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.
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.