Weighted Mean

While evaluating the arithmetic means we assume that all the items in the distribution have equal preference. But in daily life this may not be possible. If some items in the distribution are more important than others, then this point must be remembered, so that the average calculated is a proper representative of the distribution. In these cases, proper weightage should be given to various items. Data values with higher weights contribute more to the weighted mean than data values with smaller weighted mean.

Let w, be the weight attached to the item x_i, i = 1 to k. Then we define

Weighted Arithmetic Mean = ∑ w_i x_i / ∑ w_i where i = 1 to k

It may be noted that the formula this mean is same as that of the formula of a mean with f_i, (i = 1 to k), the frequencies substituted by w_i, weights.

Reasons for the usage of weighted mean:

The following are some of the reasons for why people want to use a weighted mean:

• Each data value may actually depict a value that is used by many people in the sample. The weight, then the number of people involved with that value.
• The sample may forcefully over represent or under represent certain parts of the population. To replace balance, less weight might placed on the over represented parts of the population and greater weight on the less represented parts of the population.
• Some of the values in the sample data might be known to be more variable than other values. In such cases, greater weight should be placed on those data values known to have greater precision.

Summary:

Thus, weighted mean is a mean in which every single item is being averaged is multiplied by a number based on the item's relative importance. The result is totaled and divided by the sum of the weights. These averages are used extensively in descriptive analysis such as index numbers.