6.2.3 Weighted Mean
In calculating arithmetic mean we suppose that all the items in the distribution have equal importance. But in practice this may not be so. If some items in a distribution are more important than others, then this point must be borne in mind, in order that average computed is representative of the distribution. In such cases, proper weightage is to be given to various items – the weights attached to each item being proportional to the importance of the item in the distribution. For example, if we want to have an idea about the change in cost of living of a certain group of people, then the simple mean of the prices of the commodities consumed by them will not do, since all the commodities are not equally important e.g. wheat, rice and pulses are more important than cigarettes, tea, confectionery, etc.
Let wi be the weights attached to the ith item having the value xi, i = 1, 2 …….n. Then, we define.
Weighted arithmetic mean or weighted mean = ∑ i wi xi / ∑ i wi …..…..
It may be observed that the formula for weighted mean is the similar to the formula for simple mean with fi, (i = 1,2……n) the frequencies replaced by wi, (i = 1,2,……n), the weights.
Weighted mean gives the result equal to the simple mean if the weights assigned to each of the variate values are equal. It results in higher value than the simple mean if smaller weights are given to smaller items and larger weights to larger items. If the weights attached to larger items are smaller and those should items attached to highted mean, weighted mean results in smaller value than the simple mean.