1. Graphs, charts and tables!
L Q Q Q H

2. Some reminders … Scores are the wee numbers Mean is sum of scores
number of scores
Median is the middle score.
Mode is the score which occurs most often
Range is highest score – lowest score

3. 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.

4. 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.
Draw a relative frequency table for the wine sales.
Country
Frequency
France
120
Australia 30
Italy 27
Spain 24
Germany 18
Others 21
Total
240
Country
Frequency
Relative Frequency
France
120
Australia 30
Italy 27
Spain 24
Germany 18
Others 21
Total
240
120  240
= 0.5
30  240 =
27  240 =
24  240
= 0.1
18  240 =
21  240 = 1
Note: The total of the relative frequencies is always This is a useful check.

5. P138/139 Ex1 (omit questions 3b, 5b) Total Others Germany Spain Italy
Australia
France
Country
240 21 18 24 27 30
120
Frequency
Relative Frequency
120  240
30  240
27  240
24  240
18  240
21  240
= 0.5 = =
= 0.1 = = 1
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 = 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)

6. Reading Pie Charts Page 140, 141 Ex 2 Example
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)
Won
Lost
Drawn
A full circle is 360
Won:
 24
= 8 games
120
90
150
Drawn:
 24
= 6 games
Lost:
 24
= 10 games
(Check that = 24)

7. 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 ...

8. P141/142Ex 3 Geology Survey Limestone Granite Sandstone Dolerite
Step 1: Title.
Step 2: Draw a circle.
Limestone
Step 3: Draw in start line.
Granite
58°
Step 4: Using a protractor draw in the other lines.
103°
Sandstone
74°
(you do not need to write the angles)
125°
Step 5: Label the sectors.
Dolerite
P141/142Ex 3

9. ‘43 candidates are graded 7 or less’.
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. 1 10 2 9 4 8 6 7 11 5 3
Frequency
Grade
Cumulative Frequency 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

10. 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).
Information gathered
Fixed before gathering data
Maths Students Grades 50 49 47 43 37 27 16 6 2
Cumulative Frequency 1 10 9 4 8 7 11 5 3
Frequency
Grade
Grade
CumulativeFrequency
Fixed before gathering data

11. P145,146 Ex 5 37 Maths Students Grades Cumulative Frequency Grade
Using the diagram only …
P145,146 Ex 5
How many pupils were grade 6 or less ? 37
At least 25 pupils were less than grade 5.

12. Dotplots Example A group of athletes are timed in a 100m sprint.
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 …
Each piece of data becomes a data point sitting above the number line

13. 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

14. Here are some expressions commonly used to describe distributions

15. 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

16. Example Make a five-figure-summary for the following data ... Q1 Q2 Q3
L = Q1 = Q2 = Q3 = H = 3 6 8 9 11

17. Example Q1 Q2 Q3 Make a five-figure-summary for the following data. Q1 Q2 Q3
L = Q1 = Q2 = Q3 = H = 3 6
7.5 9 11

18. P151: Ex 7 Example Make a five-figure-summary for the following data. Q1 Q2 Q3
L = Q1 = Q2 = Q3 = H = 3
5.5 7
9.5 11
P151: Ex 7

19. A boxplot is a graphical representation of a
Boxplots
A boxplot is a graphical representation of a
five-figure summary.
A suitable scale L H Q1 Q2 Q3

20. 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 97 49 32 66 10 20 30 40 50 60 70 80 90
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)
P152/153: Ex 8

21. 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
Semi-interquartile range = (Q3 – Q1)
(SIQR) Q1 Q3 10 20 30 40 50 60 70 80 90
100
Marks out of 100

22. Results of two exams
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.

23. Mr Tennent’s 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)

24. A boxplot is a graphical representation of a five-figure summary.
Boxplots
A boxplot is a graphical representation of a five-figure summary.
A suitable scale L H Q1 Q2 Q3

25. 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

26. ● 25% of the candidates got between 12 and 32 (the lower whisker)
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 97 49 32 66 10 20 30 40 50 60 70 80 90
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)

27. Example Make a five-figure-summary for the following data ... Q1 Q2 Q3
L = Q1 = Q2 = Q3 = H = 3 6 8 9 11