S6StatisticsFoundation & HigherGrades 2–5

Scatter Graphs

A scatter graph plots pairs of values — such as hours revised and test score — as points, to see whether the two quantities are related. From it you can describe the correlation, draw a line of best fit, make estimates, and judge whether those estimates can be trusted.

35 min Video by Maths Genie AQA GCSE Maths

The big picture

Correlation describes a pattern, not a cause. Ice-cream sales and sunburn cases rise together, but ice cream does not cause sunburn — hot weather drives both. AQA regularly asks you to explain exactly this distinction.

A line of best fit is only reliable inside the range of the data. Estimating within that range (interpolation) is usually sensible; going beyond it (extrapolation) assumes the pattern continues, which it may not.

Scatter GraphsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Plot scatter graphs and identify outliers
Describe positive, negative and no correlation, and its strength
Draw a line of best fit and use it to estimate values
Explain the limitations of interpolation, extrapolation and correlation

Describing correlation

$\textbf{Positive}$ correlation: as one variable increases, the other tends to increase. $\textbf{Negative}$ correlation: as one increases, the other tends to decrease. $\textbf{No}$ correlation: no clear pattern.

Describe the strength too: points close to a straight line show strong correlation; widely scattered points show weak correlation. In context: "the more hours revised, the higher the score tends to be."

Positive, negative and no correlation.

Example 1Describe the likely correlation between the age of a car and its value.

1As age increases, value tends to decrease.

Answer

Negative correlation: older cars tend to be worth less.

Your turnSame method, new numbers — try it before moving on.

What type of correlation would you expect between the outside temperature and the amount of energy used to heat homes? Type positive, negative or none.

Your answer

Show worked solution

Solution

1As temperature rises, less heating is needed.
2One increases while the other decreases: negative correlation.

Answer: negative

Lines of best fit and estimates

Draw a single straight line with a ruler that follows the trend, with roughly equal numbers of points on each side. It does not have to pass through the origin or through any particular point, and should ignore outliers.

To estimate, go up from the $x$ value to the line, then across to read $y$ (or the reverse).

Reading from the line: 5 hours of revision gives an estimated score of about 56%.

Example 1Using the graph, estimate how many hours a student revised if they scored 80%.

1Go across from 80 on the vertical axis to the line.

2Then down to the horizontal axis: about 8 hours.

Answer

About 8 hours

Tip — Leave your reading lines on the graph — they show your method.

Reliability, outliers and causation

$\textbf{Interpolation}$ (estimating within the data range) is fairly reliable. $\textbf{Extrapolation}$ (beyond it) is unreliable: using the graph to predict a score for 20 hours would give more than 100%.

An $\textbf{outlier}$ is a point that does not fit the pattern. It may be an error, or a genuine unusual case, such as a student who was ill on the test day.

Correlation does not prove causation. Two variables may be linked by a third factor, or the link may be coincidence.

Example 1A town finds a strong positive correlation between the number of ice creams sold and the number of drownings. Does eating ice cream cause drowning?

1Both increase in hot weather, when more people buy ice cream and more people swim.

Answer

No — a third factor (hot weather) explains both. Correlation does not show causation.

Why extrapolation fails

A line of best fit is a model of the relationship, and every model has a range where it works. Predicting a 25-year-old's height from a graph of children aged 2 to 12 would suggest they are over 2 metres tall — the growth pattern simply stops.

Think like an examiner

Describe correlation in context: "as the temperature increases, sales tend to increase" — "positive" alone may not score.
A line of best fit that is far off the trend, or drawn point-to-point, loses the mark.
When asked about reliability, mention interpolation/extrapolation or the number of data points.

Key ideas

\uparrow \uparrow \ \text{positive}, \quad \uparrow \downarrow \ \text{negative}

Types of correlation.

\text{interpolation} \checkmark, \quad \text{extrapolation} \times

Reliability of estimates.

\text{correlation} \ne \text{causation}

A pattern is not proof of cause.

Watch out for these

Joining the points dot-to-dot.

Draw one straight line of best fit through the trend.

Forcing the line of best fit through the origin.

It only needs to follow the trend.

"Revision causes higher scores because they are correlated."

Correlation suggests a link but does not prove cause.

Trusting an estimate far outside the data range.

Extrapolation is unreliable.

Stretch yourself

A line of best fit for score ( $y$ , %) against hours revised ( $x$ ) passes through $(2, 32)$ and $(8, 80)$ . Find its equation, use it to estimate the score for 5 hours, and explain why it should not be used for 15 hours.

Hint — Find the gradient, then the intercept, as for any straight line.

Reveal solution

SolutionGradient

= \frac{80 - 32}{8 - 2} = 8

. Using

(2, 32)

32 = 16 + c

, so

c = 16

and

y = 8x + 16

. For 5 hours:

y = 56

%. For 15 hours the model gives

136

%, which is impossible — 15 hours is far outside the data, so this is unreliable extrapolation.

\blacksquare

Questions students ask

What is the difference between strong and weak correlation?

Strong correlation has points lying close to a straight line; weak correlation has a visible trend but with points widely scattered.

Should I include the outlier when drawing the line of best fit?

No — ignore obvious outliers so that they do not pull the line away from the general trend.

Can two variables have a relationship but no linear correlation?

Yes. Some relationships are curved, which a straight line of best fit cannot describe well. At GCSE, you mainly describe linear correlation.

Key takeaways

Positive: both increase together; negative: one increases as the other decreases.
Describe strength and give context.
Line of best fit: one straight ruled line following the trend, ignoring outliers.
Interpolation is reliable; extrapolation is not.
Correlation does not imply causation.

Exam-style questions

Practise it the way it is examined

2 original questions · 5 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

A scatter graph shows the heights and shoe sizes of 12 adults. The points lie close to a straight line going up from left to right.

(a)Describe the correlation.[1]
Show mark scheme (a)
- B1Positive (strong) correlation
(b)Describe the relationship between height and shoe size.[1]
Show mark scheme (b)
- B1Taller people tend to have larger shoe sizes

Question 2Grade 43 marks

A line of best fit for ice-cream sales ( $y$ ) against temperature ( $x$ , °C) passes through $(15, 120)$ and $(30, 300)$ . The data was collected for temperatures between 12°C and 32°C.

(a)Estimate the sales when the temperature is 20°C.[2]
Show mark scheme (a)
- M1Gradient $\frac{180}{15} = 12$ per °C, or reading from a drawn line
- A1 $180$ (accept 170–190 from a graph)
Answer: $180$
(b)Explain why using the line to estimate sales at 40°C may not be reliable.[1]
Show mark scheme (b)
- B140°C is outside the range of the data (extrapolation); the trend may not continue

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

As $x$ increases, $y$ tends to increase. This is…
- positive correlation
- negative correlation
- no correlation
- causation
Show answer and explanation
Answer: positive correlation
Both go up together.
Expected correlation between age of a laptop and its value?
- negative
- positive
- none
- perfect
Show answer and explanation
Answer: negative
Older laptops tend to be worth less.
A point far from the trend is called…
- an outlier
- a median
- a line of best fit
- the mode
Show answer and explanation
Answer: an outlier
It does not fit the pattern.
Estimating a value within the range of the data is called…
- interpolation
- extrapolation
- correlation
- sampling
Show answer and explanation
Answer: interpolation
"Inter" means between.
A line of best fit should…
- follow the trend with points on both sides
- join every point
- pass through the origin
- go through the outlier
Show answer and explanation
Answer: follow the trend with points on both sides
It models the general trend.
Expected correlation between shoe size and IQ?
- none
- positive
- negative
- strong positive
Show answer and explanation
Answer: none
No link is expected.
Why is extrapolation unreliable?
- the pattern may not continue beyond the data
- it uses too many points
- it ignores the outliers
- it needs a curve
Show answer and explanation
Answer: the pattern may not continue beyond the data
There is no evidence outside the range.
Strong correlation between two variables shows that…
- they are related, but one may not cause the other
- one causes the other
- they are equal
- there are no outliers
Show answer and explanation
Answer: they are related, but one may not cause the other
Correlation is not causation.

Next step

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