Unit 18
Measures of Variation
Cumulative Frequency
The table shows the age distribution (in complete years) of the
population of Nigeria in 1991.
0 ≤ x < 15
15 ≤ x < 30
30 ≤ x < 45
45 ≤ x < 60
60 ≤ x < 75
75 ≤ x < 100
% of Pop.
32
29
20
12
6
1
Cumulative %
32
61
81
93
99
100
Age
100
90
Upper Quartile ≈ 40∙5
80
70
Estimate:
a) The lower quartile
b) The median
c) The upper quartile
60
50
Median ≈ 24
40
30
20
Lower Quartile ≈ 11
10
0
10
20
30
40
50
60
Age in Years
70
80
90
100
Unit 18
Measures of Variation
Cumulative Frequency 2
The table shows the distribution of marks on a test for 70 students.
Mark
Frequency
Cumulative
Frequency
1 – 10
2
2
11 – 20
5
7
21 – 30
9
16
31 – 40
14
30
41 – 50
16
46
51 – 60
12
58
61 – 70
8
66
71 – 80
4
70
a) Draw a cumulative frequency curve
80
70
60
50
Students Pass: 70 – 42 = 28
Mark
Frequency
Cumulative
Frequency
1 – 10
2
2
11 – 20
5
7
21 – 30
9
16
31 – 40
14
30
41 – 50
16
46
51 – 60
12
58
61 – 70
8
66
71 – 80
4
70
b) The pass mark for the test is
47. Use your graph to
determine the number of
students who passed the test
40
30
20
c) What is the probability that a
student chosen at random, had
a mark of less than or equal to
30?
10
0
10
20
30
40
Marks
50
60
70
80
Unit 18
Measures of Variation
Box and Whisker Plots
The goals scored in the first 11 football matches played by a National
Premier League team were:
1
0
4
2
2
3
1
2
5
0
1
This data can be represented using a box and whisker plot.
Construct an additional box and whisker plot for a
Record the data starting with the smallest
team with data
0
Identify:
3
0
5
1
2
1
Smallest Value
Median
1
1
2
3
2
3
2
2
5
Largest Value
0
3
5
4
6
2
Compare the two sets of data
Lower quartile
Upper quartile
1
3
Construct a box and whisker plot
0
1
2
3
4
5
6
7
5
2
1
Unit 18
Measures of Variation
Standard Deviation
The STANDARD DEVIATION (s.d.) of a set of data is a measure of the
spread of the data about the mean and is defined by
a) What is the mean (m) of each set?
S1 = {6, 7, 8, 9, 10}
m=8
S2 = {4, 5, 8, 11, 12} m = 8
S3 = {1, 2, 8, 14, 15} m = 8
b) The standard deviation for S1 is calculated as:
6
-2
4
7
-1
1
8
0
0
9
1
1
10
2
4
TOTAL
10
The STANDARD DEVIATION (s.d.) of a set of data is a measure of the
spread of the data about the mean and is defined by
S1 = {6, 7, 8, 9, 10}
S2 = {4, 5, 8, 11, 12}
S3 = {1, 2, 8, 14, 15}
m=8
m=8
m=8
c) Compare the standard deviations for S1, S2 and S3
S1 = {6, 7, 8, 9, 10}
S2 = {4, 5, 8, 11, 12}
S3 = {1, 2, 8, 14, 15}
s.d. ≈ 1∙414
s.d. ≈ 3∙162
s.d. ≈ 5∙831