M4.7.6 Chapter 6

Scatter graphs and correlation

Mei measures the height and the arm span of 12 friends. Can a graph help her guess a friend's arm span from their height?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Add up all the heights and all the arm spans.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Plot each friend as one point, height across and arm span up, and look for a pattern.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Draw one bar for each friend.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw a scatter graph and a line of best fit
Describe positive, negative or no correlation
Explain why correlation does not prove one thing causes another

Mei measures 12 friends. For each friend she writes the height and the arm span, from fingertip to fingertip, in centimetres. Each friend gives a pair of numbers.

The first friend is 142 cm tall with an arm span of 141 cm.

A scatter graph plots each pair as one point. The height is the x value and the arm span is the y value. Neither axis starts at 0 here, so each axis shows a zigzag break near its corner.

Draw it first

Draw the scatter graph and the line of best fit

1 Plot one dot for each friend: height across, arm span up.
2 Find the mean height and the mean arm span. Plot that mean point.
3 Draw one straight line through the middle of the dots and through the mean point. Aim for about as many dots on each side.

The straight line is the line of best fit. It is drawn by eye, so two people will draw slightly different lines. A good line follows the direction of the dots. It has about as many dots above it as below, and it goes through the mean point.

The line drawn here passes through (140, 141) and (170, 171). It goes up 1 for each 1 across, so its rule is y = x + 1.

Worked example

The 12 heights add up to 1 920 cm and the 12 arm spans to 1 932 cm. Find the mean point.

Mean height = 1 920 ÷ 12 = 160 cm.
Mean arm span = 1 932 ÷ 12 = 161 cm.
The mean point is (160, 161). The line of best fit should pass through it.
The line y = x + 1 passes through the mean point, because 160 + 1 = 161.

Answer: The mean point is (160, 161)

The pattern in the dots is called correlation. Here the dots rise from left to right. Taller friends tend to have longer arm spans. This is positive correlation: as one value rises, the other tends to rise.

A candle burns. The dots fall from left to right: negative correlation.

When the dots fall from left to right, as one value rises the other tends to fall. This is negative correlation. A burning candle gets shorter as the hours go by.

The number of letters in a name has nothing to do with jumping. The dots show no pattern: no correlation.

The correlation is strong when the dots lie close to a straight line, and weak when they are more spread out. The steepness of the line does not matter. A gentle slope can still have dots tight around it, as long as the line is not level.

Worked example

Use the line y = x + 1 to estimate the arm spans of friends 168 cm and 150 cm tall.

For a height of 168 cm, the line gives y = 168 + 1 = 169.
For a height of 150 cm, the line gives y = 150 + 1 = 151.
Both heights are inside the data, which runs from 142 cm to 176 cm. So the estimates are reasonable.

Answer: About 169 cm and about 151 cm

Go up from 168 cm to the line, then across to the arm span axis: about 169 cm.

A line of best fit is for estimates inside the range of the data. This is interpolation. Going beyond the data is extrapolation. It is unreliable, because the pattern may not carry on.

Solid: inside the data. Dashed: extended too far. The dashed line reaches 0 cm at about 8 hours. Beyond that it would give a candle with a negative length.

Worked example

The candle line is y = 20 − 2.6x, with y the length in cm and x the hours. Find the length at 3.5 hours. Is the line fair to use at 10 hours?

At 3.5 hours the line says 2.6 × 3.5 = 9.1 cm have burned.
The length is 20 − 9.1 = 10.9 cm.
At 10 hours the line says 2.6 × 10 = 26 cm have burned.
That is more than the 20 cm of candle, so the length would be negative. Ten hours is outside the data, so the line cannot be used there.

Answer: About 10.9 cm. No: the line gives a negative length

Now a warning. Two things can be correlated without one causing the other. On a beach, the number of ice creams sold each day and the number of cases of sunburn rise together.

See it three ways

With things

On 8 days a beach counts the ice creams sold and the cases of sunburn.

As a picture

In words

Strong positive correlation. But ice cream does not burn skin, and sunburn does not make people buy ice cream. On hot sunny days both go up. The hot sunny weather is the hidden third factor.

In symbols

Correlation does not prove cause.

A wrong idea many people have

If two things are correlated, one must cause the other.

Why it feels right: When two things rise together it feels as if one must be pulling the other.

The test that breaks it: Ice creams sold and sunburn cases are strongly correlated. Neither causes the other. Hot sunny weather makes both rise.

What is true: Correlation shows that two things go together. To say one causes the other we need other evidence, and we should look for a hidden third factor.

Try it

Measure hand span and foot length

You need: 8 people to measure, a ruler, squared paper.

For each person, measure the hand span and the foot length in centimetres.
Plot a dot for each person: hand span across, foot length up.
Find the mean point. Draw a line of best fit through it with about as many dots on each side.

What you will see: The dots rise from left to right. People with a bigger hand span tend to have a longer foot. Your own dots may be more or less spread out.

If you cannot try it, this is what happens.

A wrong idea many people have

A steeper line means stronger correlation.

Why it feels right: A steep line looks more definite than a flat one.

The test that breaks it: Draw two scatter graphs. In one, the dots hug a flat line. In the other, the dots are spread loosely around a steep line. The first has stronger correlation.

What is true: Strength depends on how close the dots are to the line. The steepness depends on the scales chosen for the axes.

Key words

scatter graph: a graph with one dot for each pair of measurements.
correlation: the pattern between two measurements: positive, negative or none.
line of best fit: a straight line drawn through the middle of the dots on a scatter graph.
extrapolation: using a line beyond the range of the data, which is unreliable.

Back to your guess

The answer is B: Plot each friend as one point, height across and arm span up, and look for a pattern.

One point for each friend, with the height across and the arm span up, makes a scatter graph. If the points fall along a line, the pattern can be used to estimate. Totals and bars lose the link between a friend's two measurements.

Keep it

Copy this into your exercise book.

Scatter graph: one dot for each pair. Correlation: positive, negative or none. The line of best fit passes through the mean point. Use it inside the data. Correlation is not cause.

Draw: A scatter graph with a line of best fit, and under it the words positive, negative and none beside small sketches.