Question

The algebraic sum of the deviations of a frequency distribution from its mean is 

A. always positive 

B. always negative 

C. 0 

D. a non-zero number

Answer 1

Suppose x₁, x_2, … , x_n are n observations with mean as x.

By definition of mean, [i.e. The mean or average of observations, is the sum of the values of all the observations divided by the total number of observations]

We have,

\(X=\frac{X_1X_2\,+.....+X_n}{n}\)

nx = x₁ + x₂ + … + x_n …[1] 

So, 

in this case we have assumed mean(a) is equal to mean of the observations(x) 

And we know that 

d_i = x_i - a 

where, 

d_i is deviation of a (i.e. assumed mean) from each of x_i i.e. observations. 

So, 

In the above case we have 

d₁ = x₁ - x 

d₂ = x₂ - x

.

.

.

d_n = x_n - x

and sum of deviations

d₁ + d₂ + … + d_n = x₁ - x + x₂ - x + … + x_n - x

= x₁ + x₂ + … + xn - (x + x + … {upto n times})

= nx - nx [Using 1]

= 0

Hence, 

sum of deviations is zero.