Suppose x₁, x₂, … , 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,

and

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 + … + x_n - (x + x + … {upto n times})

= nx - nx [Using 1]

= 0

Hence, sum of deviations is zero.