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Prepared by : NURZATY BINTI MUHAMAD NOR Page 1 B3001/UNIT3/1 Unit 3 MEASURES OF DISPERSION Understand frequency distribution graph and understand measures of dispersion. On completion of this unit, the students should be able to : 1. Determine quartile, decile and percentile from ogive graph. 2. Calculate mean deviation, variance and standard deviation for ungrouped data. 3. Calculate mean deviation, variance and standard deviation for grouped data. General Objective Specific Objective

BA201 Engineering Mathematic UNIT3 - Measures of Dispersion

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Page 1: BA201 Engineering Mathematic UNIT3 - Measures of Dispersion

Prepared by : NURZATY BINTI MUHAMAD NOR Page 1

B3001/UNIT3/1

Unit

3

MEASURES OF

DISPERSION

Understand frequency distribution graph

and understand measures of dispersion.

On completion of this unit, the students

should be able to :

1. Determine quartile, decile and

percentile from ogive graph.

2. Calculate mean deviation,

variance and standard deviation

for ungrouped data.

3. Calculate mean deviation,

variance and standard deviation

for grouped data.

General Objective

Specific Objective

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Decile (D)

Percentile

(P)

3.0 INTRODUCTION

Measures of dispersion are the measures of the degree of dispersion to

determine how far the values of data in a set of data scatter or spread out from its

average value. There are a few types of measures of dispersion :

a) Quartile

b) Decile

c) Percentile

d) Mean Deviation and Variance

e) Standard deviation

What is the Quartile,Decile and Percentile in

measures of dispersion?

Quartile (Q)

Value which divides a set of data arranged in

ascending or decending order into four equal parts.

There are first quartile (Q1), second quartile /

median (Q2) and third quartile (Q3)

Value which divides a set of data arranged in ascending or

decending order into ten equal parts. These are known as

D1, D2, D3,……,D9.

Value which divides a set of data arranged in ascending or

decending order into hundred equal part. These are known

as P1, P2, P3,……,P99.

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3.1 MEASURES OF DISPERSION

Measures of dispersion are the measures of the degree of dispersion to

determine how far the values of data in a set of data scatter or spread out from its

average value. There are a few types of measures of dispersion :

f) Quartile

g) Decile

h) Percentile

i) Variance

j) Standard deviation

3.1.1 Quartile,Decile and Percentile

Firstly, from data given , we must construct a cumulative frequency graph of

100% against class boundaries of data to determine quartile,decile and percentile

using ogive as shown below:

Example 3.1:

From data in table 3.1, determine first quartile ,third quartile fourth decile and eighty

eighth percentile through ogive.

Table 3.1

Order (RM) No. of order

10 and less than 20 85

20 and less than 30 120

30 and less than 40 225

40 and less than 50 135

50 and less than 60 105

60 and less than 70 30

INPUT

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Solution :

Step 1 : Construct cumulative frequency table as shown in Table 3.2.

Jadual 3.2

Order (RM) Cumulative

frequency (cf)

% cumulative

frequency (% cf)

less than 10 0 0

less than 20 85 12.1

less than 30 205 29.3

less than 40 430 61.4

less than 50 565 80.7

less than 60 670 95.7

less than 70 700 100.0

Step 2 : Construct ogive graph

From the graph 3.1,

Figure 3.1: Ogive Graph ‘Less Than’

(85/100) x 700

= 12.1 %

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First quartile, K1 = RM 27.00

Third quartile, K2 = RM 47.00

Fourth decile, D4 = RM 33.10

Eighty eighth Percentile , P88 = RM 54.50

ACTIVITY 3a

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3a.1 A set of grouped data shows the weight of 100 students. Find first quartile,

third quartile,fifth decile and sixtieth percentile,P60.

Berat (kg) Kekerapan

45 – 48 4

49 – 52 7

53 – 56 10

57 – 60 14

61 – 64 19

65 – 68 25

69 - 72 12

73 - 76 9

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FEEDBACK 3a

3a.1 first quartile, K1 = 57.7 kg

third quartile, K3 = 67.7 kg

fifth decile, D5 = 64.0 kg

sixtieth percentile P60 = 67.0 kg

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3.2 MEAN DEVIATION FOR UNGROUPED DATA

Mean deviation for the raw data x1, x2, x3,……………………..xn is calculated by

using the following formula.

E =

n

xxn

li

i

……………………………(3.1)

where x = mean.

Example 3.2:

Calculate mean deviation for the following data.

a) 12, 6, 15, 3, 12, 6, 21, 15, 18, 12

b) 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Solution:

a)

12 6 15 3 12 6 21 15 18 12 120

xxi 0 -6 3 -9 0 6 9 3 6 0

xxi 0 -6 3 -9 0 6 9 3 6 0 4.2

E = n

xxi =

10

42 = 4.2

b)

12 12 12 12 12 12 12 12 12 12 120

xxi 0 0 0 0 0 0 0 0 0 0

xxi 0 0 0 0 0 0 0 0 0 0 0

E = n

xxi =

10

0 = 0

INPUT

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Example 3.3:

Determine mean deviation for the following data.

5, 7, 1, 2, 4

Solution:

ix 5 7 1 2 4 19

xxi 1.2 3.2 -2.8 -1.8 0.2

xxi 1.2 3.2 2.8 1.8 0.2 9.2

Mean x = n

xn

li

i =

5

19 = 3.8

mean deviation, E:

E = n

xxn

li

i

= 5

2.9 = 1.84

3.3 MEAN DEVIATION FOR GROUPED DATA

Mean deviation for grouped data, E is calculated by using the following

E = n

fxxk

li

ii

………………………………. (3.2)

Where x = mean .

Example 3.4:

Calculate mean deviation for the following grouped data.

Class Frequency if

0 - 4

5 - 9

10 - 14

15 - 19

30

51

10

10

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Solution:

ix if ii fx xxi xxi

ii fxx

2

7

12

17

30

51

10

10

60

357

120

170

-5

0

5

10

5

0

5

10

150

0

50

100

707 300

mean, x = n

fx ii =

101

707 = 7

mean deviation, E = n

fxx ii =

101

300 = 3

Example 3.5:

Calculate mean deviation for the following grouped data.

Class Frequency, if

24 – 32

34 – 42

44 – 52

54 – 62

64 – 72

5

8

12

14

11

50

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Solution:

Mid class, ix if ii fx xxi xxi ii fxx

28

38

48

58

68

5

8

12

14

11

140

304

576

812

748

-23.6

-13.6

-3.6

6.4

16.4

23.6

13.6

3.6

6.4

16.4

118.0

108.0

43.2

89.6

180.4

50 2580 540

Mean = n

fxk

li

ii =

50

2580 = 51.6

And mean deviation = n

fxxk

li

ii

= 50

540

= 10.8

(24 +32) / 2

= 28

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ACTIVITY 3b

3b.1 Calculate mean deviation for the following data.:

Class Frequency if

0 – 4

4 – 8

8 – 12

12 – 16

16 – 20

4

16

20

16

4

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ANSWERS 3b

3b.1 Mean, E = 3.2

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3.3 VARIANCE FOR UNGROUPED DATA

Variance (S2

) for the raw data x1, x2, x3, ………xn is calculated by using the

following formula :

S2 =

1

2

n

xxn

li

i

…………………………………………(3.3)

Where x = mean.

Example 3.6:

Calculate the variance for the following data :

a) 12, 6, 15, 3, 12, 6, 21, 15, 18, 12

b) 12, 10, 12, 14, 10, 13, 12, 11, 14, 12

c) 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Solution:

a)

12 6 15 3 12 6 21 15 18 12 120

xxi 0 -6 3 -9 0 -6 9 3 6 0

2xxi 0 36 9 81 0 36 81 9 36 0 288

S2 =

1

2

n

xxi =

110

288

= 32

INPUT

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b)

12 10 12 14 10 13 12 11 14 12 120

xxi 0 -2 0 2 -2 1 0 -1 2 0

2xxi 0 36 9 81 0 36 81 9 36 0 18

S2 =

1

2

n

xxi =

110

18

= 2

c)

12 12 12 12 12 12 12 12 12 12 120

xxi 0 0 0 0 0 0 0 0 0 0

2xxi 0 0 0 0 0 0 0 0 0 0 0

S2 =

1

2

n

xxi =

110

0

= 0

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ACTIVITY 3c

3c.1 Find the mean and variance for the following data:

10, 7, 19, 13, 14, 9

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FEEDBACK 3c

3c.1 mean = 12

variance = 15.3

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3.4 STANDARD DEVIATION

Standard deviation is represented as S, square root of variance.

S= iancevar

Example 3.7:

Determine standard deviation for data in (a), (b) and (c) in example 3.6.

Solution:

Standard deviation :

(a) S2 = 32 , S = 32 = 5.66

(b) S2 = 2 , S = 2 = 1.41

(c) S2 = 0 , S = 0 = 0

Variance Formula for Ungrouped Data

You can use the following formula to find the variance for ungrouped data :

S2 =

1

2

n

xxn

li

i

………………………………(3.4)

Or

S2 =

1

22

n

xnxn

li

i

………………………………(3.5)

where x = mean.

INPUT

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Example 3.8:

Determine a variance of the following using data : 5, 7, 1, 2, 4.

Solution :

ix xxi 2xxi 2

xx

5

7

1

2

4

1.2

3.2

-2.8

-1.8

-0.2

1.44

10.44

7.84

3.24

0.04

25

46

1

4

6

19 22.8 95

mean,

x = 5

ix =

5

19 =3.8

First method – Using equation (3.4):

S2 =

1

2

n

xxn

li

i

= 4

8.22 = 5.7

Second method – Using equation (3.5):

S2 =

1

22

n

xnxn

li

i

=4

)8.3(595 2

= 4

2.7295

= 4

8.22

= 5.7

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Variance Formula for Grouped Data

You can use the following formula to find the variance for grouped data :

1

2

2

n

fxx

S

k

li

ii

………………………………(3.6)

Or

S2 =

1

22

n

xnfxk

li

ii

………………………………(3.7)

where x = mean.

Example 3.9:

Calculate the variance and standard deviation for the following distribution

frequency table using the formula in equation 3.6.

Class Frequency

0 – 4

5 – 9

10 – 14

15 – 19

30

51

10

10

Solution:

Class ix if ix if xxi 2xxi 2xxi if

0 – 4

5 – 9

10 – 14

2

7

12

30

51

10

60

357

120

-5

0

5

25

0

25

750

0

250

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15 – 19 17 10 170 10 100 1000

So,

x = 101

707 = 7

and

Variance,

1

2

2

n

fxx

S

k

li

ii

= 1101

2000

= 20

standard deviation, S = 20 = 4.47

Example 3.10:

Using formula in equation 3.7, calculate the variance and standard deviation in

the following table.

Class Frequency

0 – 4

5 – 9

10 – 14

15 – 19

30

51

10

10

Solution:

Class ix 2

ix if ix if 2

ix if

0 – 4

5 – 9

10 – 14

15 – 19

2

7

12

17

4

49

144

289

30

51

10

10

60

357

120

170

120

2499

1440

2890

101 707 6949

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7101

707

n

fxx

ii

2S =1

22

n

xnfxk

li

ii

=

100

71016949 7

20100

1000

100

49496949

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ACTIVITY 3d

3d.1 Calculate the mean and standard deviation for the following frequency

distribution table :

Class Frequency

16 – 21

22 – 27

28 – 33

24 – 39

15

16

5

5

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FEEDBACK 3d

3d.1 Mean, 5.24

x

Standard deviation, S = 6

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PRACTICES

3.a Find mean, mean deviation and standard deviation using the following data.

i) 32, 35, 36, 37, 41, 43

ii) 7.481, 7.478, 7.481, 7.483, 7.485, 7.475, 7.491, 7.488, 7.480, 7.477

3.b The sales of companies P and Q for last month are shown in the table below.

Construct an ogive from the table and determine K1, K3, D8 and P70.

Price (RM) Company P (unit) Company Q (unit)

Below 30 2 2

30 – 39 3 5

40 – 49 9 11

50 – 59 15 20

60 – 69 28 32

70 – 79 6 13

80 – 89 2 6

90 and above 1 2

Total 66 91

3.c Calculate mean deviation and standard deviation using the following data.

i) 4, 8, 12, 10, 14, 4, 8, 2, 10

ii) 23, 29, 21, 27, 33, 29, 31, 27

iii) –11, 12, -12, 11, -11, 14, 12, 13, 11

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3.d The following table shows the daily income in RM for 30 workers. Calculate

the mean, mean deviation and standard deviation for that data.

RM Bilangan

11 – 13

13 – 15

15 – 17

17 – 19

19 – 21

21 – 23

23 – 25

3

4

5

6

5

4

3

30

3.e Calculate the mean deviation and standard deviation for the following data;

kelas Frekuansi

0 – 4

4 – 8

8 – 12

12 – 16

16 - 20

4

6

10

8

4

3.f Calculate the mean deviation and standard deviation for the following data;

Kelas Frekuansi

40 – 49

50 – 59

60 – 69

70 – 79

80 – 89

90 – 99

1

4

3

5

13

4

30

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ANSWERS

3a. i). mean,

x = 37.33; mean deviation,E =3.1 and standard

deviation, S = 4.03

ii). mean,

x = 7.48; mean deviation,E =0.0039 and standard

deviation, S = 0.005

3b. K1= 30, K3=60, D8=70 and P70 = 67

3c. i). E = 28/9 ; S = 4

ii). E = 3 ; S = 3.95

iii). E = 10.44 ; S = 11.79

3d.

x = 18, E = 88/30 and S = 3.64

3e. E = 3.8 and S = 3.64

3f. E = 11.36 and S = 13.82

GOOD LUCK

“PRACTICE MAKES PERFECT”