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Weighted Index Numbers

The so-called unweighted index numbers discussed above are not unweighted in the true sense of term. They assign equal importance to all the items included in the index and as such they are in reality weighted, weighted being implicit rather than explicit. As discussed earlier, in case of unweighted indices it is possible to get different results by changing the importance of different items by quoting prices relative to different units. Implicit weighting (or the unweighted index) is far from realistic in most of the cases.

Weighted index numbers are of two types;

I. Weighted Aggregative Index Numbers.

These indices are of the simple aggregative type with the fundamental difference that weights are assigned to the various items included in the index. There are various methods of assigning weights and consequently a large number of formulae for constructing index numbers have been devised of which some of the more important ones are:

1.    Lapseyres method,

2.    Paasche method,

3.    Dorbish and Bowley’s method,

4.    Fisher’s ideal method,

5.    Marshall-Edgeworth method, and

6.    Kelly’s method.

II. Weighted Average of Relatives

In the weighted aggregative methods discussed above price relatives were not computed. However, like unweighted relative method it is also possible to compute weighted average of relatives. The steps in the computation of the weighted arithmetic mean of relatives index number are as follows:

(i)    Express each item of the period for which the index number is being calculated as percentage of the same item in the base period.

(ii)    Multiply the percentage as obtained in step (i) for each item by the weight which has been assigned to that item.

(iii)    Add the results obtained from the several multiplications carried out in step (ii).

(iv)    Divide the sum obtained in step (iii) by the sum of the weights used. The results is the index number, Symbolically,
Weighted average relatives
V = Value Weights, i.e., p0q0

Instead of using arithmetic mean the geometric mean may be used for averaging relatives. The weighted geometric mean of relatives is computed in the same manner as the unweighted geometric mean of relatives index number except that weights are introduced by applying them to the logarithms of the relatives. When this method is used the formula for computing the index is:
Weighted average relatives1

and     V = value weight, i.e., p0q0 for each item.

(i) Obtain percentage relatives or each item.

(ii) Find the logarithm of each percentage relative found in step (i)

 (iii) Multiply the logarithms by the weights assigned.

(iv) Add the results obtained in step (iii).

(v) Divide the total obtained in step (iv) by the sum of the weights.

(vi) Find the antilogarithm of the quotient obtained in step (v).

This is weighted geometric mean of relatives index number.

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