Scatter Scatter GraphsGraphs

Teach GCSE Maths

x xxx xx

xxx x

Weight and Length of Broad

Beans

Len

gth

(c

m)

3

1·5Weight (g)

0·5 1

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Scatter GraphsScatter Graphs

The data set shown in the table gives the weight and length of a sample of 10 broad beans.

Bean Numbe

r

Weight (g)

Length

(cm)

1 0·7 1·7

2 1·2 2·2

3 0·9 2·0

4 1·4 2·3

5 1·2 2·4

6 1·1 2·2

7 1·0 2·0

8 0·9 1·9

9 1·0 2·1

10 0·8 1·6

We can see if there is a link between the variables by plotting a scatter graph.

Weight and length both vary: they are called variables.

Source: Statistics for Biology by O N Bishop published by Pearson Education

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

x xxx

xxxx

x x

Weight and Length of Broad Beans

Weight (g)

Len

gth

(c

m)

Tell your partner what you notice about the lengths as the weights

increase. Ans: Generally the

heavier beans are longer.

0·7

1·7

1·2

2·2

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

We say that, for the beans, weight and length are positively correlated.“Correlated” means “there is a

relationship”.“Positive” means “as one variable increases, the other also increases”.

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

Weight (g)

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

Joining the points has no meaning but we can draw a line, the line of best fit, through the middle of the points. Tip: We draw the line “by eye” with about the same number of points above it as below.

Weight (g)

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

The line can be used to estimate the length of a bean if we are given its weight.e.g. Estimate the length of a bean

weighing 1·3 g. Ans: 2·35

cm.

1·3Weight (g)

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

We must always use the graph even if a bean of the given weight is in the table.e.g. Estimate the length of a

bean weighing 0·8 g.Ans: 1·8

cm.

0·8Weight (g)

This value must not be used as the bean may not be typical.

Can you and your partner see why it doesn’t make sense to extend the line to the y-axis ?

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

Ans: We would estimate that a bean weighing nothing is 0·9 cm long !

The line must not be used far beyond the data points as the

relationship between the variables may not hold.

Weight (g)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

In this example, the points all lie close to the line of best fit.

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

We say the correlation is strong.

Correlation can be perfect, strong or weak, or there can be no correlation.

Decide with your partner what a set of points with perfect correlation would look like.

Ans: All the points would lie on a straight line.

Weight (g)

1·60·8

2·11·0

1·90·9

2·01·0

2·21·1

2·41·2

2·31·4

2·00·9

2·21·2

1·70·7

Length

(cm)

Wt. (g)

x xxx

xxxx

x x

Weight and Length of Broad Beans

Len

gth

(c

m)

With weak correlation, the points are more scattered than here. With no correlation they are all over the graph.

Weight (g)

If one variable decreases as the other increases, we say the correlation is negative.

The scatter graph shows how, as the percentage of the population with access to clean water in Peru increases, the proportion of deaths of young children decreases.

Infa

nt

Mort

ality

(p

er

1000

bir

ths)

Access to Clean Water (%)

Infant Mortality and Access to Clean Water in Regions of Peru

Source: PAHO

Infa

nt

Mort

ality

(p

er

1000

bir

ths)

Source: PAHO

Infant Mortality and Access to Clean Water in Regions of Peru

Access to Clean Water (%)

The line of best fit slopes down to the right.

The original data actually had one more point.

We would say again that the correlation is strong, even though it is not as strong as before.

Infant Mortality and Access to Clean Water in Regions of Peru

Infa

nt

Mort

ality

(p

er

1000

bir

ths)

Source: PAHO

A point lying well off the line of best fit is called an outlier. It may have arisen because of an error in collecting or entering the data so it is sometimes missed out.

Access to Clean Water (%)

Extra point

The points on the following scatter graph do not lie on, or near, a straight line.

However, the 2 variables are related as they lie close to a smooth curve.

We say the relationship is non-linear.Zero correlation means there is no linear

relationship but there may be a non-linear one.

SUMMARY A scatter graph plots values of 2 variables

to show any relationship between them. The relationship is called

correlation.• Positive correlation means as one variable increases, the other also increases.

• Negative correlation means as one variable increases, the other decreases.

Correlation can be perfect, strong or weak. Zero correlation means there is no linear relationship between the variables.

A line of best fit through the centre of the points can be used to estimate the value of one variable from a value of the other. The line should only be used within, or close to, the range of the points. Outliers lie well away from the other points.

Exercise

1. Pick a word from each list ( or choose None ) to describe the correlation between the variables in each of the following:

(a)

(b)

(c)

List B: Positive, Negative

List A: Perfect, Strong, Weak, None

Exercise

(b) Perfect, Negative

Answers:(a)

Weak, Positive

(c) None

Exercise2. The table shows the number of accidents

to children as a percentage of those to adults, y, in 9 areas of London together with the percentage of open space in those areas, x.

A B C D E F G H IOpen Spaces (%),

x 5 1·3 1·4 7 4·5 5·2 6·3 14·6 14·8

Children’s

Accidents (%), y 46·3 42·9 40 38·2 37 33·6 30·8 23·8 17·1

(a) Plot the data on a scatter diagram and draw, by eye, the line of best fit. ( Suitable scales for squared paper are given on the next slide. A version suitable for photocopying is available at the end. )

(b)Estimate the percentage of accidents to children if the open space is 10%.

Exercise

Open Spaces (%)

Ch

ild

ren

’s A

ccid

en

ts (

%)

Childrens’ Accidents and Open Spaces

Exercise

Open Spaces (%)

Ch

ild

ren

’s A

ccid

en

ts (

%)

Childrens’ Accidents and Open Spaces

Solution:

With 10% open spaces, the percentage of accidents that happen to children is estimated as 28%.

(a)

(b)

We won’t all have drawn the line in exactly the same place. You are not wrong if your line cuts the y-axis a bit higher up (e.g. at y = 46 ).

( Your graph may give a slightly different answer. )

Children’s Accidents (%)S

catt

er

Gra

ph

Sh

ow

ing

Accid

en

ts t

o C

hild

ren

in

9 A

reas o

f Lon

don

( a

s a

perc

en

tag

e o

f all

accid

en

ts )

an

d P

erc

en

tag

e o

f O

pen

Sp

aces

Op

en

Sp

aces (

%)