When two quantities are collected in pairs, such as hours of study and test scores, a correlation is a tendency for one to change in a definite direction as the other changes.
If one tends to rise as the other rises, the correlation is positive; if one tends to fall as the other rises, it is negative; and if there is no clear tendency, there is no correlation.
Plotting each pair as a point gives a scatter diagram, and the more tightly the points cluster around a line, the stronger the correlation.
Still, two quantities moving together does not mean one causes the other. Ice-cream sales and swimming accidents both rise in summer, and often a hidden cause moves both.
A scatter diagram of study time and scores
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A scatter diagram of 25 points that gather around or spread from the trend line as the direction and strength change
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👀 Reading a scatter diagram
①Each point shows one student's (study hours, score).
②If the points tend to rise to the right, the correlation is positive.
③If they tend to fall to the right, it is negative.
④If they are scattered with no tilt either way, there is no correlation.
⑤Moving the slider towards 1 or −1 gathers the points more tightly around the dashed line. The slider value simply sets the tendency; at this level correlation is judged from the picture, not calculated.
Three kinds of correlation
Positive correlation
as x increases, y generally increases
The points head up to the right, as with height and weight or study time and grades.
Negative correlation
as x increases, y generally decreases
The points head down to the right, as with days absent and grades or winter temperature and heating costs.
No correlation
no clear tendency between x and y
The points are spread evenly, as with shoe size and maths score.
💡 Strong and weak correlation
①The more tightly the points cluster around a line, the stronger the correlation.
②If the points spread widely around the line, the correlation is weak.
③Direction (positive or negative) and strength are judged separately: there are strong negative and weak positive correlations.
Correlation is not causation
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Summer, the hidden cause that raises both ice-cream sales and swimming accidents
⚠️ Moving together is not cause and effect
①On days when a lot of ice cream is sold there are also more swimming accidents, so the two show a positive correlation.
②But eating ice cream does not cause accidents.
③Hot weather, a common cause, raises both at once.
④So even when there is a correlation, you cannot conclude cause and effect.
Practising with scatter diagrams
📊 Everyday examples
①Positive: height and weight, temperature and air-conditioner use.
②Negative: temperature and heating costs, a car's age and its second-hand price.
③None: blood type and grades, birthday and maths score.
💡 Three steps for reading a scatter diagram
①Look at the overall tendency: up to the right, down to the right, or none?
②Judge the strength by how closely the points gather around a line.
③Check for points far from the rest.
Worked examples
Example 1
Taller people tend to weigh more. State the correlation between the two variables.
1
Check whether one variable grows as the other grows.
height ↑ → weight ↑
2
They increase together, so it is a positive correlation.
positive correlation
▸ Positive correlation
If the points head up to the right in a scatter diagram, the correlation is positive.
Example 2
In winter, the lower the temperature, the higher the heating cost. State the correlation.
1
See how the heating cost changes as the temperature rises.
temperature ↑ → heating cost ↓
2
One rises as the other falls, so it is a negative correlation.
negative correlation
▸ Negative correlation
If the points head down to the right in a scatter diagram, the correlation is negative.
Exam checklist
How to judge a correlation
direction of the tilt → positive or negative · closeness to a line → strong or weak
A scatter diagram is read properly only when both direction and strength are stated.
G9 school-exam type
Over the same distance, which describes the correlation between a car's speed and its travel time?
①Strong positive correlation
②Weak positive correlation
③Negative correlation
④No correlation
⑤Cannot tell
▸ ③ Negative correlation
1
As speed increases, the time to cover the same distance decreases.
speed ↑ → time ↓
2
One rises while the other falls, so it is a negative correlation.
negative correlation
🎯 Exam points
①Points heading up to the right show positive correlation; down to the right, negative.
②The closer the points are to a line, the stronger the correlation.
③A correlation does not prove causation; this often appears in written answers.
④Look separately at any point far from the others.
⑤A table that counts pairs of values in cells is a correlation table.