x^2=320

\sum y=120

\sum xy=760

. Find

S_{xx}

and

S_{xy}

Example 2 · PMCC0 / 2 steps

S_{xx}=16

S_{yy}=25

S_{xy}=12

. Find and interpret

r

Section 5 of 14

Explore the graph

Scatter getting weaker

Loading the interactive graph…

Try this

At $s=0$ the points lie exactly on a line: $r=1$ .
Increase $s$ . The points scatter away from the line; $r$ falls towards $0$ .
Even with lots of scatter there may still be a trend. When would you call the correlation "weak"?

$r$ measures how tightly points cluster around a straight line.

In the real worldMedical research. Studies first noticed correlations between smoking and lung cancer in the 1950s. Because correlation isn't causation, researchers had to rule out other explanations with further studies before the causal link was accepted.

Section 6 of 14

Common mistakes

Incorrect

S_{xx}=\sum x^2-(\sum x)^2

CorrectDivide

(\sum x)^2

n

Otherwise it's hugely negative.

Incorrect

r=1.25

Correct

|r|\le1

Check you divided by

\sqrt{S_{xx}S_{yy}}

Incorrect"

r

near

0

means no relationship"

CorrectNo linear relationship

A curve can have

r\approx0

Incorrect"Correlation shows

x

causes

y

CorrectIt shows association

A third variable may explain both.

Ice cream sales and shark attacks: $r=0.9$

Correlation, not causation. It's just summer.

Section 7 of 14

Exam tip

Exam tipInterpret in context. "Interpret your value of

r

" needs two things: strength and direction ("strong positive"), and what it means for these variables ("taller trees tend to have wider trunks").

Section 8 of 14

Guided practice

SA-IAL-S1-CR-G012 marksL1 FoundationCalculatorS values

For

8

pairs of values,

\sum x=48

and

\sum x^2=320

. Find

S_{xx}

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-S1-CR1-0012 marksL1 FoundationNon-calculatorS values

For

10

values,

\sum y=50

and

\sum y^2=290

. Find

S_{yy}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR-0012 marksL1 FoundationCalculatorS values

For the same

8

pairs,

\sum y=120

and

\sum xy=760

. Find

S_{xy}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR1-0042 marksL1 FoundationNon-calculatorS values

For

5

pairs of values,

\sum x=15

\sum y=30

and

\sum xy=100

. Find

S_{xy}

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR1-0022 marksL1 FoundationNon-calculatorPMCC

For a set of bivariate data,

S_{xx}=16

S_{yy}=25

and

S_{xy}=12

. Find the PMCC.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR-0023 marksL2 StandardCalculatorPMCC

For

8

pairs of values,

S_{xx}=32

S_{xy}=40

\sum y=120

and

\sum y^2=1910

. Find the product moment correlation coefficient, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR-0071 markL1 FoundationCalculatorPMCC

The PMCC between the age of a car and its value is

-0.92

. This shows

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-S1-CR-0051 markL1 FoundationCalculatorPMCC

Data are coded using

c=\dfrac{x-100}{10}

and

d=y-5

. The PMCC between

c

and

d

0.82

. What is the PMCC between

x

and

y

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-IAL-S1-CR-0103 marksL4 ChallengingCalculatorPMCC

For a set of bivariate data,

S_{xx}=50

S_{xy}=40

and the PMCC is

0.8

. Find

S_{yy}

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$S_{xx}$ , $S_{yy}$ , $S_{xy}$ from the summary totals.
$r=\dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ , $-1\le r\le1$ .
Interpret strength, direction and context.
Coding leaves $r$ unchanged; correlation isn't causation.

Section 12 of 14

Quick quiz

Q1 · SA-IAL-S1-CR-Q011 markL1 FoundationCalculatorPMCC

Which value cannot be a product moment correlation coefficient?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-IAL-S1-CR1-0031 markL1 FoundationCalculatorPMCC

The PMCC between two variables is

-0.1

. This shows

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-IAL-S1-CR1-0051 markL2 StandardCalculatorPMCC

For a group of children, shoe size and reading score have

r=0.9

. The best explanation is that

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7357 A-level Mathematics
Where: Section N, Statistical distributions / Section O
Expected: AQA uses the PMCC for hypothesis tests but calculates it with a calculator.
Calculator: Permitted; some answers must still be exact.
Typical style: AQA mixes short multiple-choice questions with longer structured ones.
Open in: 7357 · Paper 3

Section 14 of 14

Past paper questions

3 questions (39 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WST01/01 · June 2025Question 314 marks
Modelling, correlation and regression from coded summary data.
Question paper, page 6 Mark scheme
WST01/01 · January 2025Question 612 marks
Correlation coefficient, a regression line and its meaning, and the effect of rescaling.
Question paper, page 18 Mark scheme
WST01/01 · June 2024Question 413 marks
Summary statistics, correlation and the regression line.
Question paper, page 14 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7357, First teaching 2017 (official specification).