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Module:5 Measures of Central Tendency: Mathematical Averages (AM, GM, HM)
Paper:15 , Quantitative Techniques for Management Decisions
Module: 20, Hypothesis Testing: Developing null and alternative hypotheses
Principal Investigator
Prof. S P Bansal Vice Chancellor
Maharaja Agrasen University, Baddi
Co-Principal Investigator Prof YoginderVerma
Pro–Vice Chancellor
Central University of Himachal Pradesh. Kangra. H.P.
Paper Coordinator
Content Writer
Prof Pankaj Madan Dean-Management
Gurukul Kangri University,Haridwar
Dr Deependra Sharma
Associate-Professor Amity University Gurgaon.Haryana
Paper:15 , Quantitative Techniques for Management Decisions
Dr. Sanjay Mishra Associate Professor, Department of Business
Administration, MJP Rohilkhand University, Bareilly.
Prof. Pankaj Madan Dean- FMS
Gurukul Kangri Vishwavidyalaya , Haridwar
Prof. YoginderVerma
Pro–Vice Chancellor
Central University of Himachal Pradesh. Kangra. H.P.
Content Writer
Paper Coordinator
Co-Principal Investigator
Prof. S P Bansal Vice Chancellor
Maharaja Agrasen University, Baddi
Principal Investigator
QUADRANT- I
Module – 5
Measures of Central Tendency: Mathematical Averages (AM, GM, HM)
Objectives
After studying this module you would be able to understand the:
Concept of measures of central tendency;
Arithmetic Mean;
Geometric Mean;
Harmonic Mean;
Methods of calculating AM, GM & HM;
Merits, demerits and uses of AM, GM & HM; and
Relation between AM, GM & HM.
Introduction
“The International Monetary Fund (IMF) on Tuesday raised projections for India’s economic
growth by 0.2 percentage points to 7.6 percent for 2016-17 and 2017-18. The projections came in
at a time when the Fund said global economic growth will be subdued this year, following a
slowdown in the US and Britain’s vote to exit the European Union. It, however, retained global
economic growth at 3.1 percent for 2016 and 3.4 percent for 2017.”
Business Standard, New Delhi October 05, 2016.
Statements like these which talk about the growth rates of
nations/states/industries/sectors/areas/etc. are quite common that we read daily in
newspapers/magazines/journals/etc. or hear it on TV channels or discussions among ourselves.
Similarly in our daily lives we often make statements like: the average income of Area “A” is Rs
15,000/- per month; the commerce students study on an average 4 hrs daily after college; average
wages of workers of Factory X are Rs 10,000/- per month; etc.
A careful analysis of these statements reveals that they are talking about some value or figure,
not extreme but some central value, around which most of the observations cluster. This central
value, around which most of the data points cluster, is used as a representative value for the data.
Hence these central values which represent the data are known as Measures of Central Tendency.
When people talk about an average value or the middle value or the most frequent value, they are
talking informally about the some measure of central tendency.
Generally, it is very difficult rather impossible for a human mind to remember the huge and
unwieldy set of numeric values which it comes across in life on daily basis; and even if it
remembers them then also it is not possible to draw some valid conclusion from these
tens/hundreds/thousands/lakhs/etc of figures. Measures of Central Tendency are the statistical
tool which helps in condensing, simplifying and making the data more understandable. Hence
Measures of Central Tendency occupy a place of pre eminence in all statistical analyses.
The measures of central tendency which we discuss here in this module are:
A. Arithmetic mean
B. Geometric mean
C. Harmonic mean
A. Arithmetic Mean The arithmetic mean or average as referred in common parlance is the most common measure of
central tendency. It is obtained by adding all the observations and then dividing the sum by the
number of observations. Depending on the type of data i.e. ungrouped (unclassified) data or
grouped (classified) data, different methods for calculating the arithmetic mean are used.
Arithmetic Mean of Ungrouped Data: There are two methods for calculating
arithmetic mean for ungrouped data.
i) Direct method
ii) Indirect or short cut method
i) Direct method:
Sum of observations
Arithmetic Mean (A.M.) =
Number of observations
If we have X1, X2, X3, ………………………………Xn observations, then
X1+ X2+ X3+………………………+Xn ∑X
A.M. (X) = =
n n
Example1:- Find the arithmetic mean of marks obtained by 10 students in a test.
The marks are as follows:- 61, 81, 87, 78, 54, 56, 67, 65, 68, 69.
Solution
A.M. = (65+81+87+78+54+56+67+65+68+69)/10
= (690)/10
= 69
The average marks are 69.
ii) Indirect or short cut method: In this method an arbitrary assumed mean
is used. Deviations of individual observations from this assumed mean
are taken for calculating arithmetic mean.
Let “A” be the arbitrary assumed mean and “di” the new variable defined
as follows:
di= xi-A, then
A.M. (X) = A+∑ 𝑑𝑖𝑛𝑖=1 /𝑛
Example2:- Find the arithmetic mean of marks obtained by 10 students in a test.
The marks are as follows:- 63, 62, 67, 68, 64, 66, 67, 65, 68, 70.
Solution
S. No. X d=x-A let “A”=60
1 63 63-60=3
2 62 62-60=2
3 67 67-60=7
4 68 68-60=8
5 64 64-60=4
6 66 66-60=6
7 67 67-60=7
8 65 65-60=5
9 68 68-60=8
10 70 70-60=10
∑d=60
A.M. (X) =A+∑ 𝑑𝑖10𝑖=1 /𝑛
= 60+60/10 = 66
Arithmetic Mean of Grouped Data: There are two methods for calculating
arithmetic mean of grouped data.
i) Direct method
ii) Indirect or step-deviation method
i) Direct method: Suppose we have data in form of X1, X2…….…….Xn
observations with corresponding frequencies f1, f2…………………fn. The
arithmetic mean will be
A.M.= 𝐟𝟏𝐗𝟏+𝐟𝟐𝐗𝟐+.…….𝐟𝐧𝐗𝐧
𝐟𝟏+ 𝐟𝟐+...…𝐟𝐧 = ∑ 𝒇𝒙/𝑵𝒏
𝒊=𝟏
Example 3:- Calculate the average number of children per family from the
following data.
No. of children 0 1 2 3 4 5 6
No. of families 30 52 60 65 18 10 5
Solution
No. of children (x)
No. of family (f) f.x
0 30 0x30=0
1 52 1x52=52
2 60 2x60=120
3 65 3x65=195
4 18 4x18=72
5 10 5x10=50
6 5 6x5=30
∑f= 240 ∑f.x= 519
A.M. = ∑ 𝑓𝑥/𝑁𝑛𝑖=1
= 519/240
= 2.1625
ii) Indirect or step-deviation method: Steps we follow in this method
are as follows-
a) First find out the mid points of different classes (X)
b) Then decide about the value of assumed mean. Let it be “A”
c) Calculate the value of dx. If class interval is denoted by ‘h’ and ‘A’ is
assumed mean then dx = (X-A)/h.
d) Multiply these deviations with corresponding frequency and calculate
the value of ∑ fdx.
e) Apply the formula-
A.M. = A+ ( ∑ 𝒇𝒅𝒙/𝑵 )h
Example:-4 The following table shows the daily income distribution of 500
workers. Find the average income.
Income 0-50 50-100 100-150 150-200 200-250 250-300
No. of Workers 90 150 100 80 70 10
Solution
Income Workers
(f)
Mid Value
(x) dx=(X-125)/50 fdx
0-50 90 25 -2 -180
50-100 150 75 -1 -150
100-150 100 125 0 0
150-200 80 175 1 80
200-250 70 225 2 140
250-300 10 275 3 30
∑f=500
∑ fdx=-80
A.M. = A+ ( ∑ 𝒇𝒅𝒙/𝑵 )h
= 125 + (-80) x 50
500
= 117
Thus, average income is Rs. 117.
Merits of Arithmetic Mean:
i) It is easy to understand and calculate
ii) It is based on all observations
iii) It is rigidly defined
iv) It is capable of further mathematical treatment
v) It is least affected by sampling fluctuation.
Demerits of Arithmetic Mean:
i) It is unduly affected by extreme values.
ii) In case of open ended classes it cannot be calculated.
B. Geometric Mean:
When we are interested in measuring average rate of change over time then we use
geometric mean. Geometric mean is defined as the nth root of the product of n items
(or) values.
Calculation of Geometric Mean (G.M.) - Individual series: If nxxxx ,.......,,, 321
be n observations studied on a variable X, then the G.M of the observations is defined
as
G.M.= nnxxxx
1
321 ........
Applying log both sides
nxxxn
MG ...........log1
.log 21
= ]log...........log[log1
21 nxxxn
=
n
i
ixn 1
log1
n
i
ixn
antiMG1
log1
log.
Calculation of G.M. - Discrete series: If nxxxx ,.......,,, 321 be n observations of a
variable X with frequencies nffff ,.......,,, 321 respectively then the G.M is defined as
G.M.= Nf
n
fffxxxx
1
3211321 ........
Where N =
n
i
if1
i.e. total frequency
Applying log both sides in (i) we get
G.M= antilog
n
i
ii xfN 1
log1
Calculation of G.M. -Continuous Series: In continuous series the G.M. is
calculated by replacing the value of ix by the mid points of the class’s i.e. im .
G.M= antilog
n
i
ii mfN 1
log1
Where im is the mid value of the ith class interval.
Merits of Geometric Mean:
1) It is rigidly defined.
2) It is based on all the observations.
3) If G1 and G2 are geometric means of two groups having n1 and n2 observations,
respectively, then the geometric mean G of the combined group of (n1+n2) values is
given by
log G = (n1log G1 + n2 log G2) / (n1 + n2)
Uses of Geometric Mean: Geometrical Mean is especially useful in the following
cases.
1) The G.M is used to find the average percentage increase in sales, production,
or other economic or business series.
For example, from 1992 to 1994 prices increased by 5%,10%,and 18% respectively,
then the average annual income is not 11% which is calculated by A.M but it is 10.9
which is calculated by G.M.
2) G.M is theoretically considered to be best average in the construction of Index
numbers.
C. Harmonic Mean:
The Harmonic Mean (H.M.) is defined as the reciprocal of the arithmetic mean of the
reciprocals of the individual observations.
Calculation of H.M -Individual series: If nxxxx .,,.........,, 321 be ‘n’ observations of a
variable X then harmonic mean is defined as
nxxx
nMH
1..............
11.
21
n
i ix
nMH
1
1.
Calculation of H.M. -Discrete series: If nxxxx .,,.........,, 321 be ‘n’ observations
occuring with frequencies nffff .,,.........,, 321 respectively, then H.M. is defined as
n
n
n
i
i
x
f
x
f
x
f
f
MH
..............
.
2
2
1
1
1
n
i i
i
n
i
i
x
f
f
MH
1
1.
Calculation of H.M – Continuous series: In case of continuous series H.M can be
calculated by taking mid values ( im ) in place of sxi ' . Hence H.M is given by
n
i i
i
n
i
i
m
f
f
MH
1
1. , where im is the mid value of the ith class interval
Example:- 5 A cyclist pedals from his house to his college at a speed of 12 km.p.h
and back from the college to his house at 15 km.p.h Find the average speed.
Solution
Let the distance from the house to the college be x kms. So the total distance
travelled by cyclist in going to college and then coming back to house is 2x kms.
Since the speed of cyclist in going from house to college is 12 km.p.h. therefore
the time taken to cover this distance is x/12 hours. Similarly the time taken to reach
house from college is x / 15 hours. Thus a total distance of 2x kms is covered in
(𝑥
12 +
𝑥
15 )hours.
Speed = Distance/Time
Hence, Average Speed = 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 =
2x
(X
12+
X
15)
= 2
(1
12+
1
15)
=13.33 km.p.h
Merits of Harmonic Mean:
1) Its value is based on all the observations of the data.
2) It is less affected by the extreme values.
3) It is strictly defined.
Demerits of Harmonic Mean:
1) It is not simple to calculate and easy to understand.
2) It cannot be calculated if one of the observations is zero.
3) The H.M is always less than A.M and G.M.
Uses of Harmonic Mean:
The H.M is used to calculate the averages where two units are involved like rates,
speed, etc.
Relation between A.M., G.M. and H.M.
The relation between A.M, G.M, and H.M is given by
MHMGMA ...
Note: The equality condition holds true only if all the items are equal in the
distribution.
Prove that if a and b are two positive numbers then MHMGMA ...
Solution:
Let a and b are two positive numbers then
The Arithmetic mean of a and b =2
ba
The Geometric mean of a and b = ab
The harmonic men of a and b =ba
ab
2
Let us assume MGMA ..
0
4
2
2
2
2
ba
abba
abba
abba
which is always true.
MGMA .. ………………………… (1)
let us assume HMGM
0
4
2
2
2
2
ba
abba
abba
ba
abab
Which is always true.
HMMG . ………………………… (2)
from (1) and (2) we get MHMGMA ...
Summary
The measures of central tendency give us an idea about the central value around which the data
values cluster. That’s why these values are considered to be representative values i.e. the values
which represent the data. Arithmetic mean is the most common measure of central tendency which
is obtained by adding all the observations and then dividing the sum by the number of observations.
Geometric mean is used for measuring the average rate of change over time. It is defined as the nth
root of the product of n items (or) values. Harmonic Mean (H.M.) is defined as the reciprocal of the
arithmetic mean of the reciprocals of the individual observations.
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