Index number is a statistical measure of average change in a variable or a group of variables with respect to time or space. The variable may be the enrolment of staff in an organization, the cost of salaries for staff, prices of a particular commodity or a group of commodities, wages of workers, volume of trade, sales, exports and imports, production, etc.

If your data science use case requires calculation of index number, following are the three tests for index number formula. Since From a theoretical viewpoint, a good index number formula is required to satisfy these tests, it would allow a data scientist or statistician to evaluate the strength of index number formula.

## 1. Time Reversal Test:

If the time subscripts 0 and n are interchanged in a price ( or quantity) index number formula wherever they appear, then the resulting price (or quantity) formula should be the reciprocal of the original index formula, ignoring the factor 100”

Since the index number is designed to measure changing values of prices or production, it is natural to expect that the formula should give the same result regardless of which the two periods is chosen as the base. The base period and the given period are only relative terms and hence should be reversed. An index number satisfying this test gives consistent results.

## 2. Factor Reversal Test:

If the factors p’s (prices) and q’s (quantities) occurring in a price (or quantity) index formula be interchanged (or reversed) so that a quantity (or price) index formula is obtained, then the product of the two index numbers should equal the value index number.” In other words, the factor reversal test requires that:

(Price index) (Quantity index) = Value index

## 3. Circular Test:

The circular test is an extension of the time reversal test, which is applicable to two years , i.e. the given year and the base year. But in circular test, three or more years are taken into account. This test is not obeyed by any of the weighted index numbers unless the weights are constant. For the purposes of illustration, we consider the weighted aggregate price number with fixed weights.