KS3 · Maths

Correlation in bivariate data

Sometimes the dots on a graph line up into a pattern. Sometimes they look like spilled sprinkles. Tell them apart, and dodge the trap that catches almost everyone.

Statistics · Correlation

Sort the point patterns

Pick a pattern, then pick the category it belongs in. Then switch the rule at the top and sort the same six patterns again.

Which description fits these points?

Still to sort

Positive correlation (0)

Points slope upwards, roughly in a straight line.

Where the line is: Loose points can still be positive. Loose means less clear, not 'no correlation'.

Negative correlation (0)

Points slope downwards, roughly in a straight line.

Where the line is: Loose points can still be negative. Look at which way they slope, not how spread out they are.

No correlation (0)

No shape or pattern, and no slope up or down.

Where the line is: If the points still tend to slope up or down along a line, even loosely, it isn't this one.

Non-linear association (0)

A clear pattern, but it's a curve, not a straight line.

Where the line is: It is a clear pattern, so it isn't 'no correlation'. It bends, so 'correlation' isn't the best word.

6 of 6 still to sort.

Six scatter graphs, described in words. Decide where each one belongs before you read why.

Watch out: Spread-out points are not the same as no pattern. Check which way they slope before you decide.

Zoom in: what one scatter graph is made of

Four ideas, in the order you'd use them when reading any scatter graph.

Predict, then check

This one looks like it should be easy. Commit to an answer before you reveal.

The points follow a clear curve. They rise steeply at first, then more gently, with hardly any wandering. Which description would you choose?

Statistics · Correlation

What does a correlation show?

Across regions of England and Wales, the number of property sales and the greenhouse gas emissions show a positive correlation. Both also correlate positively with the population of the region.

Which of these is closest to what you think the correlation shows?

More property sales cause more emissions.More emissions cause more property sales.A correlation this clear proves that one variable affects the other.They may be related somehow, and both also tend to rise with the region's population, so one doesn't have to be affecting the other.

How sure are you?

I'm sureI think soI'm guessing

More property sales cause more emissions.

This is a common misconception

It's an easy leap, and it's the one this topic is built to catch. The correlation only shows that the two tend to rise together. It doesn't show that sales affect emissions. Notice the clue in the situation: both also correlate with population.

More emissions cause more property sales.

This is a common misconception

This is the same leap, just turned round. A positive correlation can't tell you which way the effect would go, or that there is one at all. It only says the variables tend to rise together.

A correlation this clear proves that one variable affects the other.

This is a common misconception

A clearer correlation means the points follow the line more tightly. It describes the pattern better. It still only suggests the variables may be related somehow, so tight points are no proof that one variable affects the other.

They may be related somehow, and both also tend to rise with the region's population, so one doesn't have to be affecting the other.

This matches the evidence

Yes. Correlation only suggests the variables may be related. Here a third variable, population, is related to both, so the two may be associated because both tend to rise with population size, rather than because one directly affects the other. Population is a possible explanation here, not something the graph proves.

Correlation doesn't imply causation: a pattern says the variables may be related somehow, never on its own that one affects the other.

Statistics · Correlation

Write it, then mark it

Assume each pair of variables correlates. (a) Hours spent practising and exam score. (b) Time spent training and time taken to finish a race. For each, say which type of correlation you would expect, then add one sentence on what a correlation alone would not show. [3 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

(a) Positive correlation: more hours practising would be expected to go with a higher exam score. (1 mark)
(b) Negative correlation: more time training would be expected to go with less time taken to finish the race. (1 mark)
A correlation alone would not show that one variable affects the other; they may only be related somehow. (1 mark)

Model answer

(a) I would expect positive correlation: more hours spent practising would be expected to go with a higher exam score. (b) I would expect negative correlation: more time spent training would be expected to go with less time taken to finish a race. A correlation on its own would not show that one variable affects the other, only that they may be related somehow.

Decide the expected direction from the context first, and always finish with what the correlation can't show.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Direction, how clear it is, and what it can't tell you

What you need to know

A scatter graph shows two variables at once, and each point is one pair of values.
If the points form a shape instead of being random, there may be an association.
Have a goMaya plots 20 points and they're sprinkled randomly all over the grid. She announces there's definitely an association. Is she right?
No. Randomly scattered points don't suggest an association.
It's a shape or pattern in the points, rather than a random spread, that suggests there may be an association.
Correlation describes how closely the points follow a straight line going one way or the other.
Positive correlation: the points slope upwards roughly in a straight line, so as one variable increases the other also tends to increase.
Negative correlation: the points slope downwards roughly in a straight line, so as one variable increases the other tends to decrease.
Have a goGraph A's points slope upwards in a rough straight line. Graph B's slope downwards in a rough straight line. Which graph shows negative correlation, and what do the variables in it tend to do?
Graph B: as one variable increases, the other tends to decrease.
Negative correlation goes with a downward slope. Calling it 'negative' doesn't mean both variables go down.
No correlation: the points form no pattern and don't slope up or down.
A clear curve is an association that is not linear, so correlation isn't the best description.
Tight points make the correlation very clear. Looser points make it less clear, but it can still be positive or negative.
Have a goTwo graphs both slope upwards. In P the points sit tight on the line. In Q they're spread out loosely around it. Is Q 'no correlation'?
No. Q is still positive correlation, just less clear.
Looser points make the correlation less clear, but it can still be described as positive. 'No correlation' needs no pattern at all.
Correlation doesn't imply causation: the variables may be related somehow, without one directly affecting the other.
In a context, you can say which direction you'd expect, assuming the variables correlate.

The big picture

A scatter graph shows two variables at once, one point per pair of values. If the points form a pattern, there may be an association. Correlation describes how closely the points follow a straight line going one way or the other: positive when they slope upwards, negative when they slope downwards, none when there's no pattern. Tight points make the correlation very clear, looser points make it less clear, and a clear curve is a non-linear association instead. Whatever the pattern, correlation never on its own shows that one variable affects the other.

Key points

1To describe a scatter graph fully, say positive, negative or no correlation, not just the word 'correlation'.

2Very clear means the points tightly follow the line. Less clear means they follow it less tightly.

3A clear curve is a non-linear association, so correlation isn't the best word for it.

4Correlation on its own never shows that one variable affects the other.

5In a context, decide the expected direction first. More training time would be expected to go with less race time.

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Worked example

Problem

A scatter graph has horizontal axis x and vertical axis y. Eight points slope downwards from top left to bottom right. They follow a rough straight line, but are spread out loosely around it. Describe the correlation fully.

⚠ Watch out

Saying 'no correlation' just because the points are spread out. If they still slope up or down roughly along a straight line, it's positive or negative correlation, just less clear. No correlation means no pattern at all.

🧠

Memory hook

Line, then direction, then tightness. A bend is not a line. And a pattern never tells you why.

✓

Check yourself

Points rise, then level off, following a clear curve. Is that positive correlation? (No: it's a non-linear association, because correlation describes how closely points follow a straight line.)

Flashcards

(12)

What is one point on a scatter graph?

A pair of values, one for each of the two variables (bivariate data).

Points are sprinkled randomly with no shape. Does that suggest an association?

No. It's points forming a shape or pattern that suggests there may be an association.

What does correlation measure?

How closely the points on a scatter graph follow a straight line going in one direction or another.

Positive correlation: what shape do the points make?

They slope upwards roughly in a straight line, so as one variable increases the other also tends to increase.

Points fall left to right in a rough straight line. What do the two variables tend to do?

As one increases, the other tends to decrease. That's negative correlation.

Points form no pattern and slope neither up nor down. What do you say?

No correlation: there doesn't appear to be any relationship between the variables.

Points follow a clear curve. Why isn't 'correlation' the best word?

Correlation describes closeness to a straight line. A clear curve is a non-linear association.

Two graphs both slope up. The points are tight in one and loose in the other. How do the correlations compare?

Both are positive. The tight one is very clear and the loose one is less clear.

Why say 'positive correlation' rather than just 'correlation'?

To describe the graph fully you give the direction too (or say there's none) as well as the word correlation.

Two variables correlate. Does that show one variable affects the other?

No. Correlation doesn't imply causation. They may only be related somehow.

How can two variables correlate without one directly affecting the other?

A third variable can be related to both. Property sales and emissions both tend to rise with a region's population.

Heights and shoe sizes of children: what correlation would you expect, assuming they correlate?

Positive. Taller children would be expected to go with larger shoe sizes.

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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Sort the point patterns

What does a correlation show?

Write it, then mark it

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