This is the most commonly used average which you have also studied and used in lower grades. Here are two definitions given by two great masters of statistics.

Horace Sacrist : Arithmetic mean is the amount secured by dividing the sum of values of the items in a series by their number.

W.I. King : The arithmetic average may be defined as the sum of aggregate of a series of items divided by their number.

Thus, the students should add all observations (values of all items) together and divide this sum by the number of observations (or items).

Suppose, we have 'n' observations (or measures) x₁ , x₂ , x₃, ......., x_n then the Arithmetic mean is obviously

We shall use the symbol x (pronounced as x bar) to denote the Arithmetic mean. Since we have to write the sum of observations very frequently, we use the usual symbol ' S ' (pronounced as sigma) to denote the sum. The symbol x_i will be used to denote, in general the 'i' th observation. Then the sum, x₁ + x₂ + x₃ + .......+ x_n will be represented by or simply

Therefore the Arithmetic mean of the set x₁ + x₂ + x₃ + .......+ x_n is given by,

This method is known as the ''Direct Method".

Example A variable takes the values as given below. Calculate the arithmetic mean of 110, 117, 129, 195, 95, 100, 100, 175, 250 and 750.

Solution: Arithmetic mean =

= 110 + 117 + 129 +195 + 95 +100 +100 +175 +250 + 750 = 2021

and n = 10

Indirect Method (Assumed Mean Method)

A = Assumed Mean =

Calculations:

Let A = 175 then

    Su_i = -65, -58, -46, +20, -80, -75,-75, +0, + 75, +575

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Index

4.1 Introduction
4.2 Arithmetic Mean
4.3 Properties of Arithmetic Mean
4.4 Median
4.5 Mode
4.6 Empirical relation between mean, median & mode

Chapter 5