**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

**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

x̄₁ and x̄₂ are 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 + ........................

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.

**Problem 1 : **

For the following data, compute AM.

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

**Solution : **

For the given data, the formula for AM is 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 AM is given by

**x̄ = ∑fx / N**

Here,

∑fx = 2211

N = 36

Then, we have

x̄ = 2211 / 36

**x̄ = 61.42**

**Problem 3 : **

Find the AM for the following distribution.

**Solution :**

**Computation of AM**

For the given data, the formula for AM 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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