CHAPTER 7

Correlation & Regression

AQA

MATHEMATICAL STUDIES

AQA Mathematical Studies · Paper 2A

Lines of Best Fit  ·  Regression Lines  ·  Pearson's PMCC

AQA MATHEMATICAL STUDIES

LEVEL 3 CERTIFICATE

AQA MATHEMATICAL STUDIES — PAPER 2A

Chapter 7: Overview & Slide Structure

PREREQUISITES

Recognise correlation

·

Calculate mean (x̄ and ȳ)

·

Use scatter graphs

·

Find gradient of a straight line

SECTION 7.1 — SLIDES 3–9

Lines of Best Fit

Scatter graphs, mean point

(x̄, ȳ)

, drawing by eye, outliers, gradient interpretation,

interpolation

vs

extrapolation

.

SECTION 7.2 — SLIDES 10–16

Regression Lines

Equation

y = a + bx

, using the calculator to find a and b, making predictions, and

understanding the limitations of the model.

SECTION 7.3 — SLIDES 17–24

Pearson's PMCC

Formula

r = s

xy

/ (s

x

·s

y

)

, range

−1 ≤ r ≤ +1

, interpreting strength and direction of

correlation.

EVERY SECTION FOLLOWS THIS STRUCTURE

Slide-by-Slide Format

Example 1

→

Example 2

→

Key Skill

→

Exercise Q

→

Exercise A

→

Past Paper Q

→

Past Paper A

7.1 Example 1 — Delivery Van: Lines of Best Fit

AQA 7.1

JOURNEY DATA (10 JOURNEYS)

Outlier excluded — accident caused unusually long time (296 min), distorting the line.

9 points used: x̄ = 1461 ÷ 9 = 162.3 mi  |  ȳ = 1736 ÷ 9 = 193.0 min

SCATTER GRAPH WITH LINE OF BEST FIT

MEAN POINT

(162.3, 193.0)

Line MUST pass through this

GRADIENT

≈ 1.31 min/mile

(273−130) ÷ (226−117) = 143/109

PREDICTION: 150 MILES

≈ 177 minutes

193 − 1.31 × (162.3 − 150)

7.1 Example 2 — Pocket Money vs Age

Average weekly pocket money (£) for children aged 5–16 · AQA Mathematical Studies 2015 data

FULL DATA TABLE

Calculations

x̄ = (5+6+…+16) ÷ 12 = 126 ÷ 12 = 10.5

ȳ = (3.28+4.00+…+10.27) ÷ 12 = 75.95 ÷ 12 = £6.33

Gradient = (10.27 − 3.28) ÷ (16 − 5)

           = 6.99 ÷ 11 ≈ £0.64 per year

SCATTER GRAPH WITH LINE OF BEST FIT

MEAN POINT

(10.5, £6.33)

Line MUST pass through this point

GRADIENT

≈ £0.64 / year

64p more pocket money per year older

CORRELATION

Strong Positive

As age ↑, pocket money ↑

INTERPOLATION

Within range ✓

Extrapolation (outside 5–16) = unreliable

7.1 Key Skill — Lines of Best Fit

Golden Rule: The line of best fit MUST always pass through the mean point (x̄, ȳ) — this is non-negotiable.

4-Step Method

1

Plot all data points on a scatter graph with sensible scales and clearly labelled axes.

2

Calculate the mean of x-values (x̄) and the mean of y-values (ȳ) separately.

3

Mark the mean point (x̄, ȳ) clearly on the graph — use a cross ✕ to distinguish it from data points.

4

Draw a straight line through (x̄, ȳ) with roughly equal numbers of points above and below the line.

Gradient = Δy ÷ Δx — always state units and interpret in context.

e.g. "for each extra mile, journey takes 1.31 min longer"

Outlier: a point far from the line — may be due to error or unusual circumstance. Always state your reason for excluding it.

Key Concepts

Interpolation vs Extrapolation

Interpolation — predicting within the data range → reliable .

Extrapolation — predicting outside the data range → unreliable (trend may not continue).

Correlation Types (r value)

Strong positive (r ≈ +1)

Weak positive (r > 0)

No correlation (r ≈ 0)

Weak negative (r < 0)

Strong negative (r ≈ −1)

WORKED EXAMPLE — DELIVERY VAN

Gradient = 1.31 minutes per mile

Interpretation: "For each extra mile driven, the journey takes approximately 1.31 minutes longer ."

Outlier (138 mi, 296 min) excluded — accident caused unusually long time.

Exercise 7A — Questions

7.1 LINES OF BEST FIT

AQA Mathematical Studies · Chapter 7.1 · Exercise 7A  |  All data provided — no textbook required

Q1 — DESCRIBE THE CORRELATION

For each pair of variables, describe the expected correlation and explain what it means in context.

(a)

Maximum daily temperature and ice cream sales

(b)

Height and IQ

(c)

Marathon training time and race time

(d)

Spring extension and mass attached (Hooke's Law)

(e)

Engine size and time to accelerate to 60 mph

(f)

Height of horse chestnut tree and trunk circumference

State: positive / negative / no correlation — and give a reason.

Q2 — POCKET MONEY VS AGE (2015)

Average weekly pocket money (£) for children aged 5–16 in 2015. All data below.

Mean age x̄ = 126 ÷ 12 = 10.5 years  |  Mean pocket money ȳ = 75.95 ÷ 12 = £6.33

TASKS:

(a)

Draw a scatter graph with age on the x-axis and pocket money on the y-axis.

(b)

Calculate the mean point (x̄, ȳ) — shown above.

(c)

Draw a line of best fit passing through the mean point (10.5, £6.33).

(d)

Describe the correlation between age and pocket money.

(e)

Use the gradient to find how much extra pocket money children receive per year older.

Q3 — POSITIVE OR NEGATIVE?

For each pair, state whether the correlation is positive or negative and justify your answer.

(a)

Height and shoe size

Think: taller people tend to have…

(b)

Butter consumption and margarine consumption

Think: are these substitutes or complements?

(c)

Weight of loaded lorry and time to accelerate to 50 mph

Think: heavier lorry → faster or slower?

(d)

Height from which a ball is dropped and height of first bounce

Think: higher drop → higher or lower bounce?

(e)

Average speed and time taken to travel between two towns

Think: faster speed → more or less time?

State: positive or negative — give a real-world reason for each.

Exercise 7A — Answers

CHAPTER 7 · CORRELATION

For every answer, state the direction of correlation AND give a real-world reason — not just "positive" but "positive because…"

⚠ KEY RULE

Always state the direction of correlation AND give a real-world reason — not just "positive" but "positive because hotter days lead to more ice cream sales."

Q1 (A) & (B)

Ice cream sales & Height vs IQ

(a) Positive — hotter days lead to more ice cream sold.

(b) No / zero correlation — height and IQ are unrelated.

Q1 (D) & (E)

Spring extension & Engine size

(d) Positive — greater mass → greater spring extension (Hooke's Law).

(e) Negative — larger engine → faster acceleration → less time to reach 60 mph.

Q1 (C) & (F)

Marathon training & Tree height

(c) Negative — more training → faster marathon → less time taken.

(f) Positive — taller tree → larger trunk circumference.

Q3 (A) – (E)

Feet, Butter, Lorry, Bounce, Speed

(a) Positive — taller people tend to have larger feet.

(b) Negative — as butter rises, margarine falls (substitutes).

(c) Negative — heavier lorry → slower → more time.

(d) Positive — greater drop height → higher first bounce.

(e) Negative — higher speed → less time for same journey.

Q2 — POCKET MONEY

Mean point & Line of Best Fit

Mean point = (10.5, £6.33) . Line of best fit passes through this point.

Gradient ≈ £0.64 per year — children receive ~64p more pocket money per week for each year older.

Strong positive correlation — as age increases, pocket money increases.

Past Paper — Lines of Best Fit (AQA)

AQA Mathematical Studies 1350 · Paper 2A

DATA: AVERAGE WEEKLY POCKET MONEY (2015)

EXAM QUESTIONS

(a)

Draw a scatter graph and a line of best fit for this data.

2 marks

(b)

Describe the correlation shown in the scatter graph.

1 mark

(c)

How much extra pocket money do children receive for each year they get older? Interpret the gradient.

2 marks

(A) LINE OF BEST FIT [2 MARKS]

Plot all 12 points correctly. Draw a straight line through the mean point (10.5, £6.33) with roughly equal points above and below.

(B) CORRELATION [1 MARK]

Strong positive correlation — as age increases, weekly pocket money increases.

(C) GRADIENT [2 MARKS]

Gradient ≈ £0.64 per year . Each year older, children receive approximately 64p more pocket money per week.

Exam Tip: Always calculate the mean point first: x̄ = (5+6+…+16)÷12 = 10.5, ȳ = (3.28+4.00+…+10.27)÷12 = £6.33. Your line of best fit must pass through (10.5, £6.33) . The gradient tells you the rate of change — always state units and context.

Scatter Graph with Line of Best Fit — Mean Point (10.5, £6.33) marked ✕

Past Paper — Lines of Best Fit: Answers

Total: 5 Marks

1

EXAM TIP

Always show the mean point calculation. State x̄ and ȳ explicitly — the line must pass through (x̄, ȳ).

2

EXAM TIP

State 'strong' or 'weak' when describing correlation — not just 'positive'. Describe the direction AND strength.

3

EXAM TIP

Interpret the gradient in context — say what the numbers mean with units, not just the calculation.

a

Plot points & draw line of best fit

[2 marks]

✓ Plot all 12 data points correctly on the scatter graph.

✓

Calculate the

mean point

:

x̄ = (5+6+…+16) ÷ 12 = 126 ÷ 12 =

10.5

ȳ = (3.28+4.00+…+10.27) ÷ 12 = 75.95 ÷ 12 =

£6.33

Draw a straight line

through

(10.5, £6.33)

with roughly equal points above and below.

b

Describe the correlation

[1 mark]

✓Strong positive correlation — as age increases, weekly pocket money increases. The points lie close to a straight line.

c

Interpret the gradient

[2 marks]

✓

Gradient calculation:

Gradient = (10.27 − 3.28) ÷ (16 − 5) = 6.99 ÷ 11 ≈

£0.64 per year

✓

Interpretation: For each year older, children receive approximately

64p more

pocket money

per week.

SCATTER GRAPH WITH LINE OF BEST FIT & MEAN POINT

7.2 Example 1 — UK Overseas Visits: Regression Lines

n = number of visits (thousands)  |  P = expenditure (£ millions)  |  Data: 2014

FULL DATA TABLE

REGRESSION EQUATION

P = −186 + 0.628n

Found using calculator (3 s.f.)

GRADIENT B = 0.628

+£0.628m per 1 000 visits

Each extra 1 000 visits → £628 000 more expenditure

Y-INTERCEPT A = −186

Not meaningful

n = 0 is outside the data range

PREDICTION: N = 7 000

P = £4 210m ✓

−186 + 0.628 × 7000 = 4210 (interpolation —reliable)

EXTRAPOLATION WARNING

n = 2 000 or 10 000 ✗

Outside data range — unreliable, do not use

MEAN POINT

(5 078, 3 005)

Line of best fit must pass through (n̄, P̄)

Scatter Plot with Regression Line  P = −186 + 0.628n

7.2 Example 2 — Oral & Written Test Marks

12 students · each test out of 40 · Ed absent for written test

STUDENT DATA (ORAL / WRITTEN MARKS)

Ed excluded when finding regression line — no written mark to include.

STEP 1 — EXCLUDE ED

Use only the 12 complete pairs . Ed has no written mark, so he cannot be included in the regression calculation.

STEP 2 — REGRESSION LINE

Enter 12 data pairs into

calculator →

y = 10.8 + 0.574x

Mean point: (26.1, 25.8)

STEP 3 — INTERPRET

b = 0.574 : each extra oral mark→ +0.574 written marks.

a = 10.8 : y-intercept (not meaningful here).

STEP 4 — PREDICT ED

x = 34 →

y = 10.8 + 0.574×34

= 10.8 + 19.5 = 30.3 ≈ 30 marks

7.2 Key Skill — Regression Lines

How to find and use the equation y = a + bx

WORKED EXAMPLE

Regression line: y = 10.8 + 0.574x

When x = 34:   y = 10.8 + 0.574 × 34 = 10.8 + 19.5 = 30.3 ≈ 30

Ed's predicted written mark ≈ 30 (interpolation —reliable)

4-Step Method

1

Enter data — input all x and y values into your calculator's statistics mode.

2

Run regression — use the built-in regression function to find a (y-intercept) and b (gradient).

3

Write the equation — state as y = a + bx, rounding sensibly to 3 significant figures.

4

Plot the line — draw through the mean point (x̄, ȳ) and one other calculated point.

Interpreting & Using the Line

b = gradient — the increase in y for each 1-unit increase in x. Always state units and context.

a = y-intercept — the predicted y when x = 0. Check whether x = 0 is realistic; it may not be meaningful.

Predict: substitute x into the equation. Only predict within the data range— interpolation is reliable; extrapolation is not.

Missing values: exclude any data point missing one value when finding the line, then use the line to predict the missing value.

Exercise 7B — Questions

ALL DATA PROVIDED

Q1

Memory Test — Words Remembered

A volunteer memorises 100 words and is tested weekly. The number of words correctly recalled (y) is recorded each week (x).

TASKS

(a) Draw a scatter graph of the data.

(b)(i) Find the equation of the regression line of y on x.

(b)(ii) Plot the regression line on your scatter graph.

Q2

Oral & Written Test Marks (out of 40)

12 students sat both an oral test and a written test. Ed scored 34 on the oral test but was absent for the written test.

⚠ Ed: oral = 34, written = absent. Ignore Ed when finding the regression line.

TASKS

(a) Find the regression line of written mark (y) on oral mark (x). Ignore Ed.

(b) Draw a scatter graph and plot the regression line.

(c) Use the line to predict Ed's written mark.

Q3

Premiership Season Tickets 2013–14 (£)

Cheapest (x) and most expensive (y) season ticket prices for 20 Premier League clubs.

TASKS

(a) Find the regression line of most expensive (y) on cheapest (x).

(b) Find y when x = 0. Explain why this gives no useful information.

(c) Interpret the gradient of the regression line in context.

Exercise 7B — Answers

✓ Full Worked Solutions

Q1 — Memory Test (Words Recalled)

REGRESSION LINE

y = 100.5 − 4.93x

where x = week number, y = words recalled

Coefficients to 3 s.f. using calculator regression function

GRADIENT INTERPRETATION

b = −4.93: The volunteer forgets approximately 5 words per week . The negative gradient shows memory declines over time.

Y-INTERCEPT CHECK (X = 0)

When x = 0: y ≈ 100.5 This makes sense — the volunteer started with 100 words memorised, so the intercept is meaningful here.

Q2 — Oral & Written Marks (out of 40)

REGRESSION LINE (12 STUDENTS, EXCLUDING ED)

y = 10.8 + 0.574x

where x = oral mark, y = written mark

Mean point: x̄ = 313 ÷ 12 = 26.1 , ȳ = 309 ÷ 12 = 25.8

Line passes through mean point (26.1, 25.8) ✓

ED'S PREDICTED WRITTEN MARK (ORAL = 34)

y = 10.8 + 0.574 × 34

y = 10.8 + 19.516

y = 30.316

Ed's predicted written mark ≈ 30 marks

NOTE

Ed was excluded when finding the regression line (missing written mark), but the line is then used to predict his result.

Q3 — Premiership Season Tickets 2013–14

REGRESSION LINE (20 CLUBS)

y = 175 + 1.35x

where x = cheapest ticket (£), y = most expensive ticket (£)

Coefficients to 3 s.f. using calculator

(B) WHEN X = 0: Y = 175 — NOT MEANINGFUL

No Premier League club offers free season tickets (x = 0). This value lies well outside the data range , so extrapolation gives no useful information here.

(C) GRADIENT INTERPRETATION

Gradient = 1.35: For every £1 increase in the cheapest ticket price, the most expensive ticket increases by approximately £1.35 .

Past Paper — Regression Lines

AQA MATHEMATICAL STUDIES

5 marks total

(A) REGRESSION LINE [2 MARKS]

y = 10.8 + 0.574x

Mean point: x̄ = 26.1, ȳ = 25.8  |  Line passes through (26.1, 25.8)

(B) SCATTER GRAPH [2 MARKS]

Plot 12 points → draw line through (26.1, 25.8)

Line extends across the data range (x ≈ 16 to 37)

(C) ED'S PREDICTION [1 MARK]

y = 10.8 + 0.574 × 34 = ≈ 30 marks

10.8 + 19.516 = 30.3 → round to 30

EXAM TIP

When a student is absent for one test, exclude them from the regression calculation. Then use the completed line to predict their missing result by substituting their known score.

QUESTION CONTEXT

The table shows marks achieved by students in an oral test and a written test (each out of 40 marks). Ed scored 34 on the oral test but was absent for the written test — exclude him when finding the regression line.

Ed's data: oral = 34, written = absent (use line to predict)

STUDENT MARKS (ORAL X, WRITTEN Y)

TASKS

(a)

Find the equation of the regression line of written mark (y) on oral mark (x). Use the 12 students only.

[2 marks]

(b)

Draw a scatter graph of the data and plot the regression line on your graph.

[2 marks]

(c)

Ed scored 34 on the oral test. Use the regression line to predict his written mark.

[1 mark]

SCATTER GRAPH — ORAL VS WRITTEN MARKS (WITH REGRESSION LINE)

Past Paper — Regression Lines: Answers

AQA Mathematical Studies 1350 · Paper 2A

(a)

[2 marks] — Regression line of written (y) on oral (x)

Use calculator with all 12 data points (excluding Ed).

x̄ = (18+32+36+23+28+37+24+31+24+16+27+17) ÷ 12 = 313 ÷ 12 = 26.1

ȳ = (21+30+32+27+26+27+31+33+22+23+23+14) ÷ 12 = 309 ÷ 12 = 25.8

Regression line:

y = 10.8 + 0.574x

   Mean point:

(26.1, 25.8)

(b)

[2 marks] — Scatter graph + regression line

Plot all 12 data points on axes: oral mark (x) vs written mark (y).

Draw the regression line passing through the mean point (26.1, 25.8) .

Check: when x = 10 → y = 10.8 + 5.74 = 16.5  |  when x = 40 → y = 10.8 + 22.96 = 33.8

See scatter plot →

(c)

[1 mark] — Predict Ed's written mark (oral = 34)

Substitute x = 34 into regression line:

y = 10.8 + 0.574 × 34 = 10.8 + 19.516 = 30.316

Ed's predicted written mark:

≈ 30 marks

Total Marks Available

5 marks  (a: 2 + b: 2 + c: 1)

Scatter Graph: Oral vs Written Marks (12 students) with Regression Line & Ed's Prediction

12 Students

Regression line: y = 10.8 + 0.574x

Ed's prediction (34, 30)

Mean point (26.1, 25.8)

7.3 Example 1 — Airliners: Pearson's PMCC

Wingspan (w metres) vs Length (l metres) for 10 commercial airliners — regression line & correlation coefficient

AIRLINER DATA — ENTER INTO CALCULATOR

PMCC SCALE: R RANGES FROM −1 TO +1

−1 Perfect negative

0 No correlation

+1 Perfect positive

r = 0.960 ▲ (close to +1 → strong positive correlation)

SCATTER GRAPH WITH REGRESSION LINE

REGRESSION LINE

w = 7.79 + 0.647l

Gradient 0.647: wingspan increases by 0.647 m per 1 m increase in length

PMCC (R)

r = 0.960

Strong positive correlation — length and wingspan closely related

Comparison: Light aircraft gave r = 0.625 — less strongly correlated than airliners (0.625 < 0.960). The relationship between length and wingspan is stronger for airliners .

Wingspan w vs Length l — w = 7.79 + 0.647l

7.3 Example 2 — House Prices & Rents: PMCC

AQA Chapter 7.3 · England 2011

DATA: AVERAGE HOUSE PRICE (£P THOUSANDS) & WEEKLY RENT (£R)

GRADIENT INTERPRETATION

b = 0.107 : for every £1,000 increase in average house price, weekly rent increases by approximately £0.107 (about 11p per week). London has the highest prices and highest rents.

REGRESSION LINE

R = 55.1 + 0.107P

Use calculator · 3 s.f.

PMCC (R)

r = 0.981

Very strong positive correlation

MEAN POINT

(228.6, 76.7)

Line passes through (P̄, R̄)

R CLOSE TO +1 MEANS…

Strong +ve

As P ↑, R ↑ very strongly

Scatter Graph: Weekly Rent (R) vs House Price (P) with Regression Line

7.3 Key Skill — Pearson's PMCC

How to find and interpret the Product Moment Correlation Coefficient using your calculator

PMCC Scale & Interpretation

FORMULA

r = s xy / (s x · s y )

Use your calculator's built-in PMCC function — do not calculate by hand

THE R SCALE: ALWAYS −1 ≤ R ≤ +1

−1

−0.5

0

+0.5

+1

Step-by-Step Calculator Method

1

Enter data — input all x and y values into your calculator's statistics mode

2

Find regression line — use the regression function to obtain a and b (the line y = a + bx)

3

Find r — use the PMCC function on your calculator to obtain the value of r

4

State r to 3 s.f. — write your answer rounded to 3 significant figures

5

Interpret r — state the direction (positive/negative) AND strength (strong/moderate/weak) in context

KEY PROPERTIES

Sign of r = same as gradient of regression line

|r| close to 1 = strong; |r| close to 0 = weak/none

Always interpret in context: State both the direction (positive/negative) and the strength (strong/moderate/weak) using the actual variable names — e.g. "There is a strong positive correlation between house price and weekly rent."

R VALUE

INTERPRETATION

r = −1

Perfect negative correlation

−1 < r < −0.7

Strong negative correlation

−0.7 < r < −0.3

Moderate negative correlation

−0.3 < r < +0.3

Weak / no linear correlation

+0.3 < r < +0.7

Moderate positive correlation

+0.7 < r < +1

Strong positive correlation

r = +1

Perfect positive correlation

Exercise 7C — Questions

ALL DATA PROVIDED

Q1

House Prices & Weekly Rents — England 2011

Average house price P (£thousands) and average weekly rent R (£) for 9 regions of England.

TASKS

(a)

Find the regression line of R on P and the PMCC.

(b)

Draw a scatter diagram and add the regression line.

Use your calculator's regression & PMCC functions.

Q2

Coal & Natural Gas Usage — UK (Mtoe)

Monthly UK energy usage in million tonnes of oil equivalent (Mtoe). x = coal, y = natural gas.

TASKS

(a)

Find the regression line of y on x and the PMCC.

(b)

Draw a scatter diagram and add the regression line.

Q3

Car Insurance Cost vs Age

Annual car insurance cost C (£) for drivers of different ages n (years).

TASKS

(a)

Using your calculator:

(i)

Find the regression line of C on n .

(ii)

Interpret the gradient in context.

(b)

Using your regression line:

(i)

Predict the cost for a 40-year-old driver.

(ii)

Draw a scatter graph of the data.

(c)

Comment on the suitability of linear regression for this data.

Exercise 7C — Answers

FULL WORKED SOLUTIONS

All regression lines and PMCC values found using calculator — enter data, use regression function for a & b, use PMCC function for r. Always state r to 3 s.f. and interpret in context.

Q1

House Prices & Weekly Rents (England)

REGRESSION LINE

R = 55.1 + 0.107P

Where P = house price (£000s), R = weekly rent (£)

PMCC

r = 0.981

Very strong positive correlation (3 s.f.)

As house prices increase across English regions,

weekly rents also increase very strongly.

Q2

Coal vs Natural Gas Usage (UK)

REGRESSION LINE

y = −0.0476 + 2.60x

Where x = coal usage, y = gas usage (million tonnes oil equiv.)

PMCC

r = 0.960

Strong positive correlation (3 s.f.)

When more coal is used, more gas is also used — both

fuels peak together in winter months.

Q3

Car Insurance Cost vs Age

(A)(I) REGRESSION LINE

C = 1640 − 20.5n

Where n = age (years), C = insurance cost (£)

(A)(II) GRADIENT INTERPRETATION

Gradient = −20.5 — for each additional year of age, insurance cost decreases by approximately £20.50 .

(B)(I) PREDICTION: N = 40

C = 1640 − 20.5 × 40 = 1640 − 820 = £820

(C) MODEL COMMENT

The scatter graph shows a curved relationship —insurance falls rapidly for young drivers then levels off. Linear regression is not appropriate ; a curve would be a better model.

Past Paper — Pearson's PMCC (AQA)

AQA Mathematical Studies · Paper 2A

QUESTION — HOUSE PRICES & RENTS (ENGLAND, 2011)

The table gives average house prices (£P thousands) and weekly rents (£R) in regions of England in 2011.

2 marks

(a) Find the equation of the regression line of R on P.

1 mark

(b) Find the PMCC.

2 marks

(c) Draw a scatter diagram showing the data and regression line.

SCATTER DIAGRAM WITH REGRESSION LINE

✓ (A) REGRESSION LINE [2 MARKS]

R = 55.1 + 0.107P

Mean point: P̄ = 228.6, R̄ = 76.7

✓ (B) PMCC [1 MARK]

r = 0.981

Very strong positive correlation

✓ (C) SCATTER DIAGRAM [2 MARKS]

9 points + line

Line passes through mean point (228.6, 76.7)

Exam Tip: Always state the PMCC value AND interpret it in context — give both the direction (positive/negative) and strength (strong/moderate/weak) using the variable names. E.g. "As house prices increase, weekly rents also increase very strongly."

R on P — House Prices vs Weekly Rents

Data points

Regression line

Mean point (228.6, 76.7)

Past Paper Answers & Key Formulae Summary

AQA Mathematical Studies · Chapter 7

MARK SCHEME — HOUSE PRICES & RENTS

(a) Regression Line

2 marks

Use calculator with all 9 data points entered.

R = 55.1 + 0.107P

 (3 s.f.)

Mean point: P̄ = 228.6, R̄ = 76.7 — regression line must pass through this point.

(b) PMCC

1 mark

r = 0.981

 (3 s.f.)

Very strong positive correlation — as house prices increase, weekly rents also increase very

strongly.

(c) Scatter Diagram

2 marks

Plot all 9 data points correctly on axes (P on x-axis, R on y-axis).

Draw regression line passing through mean point

(228.6, 76.7)

.

Total Marks

5 marks

KEY FORMULAE — CHAPTER 7

1

Line of Best Fit

Must pass through the mean point

(x̄, ȳ)

. Gradient =

Δy / Δx

.

2

Regression Line

y = a + bx

— use calculator.

b

= gradient (increase in y per unit x).

a

= y-intercept

(may not be meaningful).

3

Pearson's PMCC

r = s

/ (s

· s

)

— use calculator. Range:

−1 ≤ r ≤ +1

.

Sign of r = direction  |  |r| close to 1 = strong  |  |r| close to 0 = weak/none

4

Interpolation vs Extrapolation

Within range

= interpolation → reliable.  

Outside range

= extrapolation →

unreliable.

5

Always Interpret in Context

State gradient with units (e.g. "rent increases by £0.107 per £1000 of house price"). State r direction AND strength with reference to the variables.

Exam tip: Always state the PMCC value AND interpret it in context — say what the strength and direction mean for the specific variables being studied.

xy

x

y

Interactive Quiz — Chapter 7: Correlation & Regression

5 Questions

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QUESTION

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OF 5

A scatter graph shows the relationship between hours of sunshine and ice cream sales. Describe the expected correlation and explain what it means.

Reveal Answer

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TIME REMAINING

Think before revealing!

PROGRESS TRACKER

Q1

Describing correlation

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Q2

Mean point rule

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Q3

Gradient interpretation

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Q4

PMCC strength & direction

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Q5

Extrapolation

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EXAM TIP

Always state BOTH the direction (positive/negative) AND the strength (strong/moderate/weak) when describing correlation. Interpret in context of the variables.

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