Paper: 15-Quantitative Techniques for Management Decisions

Module:7-Measures of Dispersion: Mean Absolute Deviation, Standard

Deviation, Variance, Coefficient of Variation

Principle Investigator Prof. S. P. Bansal Vice Chancellor

Maharaja Agrasen University, Baddi

Co-Principle Investigator Prof. YoginderVerma Pro–ViceChancellor Central University of Himachal Pradesh. Kangra. H.P.

Paper Coordinator Prof. Pankaj Madan

Dean- FMS

Gurukul Kangri Vishwavidyalaya, Haridwar

Content Writer Prof. Pankaj Madan

Dean-FMS

GurukulKangriVishwavidyalay, Haridwar

Items Description of Module

Subject Name Management

Paper Name Quantitative Techniques for Management Decisions

Module Title Measures of Dispersion: Mean Absolute Deviation, Standard Deviation,

Variance, Coefficient of Variation

Module Id 7

Pre- Requisites Basic mathematical operations

Objectives Introduction

Range

Mean Absolute Deviation

Computation of Mean Deviation

Characteristics of mean deviation

Uses of mean deviation

Standard Deviation

Computation of Standard Deviation

Characteristics of Standard Deviation

Uses of Standard Deviation

Quartile deviation or Semi Inter-Quartile range

Variance

Relative measures of dispersion

Coefficient of dispersion

Coefficient of variation

Standard error

Expression for the standard error of mean

Probable error

Summary

Self-check exercise with solutions

Keywords Range, Mean Deviation, Standard Deviation, Variance, Quartile Deviation,

Semi Inter Quartile Range, Standard Error, Probable Error

Module-7Measures of Dispersion: Mean Absolute Deviation, Standard Deviation,Variance,

Coefficient of Variation

Introduction

Range: Definition, computation of range, merits and demerits of range estimation

Mean Absolute Deviation: Definition, computation of mean deviation, characteristics of mean deviation,

uses of mean deviation

Standard Deviation: Definition, computation of standard deviation, characteristics of standard deviation,

uses of standard deviation, Quartile Deviation

Variance: Definition, computation of variance

Relative measures of dispersion: Coefficient of dispersion, Coefficient of variation

Standard Error: Definition, Expression for the standard error of mean

Probable Error

Summary

Self-Check Exercise with solutions

Quadrant-I

Measures of Dispersion: Mean Absolute Deviation, Standard Deviation, Variance, Coefficient of

Variation

Learning Objectives:

After the completion of this module the student will understand:

Range

Mean Absolute Deviation

Computation of Mean Deviation

Characteristics of mean deviation

Uses of mean deviation

Standard Deviation

Computation of Standard Deviation

Characteristics of Standard Deviation

Uses of Standard Deviation

Quartile deviation or Semi Inter Quartile range

Variance

Relative measures of dispersion

Coefficient of dispersion

Coefficient of variation

Standard error

Expression for the standard error of mean

Probable error

1. Introduction

Any measure of central tendency or average has its own limitations and gives us an idea

only about that central value of the set of observations around which all the observations

have a tendency to lie, but it fails to give any idea about the way in which they are

distributed. There can be a number of series each of which has the same mean but differs

from others in respect of the pattern in which the observations are distributed. To follow

this point Consider the following series.

Series A 9 9 9 9 9 9 9

Series B 6 7 8 9 10 11 12

Series C 1 2 4 5 11 13 27

Series D 3 15

In the above series, we observe that arithmetic mean of every series is 9, but the pattern

in which the observations are distributed is different in different series. In series A the

mean is 9 and all the observations are same. In series B also, the mean is 9 and the

observations are scattered ranging from 6 to 12 but not very much scattered. In series C,

the mean is the same value 9 but the observations are too much scattered ranging from 1

to 27. In series D there are only two observations the mean of which is 9.

From the above example it is quite obvious that for studying a series, a study of the extent

of scattering of the observations of dispersion is also essential along with the study of the

central tendency in order to throw more light on the nature of the series. The following

are the different measures of dispersion which are in common use.

2. Range

2.1. Definition:

The range is the simplest measure of dispersion. It is the difference between the highest

and lowest terms of a series of observations.

2.2.Computation of Range:

𝑅𝑎𝑛𝑔𝑒 = 𝑋𝐻 − 𝑋𝐿

Where, XH = Highest variate value

and XL = Lowest variate value

2.3.Merits and demerits of range

(i) Its value usually increases with the increase in the size of the sample.

(ii) It is usually unstable in repeated sampling experiments of the same size and large

ones.

(iii) It is a very rough measure of dispersion and is entirely unsuitable for precise and

accurate studies.

(iv) The only merit possessed by ‘Range’ are that it is (i) simple (ii) easy to understand

and (iii) quickly calculated. It is often used in certain industrial work.

3. Mean Deviation

3.1. Definition:

If the deviations of all the observations from their mean are calculated, their algebraic

sum will be zero. When this sum is always zero, it is impossible to get the average of

these deviations. In order to overcome this difficulty, these deviations are added

irrespective of plus or minus sign and then the average is calculated. The deviations

without any plus or minus sign are known as absolute deviations. The mean of these

absolute deviations is called the mean deviation. If the deviations are calculated from the

mean, the measure of dispersion is called mean deviation about the mean. As a matter of

fact mean deviation can be calculated from any average, and for that, the absolute

deviations from that average will be calculated.

3.2.Computation of mean deviation:

𝑀𝑒𝑎𝑛 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡ℎ𝑒 𝑚𝑒𝑎𝑛 = 1

𝑁∑⃓𝑥⃓ =

1

𝑁∑⃓𝑋 − 𝑋 ⃓

x= Deviation from the mean= X- X͞ ⃓x⃓= Absolute deviation

N=Number of observations

Example:

Classes Frequency

(f)

Mid values

(X)

X- X͞

(x)

fx ⃓fx⃓

0-10 1 5 -22 -22 22

10-20 3 15 -12 -36 36

20-30 5 25 -2 -10 10

30-40 4 35 +8 +32 32

40-50 2 45 +18 +36 36

Total 15 - - 0 136

Calculations:

Mean deviation about the mean= 1

𝑁∑⃓𝑓𝑥⃓=

1

15 × 136= 9.07

Mean deviation about other averages:

M.D. about A= 1

𝑁∑⃓𝑓(𝑋 − 𝐴)⃓

M.D. about Md.=1

𝑁∑⃓𝑓(𝑋 −𝑀𝑑. )⃓

M.D. about Mo. = 1

𝑁∑⃓𝑓(𝑋 −𝑀𝑜. )⃓

3.3.Characteristics of mean deviation:

(i) A notable characteristic of mean deviation is that it is the least when calculated

about the median.

(ii) Standard deviation is not less than the mean deviation in a discrete series i.e. it is

either equal to or greater than the M.D. about mean.

(iii) When an average other than the A.M. is calculated as a measure of central

tendency, M.D. about that average is the only suitable measure of dispersion.

4. Standard Deviation

4.1.Definition:

Calculation of standard deviation is also based on the deviations from the arithmetic

mean. In thecase of mean deviation the difficulty, that the sum of the deviations from the

arithmetic mean is always zero, is solved by taking these deviations irrespective of plus

or minus signs. But here, that the difficulty is solved by squaring them and taking the

square root of their average. It is thus defined by thefollowingexpression.

Standard Deviation (S.D.) = √∑(𝑋−𝜇)2

𝑁 …………. (1)

Where, X= An observation or variate value

µ = Arithmetic mean of the population

N= Number of given observations.

According to the expression given in (1), thepopulationmeanµ is required for finding the

standard deviation (S.D.) of a given set of observations. Generally, µ is not known. Therefore it

is replaced by X͞, which is the mean of the given set of observations, and then the S.D. of the

given data is given by

Standard Deviation (S.D.) = √∑(𝑋−𝑋 )2

𝑁……………(2)

(X − X͞)2 = deviation from mean

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Here, it should be noted that formula (2) gives the S.D. of the given set of data which itself is

assumed to be the population with µ=X͞. Therefore we shall this S.D. as the ‘population S.D.’

Thus,

Population S.D. (=σ) = √∑(𝑋−𝑋 )2

𝑁 …………… (3)

In case of frequency distribution-

Population S.D. (=σ) = √∑𝑓(𝑋−𝑋 )2

𝑁 …………… (4)

Sample S.D.: In case, when the given set of data is not a population but is a sample drawn from

a large population, the population mean µ is not known. Therefore, in its place, we use X͞ which

is the estimate ofµ obtained from the sample observations. The result is that we cannot calculate

the population S.D. (σ), but, in its place, we calculate its estimate (S). We represent the estimates

of population parameters, µ and σ, in the following way:

X͞= Estimate of (µ)

S= Estimate of (σ)

The best estimates (S) of the population S.D. (σ) is given by

S (sample S.D.) = √∑𝑓(𝑋−𝑋 )2

𝑁−1 …………… (5)

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4.2.Computation of Standard Deviation

For computing S.D., in every case, we have to calculate the arithmetic mean, which

increases the labor of calculation work. Therefore, to avoid it, theshort cut method should

be used in which any value of the variate is chosen as the arbitrary mean and then the

standard deviation is calculated by the following process:

Suppose, A is the arbitrary mean and d is the deviation of the variate value from A.

i.e. d = X-A

we have, ∑𝑓(𝑋 − 𝑋 )2 =∑𝑓𝑑2 −(∑𝑓𝑑)2

𝑁

Therefore, for this, we require the columns of d, fd, and fd2. In the column of d we shall

find a factor equal to the width of the class interval “i” common to all the figures in that

column. After taking out this factor as common, the columns now will be of d/I, fd/I and

fd2/i2. With the help of these symbols, the values of ∑𝑓(𝑋 − X͞)2 and S.D. will be

calculated as given bellow.

∑𝑓𝑑 = 𝑖 ×∑𝑓𝑑

𝑖

∑𝑓𝑑2 = 𝑖2 ×∑𝑓𝑑2

𝑖2

∑𝑓(𝑋 − 𝑋 )2 = 𝑖2 × {∑𝑓𝑑2

𝑖2−(∑

𝑓

𝑑)2

𝑁}

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If we use the symbol D for d/I, the above expressions will be written as

∑𝑓𝑑 = 𝑖 × ∑𝑓𝐷

∑𝑓𝑑2 = 𝑖2 × ∑𝑓𝐷2

∑𝑓(𝑋 − 𝑋 )2 = 𝑖2 × {∑𝑓𝐷2 − (∑𝑓𝐷)2

𝑁}

𝑆. 𝐷. = 𝑖 × [√1

𝑁{∑𝑓𝐷2 −

(∑𝑓𝐷)2

𝑁}]

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Example:

Calculation of S.D.

Class Frequency

f

Mid values

X

d= X-A d/I or D fD fD2

0-10 1 5 -20 -2 -2 4

10-20 3 15 -10 -1 -3 3

20-30 5 25 0 0 0 0

30-40 4 35 +10 +1 +4 4

40-50 2 45 +20 +2 +4 8

Total 15 - - - +3 19

Here,

∑𝑓(𝑋 − 𝑋 )2 = 𝑖2 × {∑𝑓𝐷2 − (∑𝑓𝐷)2

𝑁}

= 102 × [9 −32

15]

= 100 × [276

15]

= 1840

Population S.D. =√∑𝑓(𝑋−𝑋 )2

𝑁 , (Here, µ= X͞)

σ = √1840

15 = 11.07

Sample S.D. = √∑𝑓(𝑋−𝑋 )2

𝑁−1 = √

1840

14

Or S= 11.46

4.3.Characteristics of Standard Deviation

It is rigidly defined.

Its computation is based on all the observations.

If all the variate values are the same, S.D.=0

S.D. is least affected by fluctuations of sampling.

It is affected by the change of scale, but not affected by the change of origin.

4.4.Uses of Standard Deviation

It is used in computing different statistical quantities like, regression coefficient,

correlation coefficient, etc. and in connection with business cycle analysis.

It is also used in testing the reliability of certain statistical measures.

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4.5.Quartile Deviation or Semi inter quartile range

The measure of dispersion is expressed in terms of quartiles and known as quartile

deviation or semi inter quartile range.

Quartile Deviation= 𝑄3−𝑄1

2

Where,Q1 = Lower quartile

Q3= Upper Quartile

It is not a measure of the deviation from any particular average. For symmetrical and

moderately skew distributions the quartile deviation is usually two-third of the standard

deviation.

Q.D.=2

3× (𝑆. 𝐷. )

5. Variance

Variance is the square of the standard deviation.

Variance = (S.D.)2

The variance of a population is generally represented by the symbol σ2and its unbiased

estimate calculated from the sample, by the symbol S2.

6. Relative Measures of Dispersion

The measures of dispersion, which we studied so far, are the absolute measures of

dispersion, and are represented it’s the same units in which the observations are

represented, e.g., gms., cm., meters, hectares, etc. When we have to compare the

dispersions of two or more distributions, it will not be proper to compare their absolute

measures of dispersions, because, the distributions or the data may differ from one

another.

(i) With respect to their averages

(ii) With respect to their dispersions

(iii)With respect to their averages and dispersions both

(iv) With respect to their units

Therefore, they will not be comparable. Under such circumstances, their comparison is

possible with the help of relative measures of dispersion.

6.1.Coefficient of Dispersion

It is computed by the following expression:

Coefficient of Dispersion = 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑠𝑜𝑓𝐷𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛

𝑅𝑒𝑙𝑎𝑡𝑒𝑑𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑠𝑜𝑓𝐶𝑒𝑛𝑡𝑟𝑎𝑙𝑇𝑒𝑛𝑑𝑒𝑛𝑐𝑦

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6.2.Coefficient of Variation (C.V.)

This is also a relative measure of dispersion. It is especially important on account of the

widely used measures of central tendency and dispersion i.e., Arithmetic mean, and

Standard Deviation. It is given by

C.V. = 𝑆.𝐷.

𝐴.𝑀.× 100

It is expressed in percentage and used to compare the variability in the two or more

series. Lesser value of thecoefficient of variation indicates more consistency.

7. Standard Error

7.1.Definition

The Standard deviation of the sampling distribution of a statistic (estimate) is known as

the standard error of that statistic (estimate).

If we take all possible samples from the population of the same size and get a sampling

distribution of means, it can be proved that the mean of this sampling distribution of

means is the population mean and its standard deviation, the standard error of the mean.

As it is not possible to draw and study all possible samples, we have to get and we get the

estimate of the standard error from a single sample. If S be the standard deviation of the

sample of size N, the estimate of the standard error of mean is given by 𝑆

√𝑁 .

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7.2.Expression for the standard error of mean

Let there be a sample of N observations, X1, X2, X3………XN which have been drawn at

random from a population, the variance of which is σ2.

Now, Mean X͞ = 1

𝑁 (𝑋1 + 𝑋2 + 𝑋3 +⋯+ 𝑋𝑁

Variance of mean

V(X͞)= 1

𝑁2{𝑉(𝑋1 + 𝑋2 + 𝑋3 +⋯+ 𝑋𝑁)}

= 1

𝑁2 [𝑉(𝑋1) + 𝑉(𝑋2) + ⋯+ 𝑉(𝑋𝑁)]

Since, V(X1) =V(X2) = V(X3)=………………….= V(XN)= σ2

V(X͞)= 𝑁𝜎2

𝑁2=

𝜎2

𝑁

S.E. of X͞ = 𝜎

√𝑁

But since, in practice, σ is not known, it is replaced by its unbiased estimate S.

S.E. of X͞ = 𝑆

√𝑁

8. Probable Error

The quartile deviation of the sampling distribution of means is known as aProbable error

and is 0.67449times the standard error.

P.E. = 0.67449 (S.E.)

Three times the probable error is roughly twice the standard error. This measure of

dispersion has no particular advantage and moreover involves a troublesome factor

0.67449. This is why it has gone out of use and has given place to standard error.

9. Summary

This module provides an overview to students to understand the techniques that are used

to measure the extent of variation or the deviation (also called thedegree of variation) of

each value in the dataset from a measure of central tendency, usually the mean or median.

Such statistical techniques are called measures of dispersion (or variation). A small

dispersion among values in the data set indicates that data are clustered that data are

clustered closely around the mean. The mean is therefore considered representative of the

data, i.e. mean is reliable average. Conversely, a large dispersion among values in the

data set indicates that the mean is not reliable, i.e. it is not representative of data. The

symmetrical distribution of values in two or more sets of data may have same variation

but differ greatly in terms of A.M. On the other hand, two or more sets of data may have

the same A.M. values but differ in variation.

10. Self-check exercise with solution

Q.1.Thefollowing data give the number of passengers traveling by airplane from one city to

another in one week.

115, 122, 129, 113, 119, 124, 132, 120, 110, 116

Calculate the mean and standard deviation and determine the percentage of class that lie

between (i) µ±σ (ii) µ±2σ and (iii) µ±3σ. What percentage of cases lie outside these

limits.

Calculation of Mean and Standard Deviation

X X- X͞ (X- X͞)2

115 -5 25

122 2 4

129 9 81

113 -7 49

119 -1 1

124 4 16

132 12 144

120 0 0

110 -10 100

116 -4 16

Solution:

µ = ∑𝑋

𝑁=

1200

10= 120 and σ2=

∑(𝑋−X͞)2

𝑁 =

436

10 =43.6

The percentage of cases that lie between a given limit are as follows:

Interval Values within interval Percentage of

population

Percentage

falling Outside

µ±σ = 120±6.60

= 113.4 and 126.6

113, 115, 116, 119,

120, 122, 124

70% 30%

µ±2σ = 120 ± 2

= 106.80 and 133.20

110, 113, 115, 116,

119, 120, 122, 124,

129, 132

100% nil

Q.2. What do you understand by dispersion?

A.2. A measure of dispersion is designed to state numerically the extent to which individual

observations vary on the average.

Q.3. What are the different measures of dispersion?

A.3. (1) Absolute measures: (i) Mean Deviation (ii) Standard Deviation (ii) Quartile

Deviation, Range.

(2) Relative measures: (i) Coefficient of variation (ii) Coefficient of Mean deviation