EXAMPLE 9
Based on the following table, calculate
the median of the data.
Height (cm) Number of trees
15 – 19 3
20 – 24 10
25 – 29 26
30 – 34 24
35 – 39 20
40 – 44 12
45 – 49 5
44
SOLUTION
Height (cm) Number of trees Cumulative
frequency
15 – 19 3
20 – 24 10 3
25 – 29 26 13
30 – 34 24 39
35 – 39 20 63
40 – 44 12 83
45 – 49 5 95
100
Therefore, = + ൗ −
= 29.5 + 100ൗ2 − 39 (5)
24
= .
45
LET'S TRY
QUESTION 1
The table shows the number of books read by
the students of a class in one year. Calculate
the median of the data.
Number of books Number of students
10 - 14 3
15 - 19 7
20 - 24 9
25 - 29 12
30 - 34 5
46
LET'S TRY
QUESTION 2
The table shows the time taken (in seconds)
by a group of 40 students to complete a race.
Calculate the median of the data.
Time (in seconds) Number of
students
41 - 45 3
46 - 50 7
51 - 55 9
56 - 60 5
47
ෝ = +
+
L = lower boundary mode class
d1 = difference between the
mode class and frequency
before it
d2 = difference between the
mode class and frequency
after it
c = size of class
48
EXAMPLE 10
The following table are the marks
obtained by 48 students of class 3A
in a Mathematics test. Find the
mode for the data.
Test marks Number of students
50 – 59 4
60 – 69 8
70 – 79 18
80 – 89 16
90 – 99 2
49
SOLUTION
Test marks Number of students
50 – 59 4
60 – 69 8
70 – 79
80 – 89 = − =
90 – 99
18
16 = − =
2
Therefore, ෝ = +
+
10
ො = 69.5 + 10 + 2 (10)
=
50
EXAMPLE 11
Calculate the mode for the
following grouped data.
Class Number of students
21 – 30 12
31 – 40 20
41 – 50 28
51 – 60 16
61 – 70 14
71 – 80 10
51
SOLUTION
Class Number of students
21 – 30 12
31 – 40 20 = − =
41 – 50 28
51 – 60 16 = − =
61 – 70 14
71 – 80 10
Therefore, ෝ = +
+
8
ො = 40.5 + 8 + 12 (10)
= .
52
LET'S TRY
QUESTION 1
The table shows the scores of 200 students
who participated in a Mathematics Quiz. Find
the mode of the score.
Score Number of participants
1 - 20 7
21 - 40 45
41 - 60 94
61 - 80 36
81 - 100 18
53
LET'S TRY
QUESTION 2
The table shows the weight of 16 students in
a class. Find the mode of the weight.
Weight Number of students
30 - 34 3
35 - 39 5
40 - 44 6
45 - 49 2
54
SUBTOPIC 1.4
WHAT IS
MEASURE OF
DISPERSION?
The measure of dispersion shows the
scatterings of the data. It tells the variation
of the data from one another and gives a
clear idea about the distribution of the data.
The types of measures of dispersion are
mean deviation, variance, standard deviation,
quartiles, deciles and percentiles.
56
MEASURES OF DISPERSION
= σ − ഥ
σ − ഥ
=
=
57
EXAMPLE 12
Find the mean, deviation
for the set data below:
2, 7, 6, 3, 4, 8
58
SOLUTION
Step 1: Calculate the mean
σ 2 + 7 + 6 + 3 + 4 + 8
ҧ = = 6 = 5
Step 2: Form a table
− ഥ
23
72
61
32
41
83
σ − ഥ = 12
Step 3: Substitute into the formula
= σ − ҧ 12
= 6 =
59
EXAMPLE 13
Find the variance and
standard deviation for the set
data below:
1, 4, 3, 2, 7, 8, 4, 8, 8
60
SOLUTION
Step 1: Calculate the mean
σ 1 + 4 + 3 + 2 + 7 + 8 + 4 + 8 + 8
ҧ = = 9 = 5
Step 2: Form a table − ഥ
16
− ഥ 1
14 4
41 9
32 4
23 9
72 1
83 9
41 9
83
83 σ − ഥ = 62
σ − ഥ = 22
61
Step 3: Substitute into the formula
σ − ҧ 2 62
9
Therefore, 2 = = = .
= = 6.89 = .
62
EXAMPLE 14
Find the mean deviation,
variance and standard deviation
for the set data below:
3, 4, 7, 8, 10
63
SOLUTION
Step 1: Calculate the mean
σ 3 + 4 + 7 + 8 + 10
ҧ = = 5 = 6.4
Step 2: Form a table − ഥ
11.56
− ഥ 5.76
3 3.4 0.36
4 2.4 2.56
7 0.6 12.96
8 1.6
10 3.6 σ − ഥ = 33.2
σ − ഥ = 11.6
64
Step 3: Substitute into the formula
Therefore, = σ − ҧ 11.6
= 5 = .
σ − ҧ 2 33.2
5
2 = = = .
= = 6.64 = .
65
LET'S TRY
QUESTION 1
Find the mean deviation, variance and
standard deviation for the following data:
a) 4, 5, 7, 10
b) 5, 3, 11, 9, 13, 8
66
MEASURES OF DISPERSION
σ − ഥ
= σ
= σ − ഥ
σ
=
67
EXAMPLE 15
The data below shows the age of
participants in a chess competition.
Find the mean deviation of the data.
Age (years) Number of participants
10 – 14 4
15 – 19 12
20 – 24 18
25 – 29 9
30 – 34 12
35 – 39 5
68
SOLUTION
Step 1: Calculate the mean
Age (years) Midpoint,
10 – 14 4 12 48
15 – 19 12 17 204
20 – 24 18 22 396
25 – 29 9 27 243
30 – 34 12 32 384
35 – 39 5 37 185
σ = 60 σ = 1460
σ 1460
Therefore, ҧ = σ = 60 = 24.3
69
Step 2: Add 1 column and calculate
Age (years) Midpoint, − ഥ
10 – 14 4 12 48 49.2
15 – 19 12 17 204 87.6
20 – 24 18 22 396 41.4
25 – 29 9 27 243 24.3
30 – 34 12 32 384 92.4
35 – 39 5 37 185 63.5
σ = 60 σ = 1460 σ − ҧ = 358.4
Step 3: σ − ҧ 358.4
Substitute into = σ = 60
the formula = .
70
EXAMPLE 16
The frequency distribution table below
show the marks of a pupils in a Quiz.
Find the variance of the data.
Mark Number of pupils
0–9 6
10 – 19 7
20 – 29 12
30 – 39 11
40 – 49 4
71
SOLUTION
Step 1: Calculate the mean
Age (years) Midpoint,
0–9 6 4.5 27
10 – 19 7 14.5 101.5
20 – 29 12 24.5 294
30 – 39 11 34.5 379.5
40 – 49 4 44.5 178
σ = 40 σ = 980
σ 980
Therefore, ҧ = σ = 40 = 24.5
72
Step 2: Add 1 column and calculate
Age (years) Midpoint, − ഥ
27 2400
0 – 9 6 4.5 700
0
10 – 19 7 14.5 101.5 1100
1600
20 – 29 12 24.5 294
379.5 σ − ҧ 2 = 5800
30 – 39 11 34.5
40 – 49 4 44.5 178
σ = 40 σ = 980
Step 3: 2 = σ − ഥ
Substitute into σ
the formula 5800
= 40
=
73
EXAMPLE 17
Find the standard deviation of the
data below:
Score Frequency
11 – 20 7
21 – 30 16
31 – 40 18
41 – 50 9
74
SOLUTION
Step 1: Calculate the mean
Age (years) Midpoint,
11 – 20 7 15.5 108.5
21 – 30 16 25.5 408
31 – 40 18 35.5 639
41 – 50 9 45.5 409.5
σ = 50 σ = 1565
σ 1565
Therefore, ҧ = σ = 50 = 31.3
75
Step 2: Add 1 column and calculate
Score Midpoint, − ഥ
11 – 20 1747.48
7 15.5 108.5
21 – 30 16 25.5 408 538.24
31 – 40 18 35.5 639 317.52
41 – 50 9 45.5 409.5 1814.76
σ = 50 σ = 1565 σ − ҧ 2 = 4418
Step 3: Calculate 2 = σ − ഥ = 4418
the variance σ 50
Step 4: = 88.36
Find the standard
= = 88.36
deviation = .
76
LET'S TRY
QUESTION 1
The table shows the height of a group
of students. Find the mean deviation
and variance for the data.
Height (cm) Number of students
100 - 109 3
110 - 119 5
120 - 129 7
130 - 139 6
140 - 149 4
77
LET'S TRY
QUESTION 2
The table shows the number of car sold by a
dealer in 100 days. Find the standard deviation
for the number of cars sold in a day.
Number of cars Number of days
50 - 59 2
60 - 69 10
70 - 79 18
80 - 89 26
90 - 99 20
100 - 109 19
110 - 119 5
78
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