# Arithmetic Mean

Suppose the monthly income (in Rs) of six families is given as:
1600, 1500, 1400, 1525, 1625, 1630.

The mean family income is obtained by adding up the incomes and dividing by the number of families.
=
= Rs 1,547

It implies that on an average, a family earns Rs 1,547. Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by x . In general, if there are N observations as X1, X2, X3, ..., XN, then the Arithmetic Mean is given by

=

Where, = sum of all observations and N = total number of observations.

How Arithmetic Mean is calculated?
The calculation of arithmetic mean can be studied under two broad categories:
1. Arithmetic Mean for Ungrouped Data.
2. Arithmetic Mean for Grouped Data.

# Arithmetic Mean for Series of Ungrouped Data

Direct Method
Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations.

Example 1
Calculate Arithmetic Mean from the data showing marks of students in a class in an economics test: 40, 50, 55, 78, 58.
=
= = 56.2

The average marks of students in the economics test are 56.2.

# Assumed Mean Method

• If the number of observations in the data is more and/or figures are large, it is difficult to compute arithmetic mean by direct method.
• The computation can be made easier by using assumed mean method. In order to save time of calculation of mean from a data set containing a large number of observations as well as large numerical figures, you can use assumed mean method.
• Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observation.
• You can, then, take the summation of these deviations and divide it by the number of observations in the data.
• The actual arithmetic mean is estimated by taking the sum of the assumed mean and the ratio of sum of deviations to number of observations.
• Symbolically, Let, A = assumed mean X = individual observations N = total numbers of observations d = deviation of assumed mean from individual observation, i.e. d = X - A

Then sum of all deviations is taken as
Then find
Then add A and to get
Therefore, = A +
You should remember that any value, whether existing in the data or not, can be taken as assumed mean. However, in order to simplify the calculation, centrally located value in the data can be selected as assumed mean.

Example 2
The following data shows the weekly income of 10 families.

 Family A B C D E F G H I J Weekly Income(Rs) 850 700 100 750 5000 80 420 2500 400 360

Compute mean family income.

TABLE 5.1
Computation of Arithmetic Mean by Assumed Mean Method

Arithmetic Mean using assumed mean method
= A + = 850 + (2.660)/10
= Rs 1.116
Thus, the average weekly income of a family by both methods is Rs 1,116. You can check this by using the direct method.

# Step Deviation Method

The calculations can be further simplified by dividing all the deviations taken from assumed mean by the common factor 'c'. The objective is to avoid large numerical figures, i.e., if
d = X - A is very large, then find d'.

This can be done as follows:

The formula is given below:
= A +

Where d' = (X - A)/c, c = common factor, N = number of observations, A= Assumed mean.
Thus, you can calculate the arithmetic mean in the example 2, by the step deviation method, X = 850 + (266)/10 Ã— 10 = Rs 1,116.

# Calculation of arithmetic mean for Grouped data Discrete Series

Direct Method
In case of discrete series, frequency against each of the observations is multiplied by the value of the observation. The values, so obtained, are summed up and divided by the total number of frequencies.

Symbolically,

Where, = sum of product of variables and frequencies.
= sum of frequencies.

Example 3
Calculate mean farm size of cultivating households in a village for the following data.

 Farm size (in acres) 64 63 62 61 60 59 No of cultivating house holds 8 18 12 9 7 6

TABLE 5.2
Computation of Arithmetic Mean by Direct Method

Arithmetic mean using direct method,
= = 61.88 acres
Therefore, the mean farm size in a village is 61.88 acres.

# Assumed Mean Method

As in case of individual series the calculations can be simplified by using assumed mean method, as described earlier, with a simple modification. Since frequency (f) of each item is given here, we multiply each deviation (d) by the frequency to get fd. Then we get S fd. The next step is to get the total of all frequencies i.e. S f. Then find out S fd/S f. Finally the arithmetic mean is calculated by
using assumed mean method.

# Step Deviation Method

In this case the deviations are divided by the common factor 'c' which simplifies the calculation. Here we estimate d' = in order to reduce the size of numerical figures for easier calculation. Then get fd' and S fd'. Finally the formula for step deviation method is given as,

Activity
• Find the mean farm size for the data given in example 3, by using step deviation and assumed mean methods.

# Continuous Series

• Here, class intervals are given. The process of calculating arithmetic mean in case of continuous series is same as that of a discrete series.
• The only difference is that the mid-points of various class intervals are taken. You should note that class intervals may be exclusive or inclusive or of unequal size.
• Example of exclusive class interval is, say, 0-10, 10-20 and so on. Example of inclusive class interval is, say, 0-9, 10-19 and so on.
• Example of unequal class interval is, say, 0-20, 20-50 and so on. In all these cases, calculation of arithmetic mean is done in a similar way.
â€‹Example 4
Calculate average marks of the following students using (a) Direct method (b) Step deviation method.

Direct Method
Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70
No. of Students 5 12 15 25 8 3 2

TABLE 5.3
Computation of Average Marks for Exclusive Class Interval by Direct Method

Steps:
1. Obtain mid values for each class denoted by m.
2. Obtain and apply the direct method formula:
= = 30.14 marks
Step deviation method
1. Obtain d' =
2. Take A = 35, (any arbitrary figure), c = common factor.
=
= 30.14 marks
An interesting property of A.M.
It is interesting to know and useful for checking your calculation that the sum of deviations of items about arithmetic mean is always equal to zero. Symbolically, S( X - X) = 0. However, arithmetic mean is affected by extreme values. Any large value, on either end, can push it up or down.

# Weighted Arithmetic Mean

Sometimes it is important to assign weights to various items according to their importance, when you calculate the arithmetic mean. For example, there are two commodities, mangoes and potatoes. You are interested in finding the average price of mangoes (p1) and potatoes (p2).

The arithmetic mean will be. However, you might want to give more importance to the rise in price of potatoes (p2). To do this, you may use as 'weights' the quantity of mangoes (q1) and the quantity of potatoes (q2). Now the arithmetic mean weighted by the quantities would be.

In general the weighted arithmetic mean is given by,

When the prices rise, you may be interested in the rise in the price of the commodities that are more important to you. You will read more about it in the discussion of Index Numbers in Chapter 8.

Activities
• Check this property of the arithmetic mean for the following example:
X: 4 6 8 10 12
• In the above example if mean is increased by 2, then what happens to the individual Observations, if all are equally affected.
• If first three items increase by 2, then what should be the values of the last two items, so that mean remains the same.
• Replace the value 12 by 96. What happens to the arithmetic mean. Comment.