ARITHMETIC MEAN

Arithmetic mean (AM) is one of the measures of central tendency which can be defined as the sum of all observations divided by the number of observations. 

Let the  variable 'x' assume 'n' values  as given below

Then, the AM of 'x' to be denoted by x̄ and it is given by

In case of a simple frequency distribution relating to an attribute, we have 

In the above frequency distribution, if the original value 'x' is changed to 'd', due to the shift of origin (assumed mean) by A units  and  scale (class length ) by C units, then the formula for AM is given by

Properties of Arithmetic Mean

Property 1 :

If all the observations assumed by a variable are constants, say 'k', then arithmetic mean is also 'k'. 

For example, if the height of every student in a group of 10 students is 170 cm, the mean height is, of course 170 cm. 

Property 2 :

The algebraic sum of deviations of a set of observations from their arithmetic mean is zero. 

That is, 

for unclassified data, ∑(x - x̄)  =  0.

And for a grouped frequency distribution, ∑f(x - x̄)  =  0.

For example,  if a variable 'x' assumes five observations, say 10, 20, 30, 40, 50, then x̄  =  30.

The deviations of the observations from arithmetic mean (x - x̄) are -20, -10, 0, 10, 20. 

Now, ∑(x - x̄)  =  (-20) + (-10) + 0 + 10 + 20  =  0

Property 3 :

Arithmetic mean is affected due to a change of origin and/or scale which implies that if the original variable 'x' is changed to another variable 'y' effecting a change of origin, say 'a' and scale, say 'b', of 'x'. That is  y = a + bx.

Then we have,

Arithmetic mean of 'y'  =  a + bx̄

For example, if it is known that two variables x and y are related by 2x + 3y + 7  =  0 and x̄  =  15, then  

Arithmetic mean of 'y'  =  (-7 - 2x̄) / 3

Plug  x̄  =  15

Arithmetic mean of 'y'  =  (-7 - 2x15) / 3

Arithmetic mean of 'y'  =  (-7 - 30) / 3

Arithmetic mean of 'y'  =  -37/ 3

Arithmetic mean of 'y'  =  -12.33

Property 4 : 

If there are two groups containing n₁ and n₂ observations with x̄₁ and x̄₂ as the respective arithmetic means, then the combined arithmetic mean is given by

x̄  =  (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

This property could be extended to more than two groups and we may write it as 

x̄  =  ∑nx̄ / ∑n

Here,

∑nx̄  =  n₁x̄₁ + n₂x̄₂ + ..............

∑n  =  n₁ + n₂ + ........................

Some Other Properties of Arithmetic Mean

1)  It is rigidly defined.

2)  It is based on all the observations.

3)  It is easy to comprehend.

4)  It is simple to calculate.

5) It is least affected by the presence of extreme observations. 

6) It is amenable to mathematical treatment or properties. 

The above properties make arithmetic mean as the best measure of central tendency. 

However, arithmetic mean has some draw backs.

They are,

1)  It is very much affected by sampling fluctuation. 

2) Arithmetic mean can not be advocated to open en classification.

For open end classification, the most appropriate measure of central tendency is median. 

Practice Questions

Problem 1 : 

For the following data, compute arithmetic mean.

58, 62, 48, 53, 70, 52, 60, 84, 75

Solution : 

For the given data, the formula for arithmetic mean given by

x̄  =  ∑x / n

Here,

x  =  58, 62, 48, 53, 70, 52, 60, 84, 75

n  =  9

x̄  =  (58 + 62 + 48 + 53 + 70 + 52 + 60 + 84 + 75)  / 9

x̄  =  562 / 9

x̄  =  62.44

Problem 2 : 

Compute the mean weight of a group of students of an university from the following data. 

Solution : 

Computation of mean weight of 36 students.

For the given data, the formula for arithmetic mean is given by

x̄  =  ∑fx / N

Here,

∑fx  =  2211

N  =  36

Then, we have

x̄  =  2211 / 36

x̄  =  61.42

Problem 3 : 

Find the arithmetic mean for the following distribution.

Solution :

Computation of arithmetic mean :

For the given data, the formula for arithmetic mean is given by

x̄  =  A + ( ∑fd / N ) x C

Here,

A  =  419.50

∑fd  =  - 43

N  =  308

C  =  20

Then, we have

x̄  =  419.50 + [ -43 / 308 ] x 20

x̄  =  419.50 - 2.79

x̄  =  416.71

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