Scatter Diagrams and Correlation

Describe the correlation on a scatter diagram, draw a line of best fit, use it to estimate, and say when an estimate cannot be trusted.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 40 real past-paper questions below

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What a Scatter Diagram Shows

A scatter diagram plots pairs of values. Each dot is one day, one person or one thing: here, the temperature and how many cold drinks a kiosk sold.

The pattern of the dots is called the correlation.

The dots	Correlation
go up to the right	positive
go down to the right	negative
make no pattern	no correlation

Key idea

Close to a straight line means strong correlation. Spread out, but still with a direction, means weak.

Key idea

Stretch your arms out wide and measure from fingertip to fingertip. For most adults this arm span is close to their height. Plot height against arm span for a group of people and you get strong positive correlation.

Plotting and Reading Dots

Most scatter questions start with “Complete the scatter diagram”: you plot the last few points from a table. First work out what one small square is worth.

Here, a café plots hours of sunshine (across) against ice creams sold (up). The next day had $6.4$ hours and $34$ ice creams.

Worked example

1
$1 \div 5 = 0.2$ — On the bottom axis, 1 hour is split into 5 small squares. So each small square is 0.2 hours.
2
$20 \div 10 = 2$ — On the side axis, 20 ice creams is split into 10 small squares. So each small square is 2 ice creams.
3
$6.4 - 6 = 0.4, 0.4 \div 0.2 = 2$ — Start at the 6 line. 0.4 more is 2 small squares to the right, because each square is 0.2.
4
$34 - 30 = 4, 4 \div 2 = 2$ — Start at the 30 line. 4 more is 2 small squares up, because each square is 2.
5
$\times at (6.4, 34)$ — Mark a small, neat cross there. Across first, then up, in the same order as the table.

Common mistake

Counting every small square as 1, or as 0.1. Always work out the scale on each axis before you plot.

Key idea

Reading a dot works backwards. “One day had 5 hours of sunshine. How many ice creams?” Find 5 on the bottom axis, go up to the dot on that line, then across. Use the dot, not the line of best fit.

Key idea

Plotting is often worth 2 marks, and 2 correct crosses can still earn 1. Check every cross against the table before you move on.

Describing the Correlation

The 1-mark question: "What type of correlation does the diagram show?" Sometimes it adds "Describe the relationship". Then say it in words about the data.

Worked example

1
$dots go up to the right$ — As the temperature goes up, the drinks sold go up too, so the dots climb from left to right. That is positive correlation.
2
$dots close to a line$ — The dots lie near one straight line, with little spread. So the correlation is strong.
3
$hotter \Rightarrow more drinks$ — “Describe the relationship” wants a sentence about the data, not one word: the hotter the day, the more cold drinks are sold.

Rule

"Type of correlation": one word, like positive. "Relationship": a sentence, like "the hotter the day, the more drinks are sold".

Common mistake

Looking at the size of the numbers. Only the direction of the dots decides positive or negative.

Sometimes there is no diagram, only a sentence. Ask: when one thing goes up, does the other go up, go down, or do neither?

Worked example

1
$burn time ↑, candle length ↓$ — The sentence: the longer a candle burns, the shorter it gets. One goes up while the other goes down.
2
$negative correlation$ — Up with down is negative. Up with up would be positive.

Key idea

If the two things have nothing to do with each other, like the day of the month you were born and your height, the answer is no correlation.

Line of Best Fit and Estimates

The 2-mark question: "Draw a line of best fit" and "Use it to estimate the drinks sold on a day of 25 degrees."

Rule

Use a ruler. Draw one straight line through the middle of the dots, with about as many dots on each side. It does not have to go through 0.

Our line passes through $(15, 32)$ and $(35, 72)$ . Go up from 25 to the line, then across.

Worked example

1
$(15, 32) and (35, 72)$ — Two points on our ruled line, read from the grid. We use them to find the line's height at 25.
2
$25 - 15 = 10, 35 - 15 = 20$ — 25 is 10 out of 20 of the way along, so it is halfway between the two points.
3
$72 - 32 = 40$ — Between the two points, the line rises 40 drinks.
4
$32 + \frac{40}{2} = 52$ — Halfway along means halfway up: 32 + 20 = 52. So about 52 drinks.

Common mistake

Joining the dots dot to dot, or forcing the line through the corner of the graph. It is one straight line through the middle of the dots.

Key idea

Given the drinks and asked for the temperature? Go across from the side axis to the line, then down. On this line, 62 drinks gives about 30 degrees.

Key idea

Draw your up and across lines on the graph. The estimate mark is for reading your line correctly, so let the examiner see how you read it.

Can You Trust the Estimate?

The 1-mark question: "Explain why this estimate may not be reliable." The data runs from 12 degrees to 36 degrees.

Worked example

1
$12 \leq 25 \leq 36$ — 25 degrees is between the coldest and hottest days in the data. There are dots on both sides, so the estimate is reliable.
2
$45 > 36$ — 45 degrees is hotter than any day in the data. The line would have to go past the last dot, and the pattern may not carry on, so it is not reliable.
3
$45 is outside the range of the data$ — Write this sentence as your answer. It is what the mark is for.

Rule

Inside the data, the line is backed by dots on both sides. Outside it, the pattern may not carry on, so do not trust the estimate. Textbooks call these interpolation (inside) and extrapolation (outside).

Key idea

A dot far from the pattern of all the others is an outlier, like a hot day when very few drinks were sold. If asked which point does not fit, ring that dot. Ignore it when you draw the line.

Common mistake

Saying one thing causes the other. Correlation shows the two change together, not why.

Which Method? Read the Question

Look at the command word.

You see	You do
Complete the scatter diagram	work out one small square, then plot crosses
Find the value for one day	read from that dot, not the line
Type of correlation	positive, negative or none
Describe the relationship	a sentence about the data
Draw a line of best fit	ruler, straight, through the middle
Which line is best?	the one that follows the dots, through the middle
Estimate	up to the line, then across
Why not reliable?	outside the data
Which point does not fit?	the outlier, far from the rest
Compare two groups	Comparing Two Sets of Data

Key idea

Always read your estimate from your line, not from the nearest dot.

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