Question

If the arithmetic mean of n numbers of observation x1, x2,?.xn is M and A is any arbitrary number, then show that the sum of squared deviation of the observation from their mean is minimum.

Answer 1

Answer:

The sum of the deviations of a given set of observations from their arithmetic mean is always zero. It is due to the property that the arithmetic mean is characterised as the centre of gravity. i.e. sum of positive deviation from the mean is equal to the sum of negative deviations.

For example:

3,4,6,8,14

x

=

5

3+4+6+8+14

=7

x

i

x

i

−

x

3 −4

4 −3

6 −1

8 1

14 7

∑(x

i

−

x

)=−8+8=0

Hence, the sum of the deviations about mean is 0.