Graphs, charts and tables!

L Q1 Q2 Q3 H

Some reminders

Scores are the wee numbers

Median is the middle score.

Mode is the score which occurs most often

Range is highest score lowest score

Relative Frequency

Frequency is a measure of how often something

occurs.

Relative Frequency is a measure of how often

something occurs compared to the total amount.

Relative Frequency is given by frequency divided

by the number of scores.

Relative Frequency is always less than 1.

Example

A supermarket keeps a record of wine sales,

noting the country of origin of each bottle. The

frequency table shows one days sales.

Draw a relative frequency table for the wine

sales.

Country Frequency Relative Frequency

France 120

Australia 30

Italy 27

Spain 24

Germany 18

Others 21

Total 240

Country Frequency

France 120

Australia 30

Italy 27

Spain 24

Germany 18

Others 21

Total 240

120 ? 240

0.5

30 ? 240

0.125

27 ? 240

0.1125

24 ? 240

0.1

18 ? 240

0.075

0.0875

21 ? 240

1

Note The total of the relative frequencies is

always 1. This is a useful check.

If the supermarket wishes to order 1000 bottles

of wine they may start by assuming that the

relative frequencies are fixed

French wines 0.5 x 1000 500

bottles Australian wines 0.125 x 1000 125

bottles.

Relative frequencies can be used as a measure of

the likelihood of some event happening, e.g. when

a customer comes in for wine, half of the time

you would expect them to ask for French wine.

P138/139 Ex1 (omit questions 3b, 5b)

Reading Pie Charts

Page 140, 141 Ex 2

A pie chart is a graphical representation of

information.

however, a pie chart can be used to calculate

accurate data.

Example

Newton Wanderers have played 24 games. The pie

chart shows how they got on.

A full circle represents 24 games.

Using a protractor we can measure the angles at

the centre. (u estimate angles)

A full circle is 360?

? 24

8 games

Won

120?

90?

150?

? 24

6 games

Drawn

? 24

10 games

Lost

(Check that 8 6 10 24)

Constructing Pie Charts

Example

A geologist examines pebbles on a beach to study

drift. She counts the types and makes a table of

information. Draw a pie chart to display this

information.

Rock Type Frequency

Granite 43

Dolerite 52

Sandstone 31

Limestone 24

Total 150

Relative Frequency Angle At Centre

360?

Now we draw the pie chart ...

Geology Survey

Step 1 Title.

Step 2 Draw a circle.

Limestone

Granite

Step 3 Draw in start line.

58

Step 4 Using a protractor

draw in the other lines.

103

74

Sandstone

125

(you do not need to write the angles)

Step 5 Label the sectors.

Dolerite

P141/142Ex 3

Cumulative Frequency

Example

Fifty maths students are graded 1 to 10 where 10

is the best grade. The grades and frequencies are

shown below.

A third column has been created which keeps a

running total of the frequencies. These figures

are called cumulative frequencies.

Cumulative Frequency

0

2

6

16

The cumulative frequency of grade 7 is 43.

27

37

This can be interpreted as 43 candidates are

graded 7 or less.

43

P143/144 Ex4

47

49

50

Cumulative Frequency Diagrams

Using the previous example we can draw a

cumulative frequency diagram.

We make line graph of cumulative frequency

(vertical) against grade (horizontal).

Maths Students Grades

Maths Students Grades

Cumulative Frequency

Grade

P145,146 Ex 5

Using the diagram only

How many pupils were grade 6 or less ?

37

At least 25 pupils were less than grade 5.

Dotplots

It is useful to get to get a feel for the

location of a data set on the number line. A

good way to achieve this is to construct a

dotplot.

Example A group of athletes are timed in a 100m

sprint. Their times, in seconds, are 10.8 10.9

11.2 11.5 11.6 11.6 11.6 11.9 12.2 12.2 12.8

Each piece of data becomes a data point sitting

above the number line

Some features of the distribution of figures

become clearer

? the lowest score is 10.8 seconds

? the highest score is 12.8 seconds

? the mode (most frequent score) is 11.6 seconds

? the median (middle score) is 11.6 seconds

? the distribution is fairly flat

Here are some expressions commonly used to

describe distributions

P147/148 EX 6

The Five-Figure Summary

When a list of numbers is put in order it can be

summarised by quoting five figures

H

Highest number

L

Lowest number

Q2

Median of the full list (middle score)

Q1

Lower quartile the median of the lower half

Q3

Upper quartile the median of the upper half

Example Make a five-figure-summary for the

following data ...

6 3 7 8 11 8 6

10 9 8 5

3 5 6 6 7 8 8

8 9 10 11

Q2

Q3

Q1

L Q1 Q2

Q3 H

3

8

9

11

6

Example Make a five-figure-summary for the

following data.

6 3 7 8 11 6 10

9 8 5

3 5 6 6 7 8 8

9 10 11

Q1

Q2

Q3

L Q1 Q2

Q3 H

3

7.5

9

11

6

Example Make a five-figure-summary for the

following data.

6 3 7 8 11 6 10

9 5

3 5 6 6 7 8 9

10 11

Q1

Q2

Q3

L Q1 Q2

Q3 H

3

7

9.5

11

5.5

P151 Ex 7

Boxplots

A boxplot is a graphical representation of a

five-figure summary.

Example Draw a box plot for this five-figure

summary, which represents candidates marks in an

exam out of 100

L Q1 Q2

Q3 H

Marks out of 100

? 25 of the candidates got between 12 and

32 (the lower whisker)

? 50 of the candidates got between 32 and

66 (in the box)

? 25 of the candidates got between 66 and

97 (the upper whisker)

P152/153 Ex 8

Comparing Distributions

When comparing two or more distributions it is

(VERY) useful to consider the following

? the typical score (mean, median or mode)

? the spread of marks (the range can be used,

but more often the interquartile range

or semi-interquartile range is used

Interquartile range Q3 Q1

Marks out of 100

These boxplots compare the results of two exams,

one in January and one in June. Note that

the January results have a median of 38 and a

semi-interquartile range of 14 the June results

have a median of 51 and a semi-interquartile

range of 23.

On average the June results are better than

Januarys (since the median is higher) but

scores tended to be more variable (a larger

semi-interquartile range). Note the longer the

box the greater the interquartile range

and hence the variability.

Mr Tennents example

Boxplots showing spread of marks in two

successive tests.

Test 2

Test 1

Which would you hope to be test 1 and which test

2?

Has the class improved? (give reasons for your

answer)

Boxplots

A boxplot is a graphical representation of a

five-figure summary.

The Five-Figure Summary

When a list of numbers is put in order it can be

summarised by quoting five figures

H Highest number

L Lowest number

Q2 Median of the full list (middle score)

Q1 Lower quartile the median of the lower half

Q3 Upper quartile the median of the upper half

Example Draw a box plot for this five-figure

summary, which represents candidates marks in an

exam out of 100

Marks out of 100

? 25 of the candidates got between 12 and

32 (the lower whisker)

? 50 of the candidates got between 32 and

66 (in the box)

? 25 of the candidates got between 66 and

97 (the upper whisker)

Example Make a five-figure-summary for the

following data ...

6 3 7 8 11 8 6

10 9 8 5

3 5 6 6 7 8 8

8 9 10 11

Q2

Q3

Q1

L Q1 Q2

Q3 H

3

8

9

11

6