Post on 31-Dec-2015
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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
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