Points falling from top left to bottom right show:

negative correlation. As one variable increases the other decreases, which is negative correlation. Its strength depends on how closely the points follow the trend.

A line of best fit should pass through:

the middle of the points and the two means. It summarises the trend, so it belongs among the points with roughly equal numbers above and below. There is no reason for it to pass through the origin.

Predicting beyond the range of the data is called:

extrapolation. It assumes the relationship continues where nothing was measured, and is the less reliable of the two.

A prediction from a line of best fit is most reliable when:

the correlation is strong and the value is inside the range. Strong correlation means the points lie close to the line, and staying inside the range means the pattern has actually been observed there.

Ice cream sales correlate with drowning deaths because:

hot weather causes both. Temperature is a third factor driving both independently — the standard illustration that correlation is not causation.

Towns with more doctors have more illness. The likely explanation is:

population size causes both. Bigger towns have more of everything. Comparing doctors and illness per head of population would test whether any real relationship remains.

On a scatter diagram showing no correlation, you should:

draw no line of best fit. There is no trend to summarise, so a line would suggest a relationship the data does not support.

Correlation between two variables means:

they tend to change together. It describes a tendency to move together and nothing more. Establishing a cause requires a controlled investigation.

Scatter Diagrams and Correlation — Mathematics Notes, Worked Examples and Quiz

Scatter Diagrams and Correlation

Looking for a relationship between two things, and not overclaiming when you find one

This topic appears in:

Plotting two variables against each other

Every other statistical diagram in the syllabus shows one variable. A scatter diagram shows two, one on each axis, with a point for each individual — so each point carries two measurements about the same thing.

The purpose is to see whether the two are related. If tall people tend to weigh more, plotting height against weight will show the points drifting upward, and that drift is what "correlation" means.

Pattern	Called	Means
points rise to the right, tightly grouped	strong positive	as one increases so does the other, reliably
points rise but widely scattered	weak positive	the same tendency, less consistently
points fall to the right, tightly grouped	strong negative	as one increases the other decreases
no pattern at all	no correlation	the two are unrelated

Select None and then turn the line of best fit on. A line can always be drawn, but on uncorrelated data it means nothing — drawing one anyway is the standard way to lose a mark.

The line of best fit

When there is correlation, a line of best fit summarises it. Drawn by eye, it should pass through the middle of the points with roughly as many above as below, and it should pass through the point of the two means.

Its purpose is prediction: reading from a known value of one variable to an estimate of the other. That works reasonably within the range of the data. Extending the line beyond that range is extrapolation, and it assumes the pattern continues where nothing was measured — which it frequently does not.

Two things to say before predicting

A prediction is trustworthy only when the correlation is strong and the value lies within the data range. Weak correlation means the points are far from the line, so the estimate could be badly out. Extrapolating means guessing beyond the evidence — revision hours predicting marks might extend to a prediction of 130%, which is nonsense. Exam questions ask you to comment on the reliability precisely because both faults are so easy to fall into.

Correlation is not causation

This is the most important idea in the topic and the one examiners test most often. Two variables moving together does not establish that one causes the other, and there are three reasons why.

There may be a third factor causing both — ice cream sales and drowning deaths both rise in hot weather. The causation may run the other way round. Or it may be coincidence, which becomes increasingly likely as more pairs of variables are tested.

The safe answer names the relationship, then says explicitly that a cause has not been established and would need a separate investigation.

Worked example

A study finds a strong positive correlation between the number of firefighters sent to a fire and the damage caused. Should fewer firefighters be sent?

The correlation is real: more firefighters really do coincide with more damage.Do not dispute the data — the fault is in the interpretation.
But the third factor is the size of the fire.Large fires cause more damage and also attract more firefighters.
Both variables are consequences of that third one, so neither causes the other.This is exactly the ice-cream-and-drowning structure in a different setting.
Sending fewer firefighters would increase damage, not reduce it.Acting on a correlation as though it were a cause can produce precisely the opposite of the intended effect.

No. Fire size causes both, so the correlation says nothing about what sending fewer would do.

Answering a scatter diagram question

These questions follow a predictable sequence, and knowing it means no marks are left behind.

Plot the remaining points carefully. Describe the correlation using two words — strength and direction, as in "strong positive". Draw a line of best fit through the middle of the points. Use it to estimate, showing the lines you read across and down. Then comment on reliability, mentioning whether the correlation was strong and whether the value was inside the data range.

Before you leave this chapter

A scatter diagram plots two variables, one point per individual.
Describe correlation with two words: strength and direction.
A line of best fit passes through the middle of the points and through the two means.
Predict only within the data range and only when the correlation is strong.
Correlation never proves causation — a third factor, reversed cause or coincidence may explain it.

What correlation does not tell you

A strong correlation says two quantities move together. It does not say one causes the other, and the distinction is examined more often than the plotting.

Ice cream sales and drowning incidents rise together, but neither causes the other — hot weather drives both. A third factor influencing two others is called a confounding variable, and it is the usual explanation when a correlation looks surprising.

The other trap is extrapolation. A line of best fit is only evidence within the range of the data collected. Extending it far beyond that range assumes the pattern continues, which nothing in the data supports — a child's height against age is nearly linear for a few years, but extending the line predicts a four-metre adult.

Correlation — the two quantities move together.
Causation — one of them actually produces the change in the other.
Confounding variable — a third factor driving both.
Interpolation — reading within the data range, which is reliable.
Extrapolation — reading beyond it, which is not.

Say why, not just "correlation is not causation"

Quoting the phrase rarely earns the mark on its own. What examiners want is a plausible alternative explanation for the particular data given — a confounder that would produce the same pattern, or the observation that the relationship was never tested outside the range collected.

Practice questions

6 questions · 20 marks · full working on every one

Try each one on paper first, then open the working. The marks are shown where they are actually awarded, because that is where they are actually lost.

Short questions

3 · 6 marks

Two marks each, in the style of the short-question section of the paper. Answer in two or three lines.

SQ1[2 marks]

Describe the correlation you would expect between a car's age and its value.

Model answer

Negative correlation — as age increases, value decreases. It would likely be fairly strong, since age is one of the main determinants of a used car's price.

Examiner tip. Give both strength and direction. "Negative" alone is usually worth one of the two marks.

SQ2[2 marks]

What does it mean if a scatter diagram shows no correlation?

Model answer

The points show no pattern — as one variable changes the other does not tend to change in any particular direction. The two appear unrelated, and no line of best fit should be drawn.

Examiner tip. Adding that no line should be drawn shows you know what follows from the observation, and it is often the second mark.

SQ3[2 marks]

Why should a line of best fit not be used to predict far outside the plotted data?

Model answer

That is extrapolation: it assumes the relationship continues into a region where nothing was measured. The pattern may change or stop, and the prediction may be impossible — such as a predicted mark above 100%.

Examiner tip. Naming extrapolation and giving an example of an impossible prediction covers both marks.

Solved numericals

2 · 8 marks

Full working, one step per line, with the marks shown where they are awarded.

N1[4 marks]

A scatter diagram of revision hours against exam mark shows strong positive correlation, with data ranging from 2 to 12 hours.

A student revises for 8 hours. Comment on the reliability of a prediction from the line.
Another revises for 30 hours. Comment on that prediction.

Full working

8 hours lies within the data range of 2 to 12interpolation[1]
and the correlation is strong, so the points lie close to the line — the prediction is reasonably reliableboth conditions must be checked[1]
30 hours is far outside the range, so this is extrapolation[1]
The relationship may not continue — marks are capped at 100%, and returns from extra revision are likely to diminish — so the prediction is unreliablea concrete reason is needed[1]

(a) reliable — inside the range and strong correlation (b) unreliable — extrapolation beyond the evidence

Examiner tip. Two checks every time: is the correlation strong, and is the value inside the range? A question giving one value inside and one outside is asking you to apply both.

N2[4 marks]

A newspaper reports that towns with more libraries have higher crime rates, and concludes that libraries cause crime. Evaluate this conclusion.

Full working

The conclusion is not justified — correlation does not establish causation[1]
A third factor explains both: population sizenaming it is the key mark[1]
Larger towns have more libraries and also more crime simply because more people live there[1]
Comparing crime and libraries per head of population would test whether any real relationship existsproposing the fix earns the fourth mark[1]

Unjustified — population size causes both. Compare per head of population instead.

Examiner tip. When a correlation involves totals across places of different sizes, population is nearly always the hidden third factor. Suggesting a per-head comparison is the strongest possible answer.

Long questions

1 · 6 marks

Theory and numerical together, as they appear in the long-question section.

LQ1[6 marks]

A teacher records the number of days each of 30 students was absent and their end-of-year mark. The scatter diagram shows moderate negative correlation.

Explain what "moderate negative correlation" means here.
Describe how to draw and use a line of best fit to estimate the mark of a student absent for 6 days.
A colleague concludes that absence causes low marks. Give two reasons this may not follow.

Mark scheme

As the number of absences increases, marks tend to decreasethe direction[1]
"Moderate" means the points are somewhat scattered about the trend, so the relationship is real but not tightthe strength[1]
Draw a straight line through the middle of the points, with roughly equal numbers above and below, passing through the point of the two means[1]
Read up from 6 on the absence axis to the line, then across to the mark axis, showing both construction linesshowing the lines is usually a mark[1]
Reason 1: a third factor such as prolonged illness or difficulty at home could cause both the absence and the lower marks[1]
Reason 2: the causation could run the other way — a student who is struggling may become discouraged and stop attendingaccept that moderate correlation means many students do not fit the trend[1]

(a) marks tend to fall as absence rises, with noticeable scatter (b) line through the middle, read up then across (c) a third factor such as illness, or reversed causation

Examiner tip. Reversed causation is the reason most often overlooked. Ask yourself whether the second variable could plausibly be causing the first — here, struggling with the work leading to absence rather than the reverse.