What are the various Tests of Adequacy of Index Number Formula?

List of various Tests of Adequacy of Index Number Formula Are:

  • Unit Test
  • Time Reversal Test
  • Factor Reversal Test
  • Circular Test

Unit Test: 

This test states that the formula for constructing an index number should be independent of the units in which prices and quantities are expressed. All methods, except simple aggregative method, satisfy this test.

Time Reversal Test:

This test guides whether the method works both ways in time forward and backward. To quote Fisher, Time Reversal Test is the test which gives the same ratio between one point of comparison and the other, for calculation of index number, no matter which of the two is taken as base. In other words, when the index numbers of the two years are constructed by reversing the base year, they should be reciprocals of each other so that their product is unity.

Symbolically the test is represented as:

P01 X P10 = 1

where P01 is the index for time “1” on time “0” as base and P10 is the index for time “0” on time “1” as base. If the product is not unity, the method suffers from time bias.

Time reversal test is satisfied by

  1. Simple aggregative method
  2. Fisher’s method
  3. Marshall-Edgeworth’s method and
  4. Kelly’s method.

Factor Reversal Test: 

Fisher has described this test in the following words: Just as each formula should permit the interchange of the two items without giving inconsistent results, so it ought to permit interchanging the prices and quantities without giving inconsistent result, i.e., the two results multiplied together should give the true value ratio.”

In simple words, the test means that the change in the price multiplied by the change in the quantity should be equal to the total change in the value. The total value of a given commodity in a given year is the product of the quantity and the price per unit (total value = p x q) . The ratio of the total value in one year to the total value in the preceding

year is p1q1 / p0q0. Symbolically the test is represented as

P01 x Q01 = Σp1q1 / Σp0qV01

Where P01 denotes price index and Q01 , quantity index number.

This test is satisfied by Fisher’s method only.

Circular Test: 

According to this, if indices are constructed for year one based on year zero, for year two based on year one and for year zero based on year two, the product of all the indices should be equal to 1.


P01 X P12 X P20 = 1

This test is satisfied by

  1. Simple aggregative method and
  2. Kelly’s method.
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