GCSE · Maths · AQA · Spec 8300

Scatter graphs, correlation, line of best fit, interpolation/extrapolation

Can a graph of used cars tell you what a 10-year-old car is worth? Drag along the line and find the exact moment it stops telling the truth.

Used cars: walk the line until the evidence runs out

Age of car (years): 6. Value (£1000s): 5

Drag the point along the line. (6, 5) means 6 years old, worth £5000. Then keep going past 8 years.

The thick stretch on the age axis: This line of best fit was drawn through data on used cars aged 1 to 8 years, and that stretch marks those ages. Read the line above that stretch and you're interpolating: there are real cars either side of your reading. Go off either end and you're extrapolating: there are no cars there at all, only the line.

Watch out: Past 8 years the line just keeps falling. At 10 years it reads −1: a car worth minus £1000. The line hasn't broken. The evidence ran out at 8 years, and the line carried on without it.

Correlation

What is the cloud of points telling you?

That car line came from a cloud of points, and the cloud tells its own story before anyone draws a line. Sort each scatter graph by direction first. Then switch the rule and sort the same graphs by strength.

Which way do the points go as you read from left to right?

Still to sort

Positive correlation (0)

Points rise from bottom-left to top-right: as one variable goes up, the other tends to go up.

Negative correlation (0)

Points fall from top-left to bottom-right: as one variable goes up, the other tends to go down.

No correlation (0)

No upward or downward drift at all.

Where the line is: A weak correlation still drifts one way overall. No correlation has no drift to find, however hard you look.

6 of 6 still to sort.

Draw a line of best fit, then use it

Problem

Ten students sat a maths test and a science test, both out of 50. Maths: 12, 18, 20, 25, 28, 31, 35, 38, 42, 45 Science: 15, 19, 24, 26, 28, 35, 36, 40, 43, 48 Draw a line of best fit and use it to estimate the science mark of a student who scored 33 in maths.

Correlation and cause

Does one thing cause the other?

A seaside café keeps records all summer. On days when it sells more ice creams, the first-aid tent on the beach treats more people for sunburn. Plotted on a scatter graph, the days show a strong positive correlation.

Which is closest to what you think the graph tells you?

Eating ice cream must make people more likely to get sunburnt.Getting sunburnt makes people go and buy ice creams.The correlation is so strong that one of them has to be causing the other.The two numbers tend to rise together, but the graph alone can't say why.

How sure are you?

I'm sureI think soI'm guessing

Eating ice cream must make people more likely to get sunburnt.

This is a common misconception

That reads cause into the graph. The graph only shows that on the same days, both numbers tended to be high. It has no way of showing that one made the other happen. And there's an obvious suspect behind both: hot, sunny days bring out ice-cream buyers and burn skin.

Getting sunburnt makes people go and buy ice creams.

This is a common misconception

Flipping the direction doesn't fix it: it's still a claim about cause, and the graph can't show cause either way round. The scatter graph would look exactly the same whichever story you told about it.

The correlation is so strong that one of them has to be causing the other.

This is a common misconception

Strength tells you how closely the points follow a line, not why they do. Something else, like hot sunny weather, can push two things up together and make a very strong correlation with no direct link between them. A strong pattern can even be a coincidence.

The two numbers tend to rise together, but the graph alone can't say why.

This matches the evidence

Exactly. Correlation means the variables tend to change together, and that's all it means. Sunny weather is a likely reason both rise, but even that would need other evidence: the graph can't prove it.

Correlation shows that two things tend to change together. On its own, it never proves that one causes the other.

Put it all together

Your turn: a full exam-style question

A garden centre records the maximum temperature (°C) and the number of bags of compost sold on 10 spring days. The temperatures in the data go from 8 °C to 20 °C. On the scatter graph, the points go up from left to right and lie close to a straight line. The line of best fit passes through (8, 22) and (20, 58).

(a) Describe the correlation. (b) Use the line of best fit to estimate the number of bags sold on a day when the maximum temperature is 14 °C. (c) The manager uses the line to predict the number of bags sold on a day when the maximum temperature is 35 °C. Explain why this prediction may not be reliable. [4 marks]

Your answer

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The mark scheme

(a) Positive correlation (also accept strong positive). (1 mark)
(b) About 40 bags (accept 38 to 42). (1 mark)
(c) 35 °C is outside the range of the data (8 °C to 20 °C), so the prediction is extrapolation. (1 mark)
(c) The trend may not continue beyond the data: there is no evidence about sales on days that hot. (1 mark)

Model answer

(a) Positive correlation: on warmer days, more bags of compost tend to be sold. (b) 14 °C is halfway between 8 °C and 20 °C, so read halfway between 22 and 58 on the line: about 40 bags. (c) 35 °C is well outside the temperatures in the data, which only go from 8 °C to 20 °C, so this is extrapolation. We have no evidence about sales on days that hot, and the trend may not carry on, so the prediction could be a long way out.

Two moves earn the reliability marks: say the value is outside the data (extrapolation), then say why that matters: the trend may not continue there.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Correlation, lines of best fit, and knowing exactly when to stop trusting a prediction.

What you need to know

Plot and read scatter graphs of bivariate data: two measurements for each item.
Describe a correlation by its direction (positive, negative or none) and its strength (strong or weak).
Draw an estimated line of best fit by eye and use it to make predictions.
Tell interpolation from extrapolation, and explain why extrapolation is risky.
Know that a correlation does not show that one variable causes the other.

The big picture

A scatter graph plots pairs of values so you can see whether two things are related. You describe the pattern by its direction and strength, draw an estimated line of best fit, and use it to predict, trusting it inside the data (interpolation) far more than beyond it (extrapolation). And a correlation never proves that one thing causes the other.

Key points

1Positive correlation: as one variable increases, the other tends to increase. Negative correlation: as one increases, the other tends to decrease.

2Strength is how closely the points cluster around a straight line, not how steep the pattern is.

3A line of best fit follows the trend with roughly equal numbers of points on each side. It needn't pass through (0, 0) or any particular point.

4To predict, go from the value you know to the line, then across (or down) to the other axis.

5Interpolation estimates inside the range of the data. Extrapolation estimates outside it, where the trend may not continue.

6Correlation shows two variables tend to change together. It does not prove that one causes the other.

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Worked example

Problem

A line of best fit for used cars passes through (2, 11) and (8, 2), with age in years across and value in £1000s up. The data covers cars aged 1 to 8 years. A car is valued at £8000. Estimate its age.

⚠ Watch out

Judging strength by steepness. A steep pattern in a loose, spread-out cloud is weak correlation; a gentle pattern with every point close to the line is strong.

🧠

Memory hook

No dots, no trust: between the dots, the line is backed by evidence. Beyond the dots, it's just guessing.

✓

Check yourself

Cover the page: name the two things you describe about any correlation, say what interpolation and extrapolation mean, and explain why a strong correlation doesn't prove cause.

Flashcards

(13)

What does a scatter graph show?

Pairs of values plotted as points, so you can see whether two variables are related.

What is bivariate data?

Data with two measurements for each item, like the height and arm span of each student.

Positive correlation

As one variable increases, the other tends to increase. The points rise from bottom-left to top-right.

Negative correlation

As one variable increases, the other tends to decrease. The points fall from top-left to bottom-right.

No correlation

No upward or downward trend: the points show no pattern either way.

Strong or weak correlation?

Strong: the points lie close to a straight line. Weak: there's a trend, but the points are loosely spread around it.

Does a steeper pattern mean a stronger correlation?

No. Strength is how tightly the points hug a line, not how steep the line is.

Does a correlation show that one variable causes the other?

No. It shows the two variables tend to change together, not that one causes the other.

Apart from cause, what can explain a correlation?

A third factor that affects both variables, or plain coincidence.

What is a line of best fit?

An estimated straight line that follows the trend, with roughly as many points above it as below.

Must a line of best fit pass through (0, 0)?

No, and it needn't pass through any particular point. It just follows the trend of the data.

Interpolation

Estimating a value inside the range of the data. A fair estimate, especially when the correlation is strong.

Extrapolation

Estimating a value outside the range of the data. Risky, because the trend may not continue there.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Used cars: walk the line until the evidence runs out

What is the cloud of points telling you?

Does one thing cause the other?

Your turn: a full exam-style question

More AQA GCSE Maths topics