Module :7-Measures of Dispersion: Mean Absolute Deviation ...
Transcript of Module :7-Measures of Dispersion: Mean Absolute Deviation ...
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 ProβViceChancellor 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:
π ππππ = ππ» β ππΏ
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 βRangeβ 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:
ππππ πππ£πππ‘πππ ππππ’π‘ π‘βπ ππππ = 1
πββπ₯β =
1
πββπ β π β
x= Deviation from the mean= X- XΝ βxβ= Absolute deviation
N=Number of observations
Example:
Classes Frequency
(f)
Mid values
(X)
X- XΝ
(x)
fx βfxβ
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
πββππ₯β=
1
15 Γ 136= 9.07
Mean deviation about other averages:
M.D. about A= 1
πββπ(π β π΄)β
M.D. about Md.=1
πββπ(π βππ. )β
M.D. about Mo. = 1
πββπ(π βππ. )β
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.) = ββ(πβπ)2
π β¦β¦β¦β¦. (1)
Where, X= An observation or variate value
Β΅ = Arithmetic mean of the population
N= Number of given observations.
According to the expression given in (1), thepopulationmeanΒ΅ is required for finding the
standard deviation (S.D.) of a given set of observations. Generally, Β΅ is not known. Therefore it
is replaced by XΝ, 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.) = ββ(πβπ )2
πβ¦β¦β¦β¦β¦(2)
(X β XΝ)2 = deviation from mean
s-10
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 Β΅=XΝ. Therefore we shall this S.D. as the βpopulation S.D.β
Thus,
Population S.D. (=Ο) = ββ(πβπ )2
π β¦β¦β¦β¦β¦ (3)
In case of frequency distribution-
Population S.D. (=Ο) = ββπ(πβπ )2
π β¦β¦β¦β¦β¦ (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 Β΅ is not known. Therefore, in its place, we use XΝ which
is the estimate ofΒ΅ obtained from the sample observations. The result is that we cannot calculate
the population S.D. (Ο), but, in its place, we calculate its estimate (S). We represent the estimates
of population parameters, Β΅ and Ο, in the following way:
XΝ= Estimate of (Β΅)
S= Estimate of (Ο)
The best estimates (S) of the population S.D. (Ο) is given by
S (sample S.D.) = ββπ(πβπ )2
πβ1 β¦β¦β¦β¦β¦ (5)
s-11
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, βπ(π β π )2 =βππ2 β(βππ)2
π
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 βiβ 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 βπ(π β XΝ)2 and S.D. will be
calculated as given bellow.
βππ = π Γβππ
π
βππ2 = π2 Γβππ2
π2
βπ(π β π )2 = π2 Γ {βππ2
π2β(β
π
π)2
π}
s-12
If we use the symbol D for d/I, the above expressions will be written as
βππ = π Γ βππ·
βππ2 = π2 Γ βππ·2
βπ(π β π )2 = π2 Γ {βππ·2 β (βππ·)2
π}
π. π·. = π Γ [β1
π{βππ·2 β
(βππ·)2
π}]
s-13
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,
βπ(π β π )2 = π2 Γ {βππ·2 β (βππ·)2
π}
= 102 Γ [9 β32
15]
= 100 Γ [276
15]
= 1840
Population S.D. =ββπ(πβπ )2
π , (Here, Β΅= XΝ)
Ο = β1840
15 = 11.07
Sample S.D. = ββπ(πβπ )2
πβ1 = β
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.
s-15
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= π3βπ1
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
3Γ (π. π·. )
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 Ο2and 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 itβs 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 = ππππ π’πππ πππ·ππ ππππ πππ
π ππππ‘ππππππ π’πππ πππΆπππ‘πππππππππππ¦
s-17
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. = π.π·.
π΄.π.Γ 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 π
βπ .
S-18
7.2.Expression for the standard error of mean
Let there be a sample of N observations, X1, X2, X3β¦β¦β¦XN which have been drawn at
random from a population, the variance of which is Ο2.
Now, Mean XΝ = 1
π (π1 + π2 + π3 +β―+ ππ
Variance of mean
V(XΝ)= 1
π2{π(π1 + π2 + π3 +β―+ ππ)}
= 1
π2 [π(π1) + π(π2) + β―+ π(ππ)]
Since, V(X1) =V(X2) = V(X3)=β¦β¦β¦β¦β¦β¦β¦.= V(XN)= Ο2
V(XΝ)= ππ2
π2=
π2
π
S.E. of XΝ = π
βπ
But since, in practice, Ο is not known, it is replaced by its unbiased estimate S.
S.E. of XΝ = π
βπ
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) Β΅Β±Ο (ii) ¡±2Ο and (iii) ¡±3Ο. What percentage of cases lie outside these
limits.
Calculation of Mean and Standard Deviation
X X- XΝ (X- XΝ)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:
Β΅ = βπ
π=
1200
10= 120 and Ο2=
β(πβXΝ)2
π =
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
Β΅Β±Ο = 120Β±6.60
= 113.4 and 126.6
113, 115, 116, 119,
120, 122, 124
70% 30%
¡±2Ο = 120 Β± 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