6 Measures of Central Tendency: Mathematical Averages (AM, GM, HM).

sanjay mishra

 

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:

 

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:

 

 

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

 

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

 

 

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

 

No. of children (x) No. of family (f) f.x
0 30 0x30=0
1 52 1×52=52
2 60 2×60=120
3 65 3×65=195
4 18 4×18=72
5 10 5×10=50
6 5 6×5=30
∑f= 240 ∑f.x= 519

A.M. = ∑  =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+ ( ∑  /   )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[1] 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+ ( ∑   /   )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:

Ifx1,x2,x3,…….,xn be n observations studied on a variable X, then the G.M of the observations is defined as

 

Applying log both sides

 

Calculation of G.M. – Discrete series:

 

If  x1 , x2 , x3 ,…….,xn be n observations of a variable X with frequencies  f1 , f 2 , f 3 ,…….,f n respectively then

the G.M is defined as

 

n

Where N =       f i  i.e. total frequency

i 1

Applying log both sides in (i) we get

 


Calculation  of  G.M.  -Continuous  Series:  In  continuous  series  the  G.M.  is calculated by replacing the value of xi by the mid points of the class’s i.e. mi

 

Where mi  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 x1 variable X then harmonic mean is defined as be ‘n’ observations of a variable X then harmonic means defined as 

 

Calculation  of  H.M.  -Discrete   series:     If X1,X2,X3—————Xn be an observations occuring with frequencies f1,f2,————-fn respectively then H.M defined as

 

Calculation of H.M – Continuous series: In case of continuous series H.M can be calculated by taking mid values ( mi ) in place of xi ‘ s . Hence H.M is given by 

 

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 (12 +15 )hours.

 

Speed = Distance/Time

 

 

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

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

 

Solution:

 

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.

 

Learn More:

  1. Business Research Methods’ Authored by Naval Bajpai, Published by Pearson’s India PHI
  2. Business Statistics Authored by Dr. K.L. Gupta, Published by Nirupam Publications.
  3. Business Statistics Authored by G.C. Beri, Published by TMH Publications.
  4. .Statistics For Managers using Microsoft Excel by David M. Levine David F. Stephan Timothy C. Krehbiel Mark L. Berenson, Published by PEARSON
  5. Business Statistics by Ken Black, Published by John Wiley & Sons, Inc.
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