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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