Here is a simple example of weighting adjustment with one auxiliary variable. Suppose on online survey has been carried out. Among the variables measured is the age of respondents. Because the population distribution is age is available, we can compare the response distribution of age with the population distribution.
The response consists for 60% of young persons, for 30% of middle-age persons and for 10% of elderly. These percentages are different in the population. For example, the population consists for 30% of young people. Clearly, the young are over-represented in the response. You can conclude the response is not representative with respect to age.
We can make the response representative with respect to age by assigning to the young a weight equal to
30.0 / 60.0= 0.500.
This weight is obtained by dividing the population percentage by the corresponding response percentage. The weight for middle-age persons becomes
40.0 / 30.0 = 1.333.
The weight for the elderly becomes
30.0 / 10.0 = 3.000.
The weight assigned to young people is smaller than 1. This is not surprising as they are over-represented in the survey. After weighting each young person does not count for 1 person any more but just for 0.5 person.
The elderly are under-represented in the survey. Therefore their weight is larger than 1. After weighting, each elderly persons counts for 3 persons.
Suppose, you use the weighted response to estimate the percentage of young people. The weighted percentage is equal to
0.500 x 60% = 30%
This is exactly equal to the percentage of young people in the population. Also the percentages for the other age categories will be estimated exactly. So, the weighted response is representative with respect to age.