Graphs, charts and tables!

Graphs,

charts and

tables!

G

A

DL Q1 Q

2 Q3 H

Some reminders …

Median is the middle score.

Mode is the score which occurs most often

Range is highest score – lowest score

Mean is sum of scores

number of scores

Scores are the wee numbers

Relative FrequencyFrequency 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 day’s sales.

Country

Frequency

France 120

Australia 30

Italy 27

Spain 24

Germany 18

Others 21

Total 240

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

120 24030 24027 24024 24018 24021 240

= 0.5

= 0.125

= 0.1125

= 0.1

= 0.075

= 0.0875

1

Note: The total of the relative frequencies is always 1. This is a useful check.

Total

Others

Germany

Spain

Italy

Australia

France

Country

240

21

18

24

27

30

120

Frequency Relative Frequency

120 24030 24027 24024 24018 24021 240

= 0.5

= 0.125

= 0.1125

= 0.1

= 0.075

= 0.0875

1

If the supermarket wishes to order 1000 bottles of wine they may start by assuming that the relative frequencies are fixed …

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

French wines = 0.5 x 1000 = 500 bottlesAustralian wines = 0.125 x 1000 = 125

bottles.

Reading Pie ChartsA pie chart is a graphical representation of information. …… however, a pie chart can be used to calculate

accurate data.

Newton Wanderers have played 24 games. The pie chart shows how they got on.

Won

Lost

Drawn

A full circle represents 24 games.

Using a protractor we can measure the angles at the centre. (u estimate angles)

15090

120

A full circle is 360

Won:360

120 24

= 8 games

Drawn: 360

90 24

= 6 games

Lost:360

150 24

= 10 games

(Check that 8 + 6 + 10 = 24)

Example

Page 140, 141Ex 2

Constructing Pie Charts

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 Frequenc

yAngle At Centre

150

43

150

52

150

31

150

24

ooof 10315036043360150

43

ooof 12515036052360150

52

ooof 7415036031360150

31

ooof 5815036024360150

24

360

Now we draw the pie chart ...

Example

Geology SurveyStep 1: Title.

Step 2: Draw a circle.

Step 3: Draw in start line.

Step 4: Using a protractor draw in the other lines.

103°74°

125°

58°

(you do not need to write the angles)

Step 5: Label the sectors.

Granite

Dolerite

Sandstone

Limestone

P141/142Ex 3

Cumulative Frequency

Fifty maths students are graded 1 to 10 where 10 is the best grade.The grades and frequencies are shown below.

50

49

47

43

37

27

16

6

2

0

Cumulative Frequency

110

29

48

67

106

115

104

43

22

01

Frequency

Grade

A third column has been created which keeps a running total of the frequencies.

These figures are called cumulative frequencies.

The cumulative frequency of grade 7 is 43.

This can be interpreted as …‘43 candidates are graded 7 or

less’.

P143/144Ex4

Example

Cumulative Frequency DiagramsUsing the previous example we can draw a cumulative frequency

diagram.We make line graph of cumulative frequency (vertical) against grade (horizontal).

50

49

47

43

37

27

16

6

2

0

Cumulative Frequency

110

29

48

67

106

115

104

43

22

01

FrequencyGrade

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10Grade

Cu

mu

lati

veFr

eq

ue

ncy

Maths Students Grades

Fixed before gathering data

Fixed before gathering data

Information gathered

0

5

10

15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10Grade

Cu

mu

lati

ve

Fre

qu

en

cy

Using the diagram only …How many pupils were grade 6 or less ? 37At least 25 pupils were less than grade 5.

P145,146 Ex 5

Maths

Students

Grades

DotplotsIt 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.

ExampleA 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

P147/148

EX 6

Here are some expressions commonly used to describe distributions

The Five-Figure Summary

When a list of numbers is put in order it can be summarised by quoting five figures:

H

L

Q2

Q1

Q3

Highest number

Lowest number

Median of the full list (middle score)

Lower quartile – the median of the lowerhalf

Upper quartile – the median of the upperhalf

ExampleMake 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

L = Q1 = Q2 = Q3 = H =

3 11

Q2Q1Q3

86 9

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

L = Q1 = Q2 = Q3 = H =

3 11

Q2Q1 Q3

7.56 9

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

L = Q1 = Q2 = Q3 = H =3 11

Q2Q1 Q3

75.5 9.5

P151: Ex 7

Boxplots

A boxplot is a graphical representation of a

five-figure summary.

A suitable scale

L HQ1Q2 Q3

Example: Draw a box plot for this five-figure summary, which represents candidates marks in an exam out of 100L = Q1 = Q2 = Q3 = H

=

0 10

20

30

40

50

60

70

80

90

100

12

974932 66

● 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

Marks out of 100

Comparing DistributionsWhen 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 rangesemi-interquartile range is used

0 10 20 30 40 50 60 70 80 90 100Marks out of 100

Interquartile range = Q3 – Q1 Semi-interquartile range = (Q3 – Q1)

(SIQR) 2Q1 Q3

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 January’s (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.

Results of two exams

Boxplots showing spread of marks in two successive tests.

0 10 20 30 40 50 60 70 80 90 100

Test 1

Test 2

Mr Tennent’s example

Has the class improved? (give reasons for your answer)

Which would you hope to be test 1 and which test 2?

Boxplots

A boxplot is a graphical representation of a five-figure summary.

A suitable scale

L HQ1 Q2 Q3

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 halfQ3 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

L = Q1 = Q2 = Q3 = H =12

974932 66

0 10 20 30 40 50 60 70 80 90 100

● 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)

Marks out of 100

ExampleMake 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

L = Q1 = Q2 = Q3 = H =

3 11

Q2Q1Q3

86 9