Question

The sum of deviations of observations around its mean is always zero. Explain this in detail with example?

Answer 1

The sum of the deviations of a given set of observations from their arithmetic mean or around mits mean is always zero because of the property that the 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.Hence there sun will be zero .

Thus sum of deviations of observations around its mean is always zero

Let be mean of given observation , then = 0

Proof -

= =           ....... (i)

Now Mean =    = n *                   ....... (ii)

Form (i) and (ii) we get

= = n *     = 0

= 0

Example

Suppose    :   x = ( 5 ,3, 3, 2 ,5 ,6, 4 )

Thus Mean = = (5+ 3 +3+2 +5+ 6 +4)/ 7 = 28 /7 = 4

hence    =   4

Thus        

x	x - = x -4
5	1
3	-1
3	-1
2	-2
5	1
6	2
4	0

Thus

= ( 1 - 1 - 1 -2 +1 +2 +0 ) = 0

Thus , = 0