Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 7.17

× = Σ × Σ ×Σ × Σ = =

Σ Σ Σ Σ

01 10 1 0 1 1 0 1 0 0

0 0 0 1 1 1 1 0

p p p q p q pq pqpq pq p q p q 1 1

Hence the test is satisfied.

3. Factor Reversal Test - An Index number formula satiesfies this test if the product of the Price Index and
   the Quantity Index gives the True value ratio, omitting the factor 100 from each index. This test is
   satisfied if the change in the price multiplied by the change in quantity is equal to the change in the
   value.
   Speaking precisely if p and q factors in a price (or quantity) index formula be interchanged, so that a
   quantity (or price) index formula is obtained the product of the two indices should give the true value
   ratio.
   Symbolically,

01 01 1 1

0 0

P ×Q = Σp q

Σp q = The True Value Ratio = TVR

Consider the Laspeyres formula of price index

= Σ

Σ

01 1 0

0 0

P p qpq

Consider the quantity index by interchange p with q & q with p

= Σ

Σ

01 1 0

0 0

Q q pq p

× = Σ × Σ ≠ Σ

Σ Σ Σ

01 01 1 0 1 0 1 1

0 0 0 0 0 0

Now P Q p q q p p qpq q p pq

This test is not met.

This test is only met by Fisher’s ideal index. No other index number satisfies this test:

Proof:

= Σ × Σ

Σ Σ

01 1 0 1 1

0 0 0 1

P p q p qpq pq

Changing p to q and q to p, we get

= Σ × Σ

Σ Σ

01 1 0 1 1

0 0 0 1

Q q p q pq p q p

× = Σ × × ΣΣ × Σ

Σ Σ Σ Σ

01 01 1 0 1 1 1 0 1 1

0 0 0 1 0 1 0 1

P Q p q p q q p q ppq pq q p q p

( )

( )

2

1 1 2 1 1

0 0 0 0

q p q p

q p q p

= Σ = Σ

Σ Σ = True Value Ratio

4. Circular Test - Circular test is an extension of time reversal test for more than two periods & is based on
   shiftability of the base period. For example, if an index is constructed for the year 2012 with the base of