Unit 18Measures of Variation

Cumulative Frequency

The table shows the age distribution (in complete years) of the population of Nigeria in 1991.

Age 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

0 10 20 30 40 50 60 70 80 90 100

Age in Years

100

90

80

70

60

50

40

30

20

10

Lower Quartile ≈ 11Lower Quartile ≈ 11

Median ≈ 24Median ≈ 24

Upper Quartile ≈ 40∙5Upper Quartile ≈ 40∙5

Estimate:a)The lower quartileb)The medianc)The upper quartile

Unit 18Measures of Variation

Cumulative Frequency 2

The table shows the distribution of marks on a test for 70 students.

Mark FrequencyCumulative Frequency

1 – 10 2 2

11 – 20 5 7

21 – 30 9 16

31 – 40 14

41 – 50 16

51 – 60 12

61 – 70 8

71 – 80 4

30

46

58

66

70

0 10 20 30 40 50 60 70 80Marks

80

70

60

50

40

30

20

10

Mark FrequencyCumulative 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

b) The pass mark for the test is 47. Use your graph to determine the number of students who passed the test

c) What is the probability that a student chosen at random, had a mark of less than or equal to 30?

Students Pass: 70 – 42 = 28Students Pass: 70 – 42 = 28

Unit 18Measures of Variation

Box and Whisker Plots

The goals scored in the first 11 football matches played by a National Premier League team were:

This data can be represented using a box and whisker plot.

Record the data starting with the smallest

Identify:

Construct a box and whisker plot

1 0 4 2 2 3 1 2 5 0 11 0 4 2 2 3 1 2 5 0 1

0 0 1 1 1 2 2 2 3 4 50 0 1 1 1 2 2 2 3 4 5

Smallest Value Largest Value

Median

Lower quartile Upper quartile

0

2

1

5

3

0 1 2 3 4 5 6 7

Construct an additional box and whisker plot for a team with data

3 5 2 1 3 3 2 5 6 2 1

Compare the two sets of data

Construct an additional box and whisker plot for a team with data

3 5 2 1 3 3 2 5 6 2 1

Compare the two sets of data

Unit 18Measures 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}

S2 = {4, 5, 8, 11, 12}

S3 = {1, 2, 8, 14, 15}

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

m = 8

m = 8

m = 8

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} m = 8

S2 = {4, 5, 8, 11, 12} m = 8

S3 = {1, 2, 8, 14, 15} m = 8

c)Compare the standard deviations for S1, S2 and S3

S1 = {6, 7, 8, 9, 10} s.d. ≈ 1∙414

S2 = {4, 5, 8, 11, 12} s.d. ≈ 3∙162

S3 = {1, 2, 8, 14, 15} s.d. ≈ 5∙831