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CONTENTS

Introduction Practical Applications of Arithmetic Mean Types of Data

Ungrouped Data Grouped Data

Methods for Calculating Arithmetic Mean

Direct Method Assumed Mean Method Step deviation Method

Weighted Mean Combined or Composite Arithmetic Mean

Properties of Arithmetic Mean

INTRODUCTION

In the present scenario of competitive world, Statistics plays a vital role. Whether it is the field of Business, Economics, Sports, Statistics can readily be applied as per requirement. Hence, it can rightly be said as the driver of Policies & decisions.

Some data that are inconvenient for diagrammatic representation can easily be interpreted with the use of different formulae. Amongst all the formulae, Arithmetic Mean is the most commonly used tool of Statistics because of its simplicity in calculation. It gives us an idea about the average point of the concentration of the data. This lets us to derive a rough idea about the average performance of the variables.

Arithmetic Mean, in simple words, is nothing but the simple concept of Average what we learnt in Primary Level. Hence, Arithmetic Mean of a group of observations is the sum of the values of all the observations divided by the number of observations. Arithmetic Mean is abbreviated as A.M. & is denoted by the symbol

x. Symbolically, in simplest term,

x = x1 + x2 + x3 + ………… + xn

n n

= xi i=1

n

where, x1, x2, ………, xn : Values of the observations n : No. of observations

: Sigma (it represents the sum of observations)

Although A.M. is subject to certain limitations, yet mainly because of its familiarity to most people, it is most popularly used measure of central tendency. It is widely used in Social, Economic & Business problems. It is the Best Measure of Central Tendency. It is called the Ideal Average because :

It is comparatively easier to calculate & understand.

It provides a good basis for comparison.

It can be determined for any value of series.

It is amenable for further Mathematical Treatment.

PRACTICAL APPLICATIONS OF ARITHMETIC MEAN

Arithmetic Mean has a wide application in various fields. Some of its

applications are enumerated as under :

In Sports, it is used in calculating values like Average Score of a batsman.

In Economics, it is applied for performing functions like calculating Per Capita Income, Average Monthly Income of general people.

In Business, it is used to calculate Average Profit per unit of Article.

In Education, it is applied to determine the Average Performance of the Students.

It is used by the common people in general to calculate the Average Monthly Budget of the Family.

Before going in details with the concept of A.M., let us first refresh our knowledge about the types of data, because the procedure of computation of A.M. can be categorized on the basis of types of data.

Following chart explains about the types of data :

TYPES OF DATA

Ungrouped Data

Individual Series

Discrete Series

Grouped Data

Frequency Distribution

Continuous Series

Discontinuous Series

After knowing about the types of data, we can now proceed to the procedure

of computation of Arithmetic Mean (or A.M.). this can be discussed as follows :

UNGROUPED DATA

(i) Individual Series It is the series of data which is in raw form. The values

do not contain any frequency assigned to them. Hence, the calculation of A.M. is quite simple. Only the thing we have to do

is to sum up the values of the observations & then divide it by the number of observations. Symbolically, n

x = xi i=1

n

Example : Compute A.M. from the following data : 10, 15, 25, 70, 35, 30, 15, 75, 5, 80

Soln :

Here,

xi = 10+15+25+70+35+30+15+75+5+80 = 360 No. of observations, n = 10 Therefore,

x = xi n = 360 10

= 36

(ii) Discrete Series

It is the series of data where the values of the observations contain frequencies assigned to them.

Here, for calculation of A.M., we have to first obtain the sum total of the observations multiplied by their respective frequencies & then divide it by the total of the frequencies. Symbolically, n

x = fixi i=1

N

Example : Compute A.M. from the following distribution : Variable : 5 6 7 8 9 Frequency : 4 8 14 11 3 Soln :

xi fi fixi

5 4 20

6 8 48

7 14 98

8 11 88

9 3 27

N = 40 fixi = 281

Therefore,

x = fixi N = 281 40 = 7.025

GROUPED DATA

(i) Frequency Distribution

Sometimes the data are so large that it becomes inconvenient to list each observation individually. In such a case, we group the items into convenient intervals & the data is presented in a frequency distribution.

The mid-value of each class is the representative of each item falling in that interval. Mid-value is also termed as „Class Mark‟. Here, the individual values lose their identity.

Mid-Value (or Class Mark) = Lower Class Limit + Upper Class Limit 2

It is important to note that Frequency Distribution may be of the following

two types : Continuous Series Discontinuous Series

(a) Continuous Series It refers to the data seires where the class intervals are

recorded without a break, i.e., the Upper Class Limit of a class is the Lower Class Limit of the succeeding class.

It is also termed as „Exclusive Series‟ because in this series, the Upper Class Limit is not included as an item of that particular class interval.

Here, for calculation of A.M., we have to first obtain the mid-value of each class (represented as xi), then obtain the sum total of the mid-values multiplied by their respective class frequencies & finally divide it by the total of the frequencies. Symbolically, n

x = fixi i=1

N

(b) Discontinuous Series It refers to the data series where the class intervals

are not continuous, i.e., the Upper Class Limit of a class is not the Lower Class Limit of the succeeding class.

It is also termed as „Inclusive Series‟ because in this series, both the Upper Class Limit & the Lower Class Limit items are included as items of that particular class interval.

Here, for calculation of A.M., we have to first make the class intervals continuous by adding & subtracting a particular value to & from the Upper Class Limit & the Lower Class Limit of the classes respectively. This value is calculated as under :

Lower Class Limit - Upper Class Limit of the preceeding class 2

Thereafter, the calculation procedure is same as that of „Continuous Series‟. Symbolically,

n

x = fixi i=1

N

Example : Compute A.M. from the following distribution : Class Interval : 1-10 11-20 21-30 31-40 41-50 Frequency : 5 10 20 25 15

Soln :

C.I. C.B. xi fi fixi

1-10 0.5-10.5 5.5 5 27.5

11-20 10.5-20.5 15.5 10 155

21-30 20.5-30.5 25.5 20 510

31-40 30.5-40.5 35.5 25 887.5

41-50 40.5-50.5 45.5 15 682.5

N = 75 fixi = 2262.5

Therefore,

x = fixi N = 2262.5 75 = 30.1667 (Approx.)

METHODS FOR CALCULATION OF ARITHMETIC MEAN

In order to make calculation of A.M. simpler & hence, save time & energy, following methods are adopted for calculation purpose :

Direct Method Assumed Mean Method Step Deviation Method

DIRECT METHOD

Direct Method is the first method of calculating A.M. . It is the method which we have used in the problems of our previous discussions.

Here, the simple total of the values of the observations or the values multiplied by

their respective frequencies, as applicable, is divided by the total of the frequencies. This method involves large numbers & hence, takes greater time for calculation.

Symbolically,

n

x = xi (in case of Individual Series) i=1

n

OR,

n

x = fixi (in case of Discrete Series) i=1

N

ASSUMED MEAN METHOD

Assumed Mean Method of calculating A.M. is used to avoid lengthy calculations. In this method, we choose an assumed mean (say A) & subtract it from each of the values of xi. The reduced value, “xi-A”, is called “Deviation of xi from A”. This may symbolically be represented as “di”.

NOTE : The Assumed Mean should be chosen in such a manner that :

It should be one of the smallest Central Values.

The Deviations are small.

One Deviation is “Zero”.

WORKING RULE : Choose Assumed Mean (say A) from the Central Values of xi.

Obtain Deviations “di (= xi-A)”. n

Obtain fidi by multiplying deviations with their respective i=1

frequencies & then summing them up.

Find A.M. by using the formula : n

x = A + fidi i=1

N

Example :

Compute A.M. from the following distribution : Class Interval : 50-60 60-70 70-80 80-90 90-100 Frequency : 8 6 12 11 13

Soln : Let the assumed Mean be A = 75

C.I. xi fi di fidi

50-60 55 8 -20 -160

60-70 65 6 -10 -60

70-80 75 (=A) 12 0 0

80-90 85 11 10 110

90-100 95 13 20 260

N = 50 fidi = 150

Therefore,

x = A + fidi N = 75 + 150

50 = 75 + 3 = 78

STEP DEVIATION METHOD Step Deviation Method of calculating A.M. is an extension of Assumed Mean

Method. Assumed Mean Method can further be simplified by dividing the deviations by width of the class interval (represented by h). The result so

obtained is represented as “ di ” (i.e., di = xi – A). h

WORKING RULE : Choose Assumed Mean (say A) from the Central Values of xi.

Obtain Deviations “ di ( = xi-A ) ”. h n

Obtain fidi by multiplying di with their respective i=1

frequencies & then summing them up.

Find A.M. by using the formula : n

x = A + fidi i=1 X h

N

Example :

Compute A.M. from the following distribution : Class Interval : 1400-1500 1500-1600 1600-1700 1700-1800 1800-1900 Frequency : 5 10 20 10 5

Soln : Let the assumed Mean be A = 1650

Therefore,

x = A + fidi x h N = 1650 + 20 x 100

60 = 1650 + 33.33 = 1683.33

C.I. xi fi di di fidi

1400-1500 1450 5 -200 -2 -10

1500-1600 1550 10 -100 -1 -10

1600-1700 1650 (=A) 20 0 0 0

1700-1800 1750 10 100 1 10

1800-1900 1850 15 200 2 30

N = 60 fidi = 20

NOTE :

Following points should be kept in mind :

We have so far discussed computation of A.M. when the class width of each class are equal. In case of unequal class width for different classes, the same procedure may be followed.

In case of open end class intervals, (i.e., Lower Class Limit of the first class & Upper Class Limit of the last class are not mentioned), it is reasonable to assume the missing values by observing the trend (i.e., magnitude of the class widths for the other classes) of the class intervals for rest of the classes.

WEIGHTED MEAN

We have so far calculated mean by assuming that all items in a distribution have equal importance. But it may so happen in certain cases that the importance of different items may not be the same.

In such a case, we assign weights to different items. The weight assigned to an item is proportional to the importance of the item.

The mean calculated in such a situation is termed as „Weighted Mean‟,

denoted by xw. Symbolically,

n

x = wixi i=1 n

wi

i =1 where, n

wixi : Sum total of items multiplied by their respective weights i=1

n

wi : Total of weights i=1

Example : Calculate Weighted Mean from the following data : Subjects : English Hindi Economics Mathematics Weights : 4 1 3 2 Marks Scored : 70 50 90 60 Soln :

Subjects wi xi wixi

English 4 70 280

Hindi 1 50 50

Economics 3 90 270

Mathematics 2 60 120

wi = 10 wixi = 720

Therefore,

wixi

xw =

wi

= 720 10 = 72

COMBINED OR COMPOSITE ARITHMETIC MEAN

If there are n1 values in a series whose A.M. is x1 & there is another series

with n2 values whose A.M. isx2, then the A.M., x, of the combined distribution is given by :

x = n1x1 + n2x2 n1 + n2

PROOF :

A.M. of series with n1 values = x1

sum of values of the series = n1x1

Similarly,

Sum of values of the series with n2 values whose A.M. is x2 = n2x2

Also, Total number of values of the combined series = n1 + n2

Hence,

A.M. of the combined series, x = n1x1 + n2x2

n1 + n2

NOTE : If three series are combined together, then their combined A.M. will be :

x = n1x1 + n2x2 +n3x3 n1 + n2 +n3

PROPERTIES OF ARITHMETIC MEAN

The Algebric Sum of the deviations of values of a variable from it‟s A.M. is Zero, i.e.,

n

(xi -x ) = 0 i=1

& n

fi (xi -x ) = 0 i=1

If the A.M. of „n‟ observations is „x „, then the mean of observations when a constant „a‟ is added, subtracted, divided & multiplied to it, the new A.M. is given by:

x + a,

x - a,

x and a

ax

respectively

The Sum of Squares of Deviations of a variable is the least if the deviations are taken from the A.M.