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Fisher has discovered weighted index number in which he try to calculate best index number because he has taken both current and past years quantity as the base of price index number.

Why it is ideal index number

1. This is derived by using the geometric mean which is considered to be the best in the construction of index number

2. It depends upon the price and quantities used in the base period as well as the current period. The upward  bias existing in laspeyre's method and downward bias existing in paasche's method gets nullified and fisher's index become free from any bias.

3 . The main quality of fisher index number is that it qualify both time reversal test and factor reversal test.

Time reversal test means

According to this test the formula should be such which works both ways both forward and backward. If for example we prepare the index number for the year 2021 with the base of 2015. Time reversal test states that the product of index number for 2021 with base of 2015 and that of index number for 2015 with base 2021 should be equal to unity( without using 100 each time ). Thus for example if the index number for the year 2021 with base 2015 is 6 ( i.e. 600 if it is multiplied by 100 ) then the formula will be said to be consistence with regard to time if the index number for 2015 with the base as 2021 is 1/6

As per fisher

The test is that formula for calculating an index number should be such that it will give the same ratio between one point of comparision and the other, no matter which of the two is taken as the base

symbolically the time reversal is said to be satisfied if

P01 X q10 = 1

P01 X P10 = 1 , It means if we changes time of fisher formula current period into base year and base year in to current year and when we multiply both index number it must equal to 1.

P01 X q10 =

Σ P1q0 /  Σ P0q0 X  Σ P1q1 /  Σ P0q1 X √  Σ P0q1 /  Σ P0q0 X  Σ P0q0/  Σ P1q0 = 1

Factor reversal test

According to this method changes in the prices multiplied by the changes in quantities should be equal to the total change in the value. This implies that if the commodity has reduced to half in the current period as compared to based period during this period quantity has reduced to 1/3 , then according to this test the total change in the value will be 1/6 of the value of the base period with regard to the value in current period.

P01 X q01 = Σ P1q1 /  Σ P0q0

As per fisher

Just as each formula should permit interchange of two times without giving inconsistent results. So it ought to permit interchanging the prices and quantity without giving inconsitent result i.e.  the two results multiplied together should give the true value ration except for a constant of proportionality

simbolically the test is applicable if

apply same test on fisher

P01 X q 01 =

√ Σ P1q0 /  Σ P0q0 X  Σ P1q1 /  Σ P0q1 X √  Σ P0q1 /  Σ P0q0 X  Σ P1q1Σ P1q0

Σ P1q1 /  Σ P0q0

= V10

If we changes price into quantity and quantity into price.Then we calculate index and then we multiply , it must equal to value Index. Fisher pass this test

So above reason fisher index is best.

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