Definition

“The standard deviation is defined as the positive square root of the mean of the squared deviations of the values from their mean.”

Characteristics of Standard Deviation:

i. The standard deviation is affected by the values of every observations.

ii. The process of squaring the deviations before adding, avoids the algebraic fallacy of disregarding signs.

iii. In general it is less affected by fluctuations of sampling than the other measures of dispersion.

iv. It has a specific mathematical meaning and could be easily adapted according to nature of algebraic treatment.

v. It has great practical utility in sampling and statistical inference.

Standard Deviation for Ungrouped Data

The standard deviation of a set of n values, clip_image002 denoted by S. The standard deviation for ungrouped data is mathematically represented as follows;

clip_image004

For Example

Find the standard deviation for the values 2, 3, 6, 8 and 11.

clip_image006

clip_image008

clip_image010

clip_image012

clip_image014

clip_image016

clip_image018

Standard Deviation for Grouped Data

In case of a frequency distribution with clip_image020 as class marks and clip_image022as the corresponding class frequencies, the standard deviation shall be calculated by using the following formula;

clip_image024

Where n = clip_image026

For Example

Find the frequency distribution for the frequency distribution of marks obtained by students provided in following Table 25

Table 25

Marks

Frequency (f)

Class Marks (X)

fX

20 – 24

1

22

22

25 – 29

4

27

108

30 – 34

8

32

256

35 – 39

11

37

407

40 – 44

15

42

630

45 – 49

9

47

423

50 – 54

2

52

104

clip_image028

clip_image030

Marks f X clip_image032 clip_image034 clip_image036 clip_image038
20 – 24 1 22 -17 -17 289 289
25 – 29 4 27 -12 -48 144 576
30 – 34 8 32 -7 -56 49 392
35 – 39 11 37 -2 -22 4 44
40 – 44 15 42 3 45 9 135
45 – 49 9 47 8 72 64 576
50 – 54 2 52 13 26 169 338
n= ∑f=50 clip_image040

clip_image042

clip_image044

clip_image046

clip_image048

For Example

Find the standard deviation of the weights distribution of 120 students in a University.

Weight (lb) Class Marks (X) Frequency (f) clip_image032[1] clip_image036[1] clip_image038[1]
110 – 119 114.5 1 -41.7 1738.89 1738.89
120 – 129 124.5 4 -31.7 1004.89 4019.56
130 – 139

134.5

17

-21.7

470.89

8005.13

140 – 149

144.5

28

-11.7

136.89

3832.92

150 – 159

154.5

25

-1.7

2.89

72.25

160 – 169

164.5

18

8.3

68.89

1240.02

170 – 179

174.5

13

18.3

334.89

4353.57

180 – 189

184.5

6

28.3

800.89

4805.34

190 – 199

194.5

5

38.3

1466.89

7334.45

200 – 209

204.5

2

48.3

2332.89

4665.78

210 -219

214.5

1 58.3 3398.89 3398.89
n = ∑f = 120 clip_image050

clip_image052

clip_image042[1]

clip_image054

clip_image056

clip_image058

Short Method for Computing Standard Deviation:

From the above methods of calculating standard deviation, both grouped and ungrouped, we have noticed that computing standard deviation requires number of calculations which makes the task laborious. To make the difficult task easy, there is also a short method for computing standard deviation. Such short formulae are explained as follows;

Short Method for Ungrouped Data

The standard deviation for ungrouped data using short method, will be computed by using the following formula;

clip_image060

For Example

Find the standard deviation using short method for the values, 2, 3, 6, 8 and 11

n = 5

∑X = 2 + 3 + 6 + 8 + 11 = 30

clip_image062

Substituting the values in the formulaclip_image064

clip_image066

clip_image068

clip_image016[1]

clip_image018[1]

Short Method for Grouped Data:

The standard deviation for grouped data using short method, will be computed by using the following formula;

clip_image070

For Example

Find the standard deviation using short method for the values given in following Table 26

Table 26

X f fX clip_image072 clip_image074
1 4 4 1 4
2 25 50 4 100
3 53 159 9 477
4 18 72 16 288
5 11 55 25 275
6 7 42 36 252
7 2 14 49 98
clip_image076 clip_image078 clip_image080

clip_image070[1]

clip_image082

clip_image084

clip_image086

clip_image088

Lets get started

Lets get started

Join us to receive instructor manuals, notes, lectures , e-book and tutorials videos, completely and a complete online course of Economics totally FREE

You have Successfully Subscribed!

Shares
Share This