Graphs, charts and tables!

1
Graphs, charts and tables!
L Q1 Q2 Q3 H
2
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
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 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.
5
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)
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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)
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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 ...
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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
9
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
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).
Maths Students Grades
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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.
12
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
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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
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Here are some expressions commonly used to
describe distributions
P147/148 EX 6
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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
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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
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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
18
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
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Boxplots
A boxplot is a graphical representation of a
five-figure summary.
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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
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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
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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.
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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)
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Boxplots
A boxplot is a graphical representation of a
five-figure summary.
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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
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)
27
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
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