What makes disproportionate stratified sampling okay to use despite the fact that it is not representative of the population? I know what disproportionate stratified sampling is and how it is used for small subgroups in order to get a large enough sample size for inference and estimates, but what makes it okay to use despite the fact that it is not representative. Does it have to do with the research question being about the groups rather than population? Or something to do with weighting being a corrective measure? Or both? 
 A: You're right, the weighting of a survey does act as a corrective measure in stratified sampling. In fact the weights do this in almost any sampling scheme which is more complicated than simple random sampling.
When a survey is designed, each unit in the sample is assigned a design weight. The design weight is almost always some number greater than 1. This is because it can be thought of as the number of units in the population the surveyed unit represents. So if a person in a survey is given a design weight of 10, this means their response represents themself and 9 others in the population. Likewise a person with a weight of 2 represents themself and 1 other person.
If you stratify and select from different strata with different sampling proportions, then this is reflected in the design weights. That is, as you increase the sampling proportion of one stratum, each unit in this stratum represents less and less of the overall population. This ensures that your sample remains "representative" in some sense and your overall estimate for the entire population remains unbiased.
You raise another point about choosing to stratify because research questions might be about groups rather than the whole population. This is true, but there are other reasons why you might want to stratify. I can think of three other main reasons:


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*There may be some known outliers in the population that don't represent very many other units. For example a small business is far more likely to be representative of 1000 other businesses than a large multinational is. If you put the large multinational in a stratum where each unit represents 1000 others, and you happen to select it, your estimate is going to be pushed way up. It's better to put these outlier units in a separate stratum and increase the sampling proportion for them so they represent less of the population each.

*It might cost a different amount to survey different subpopulations, and you want to control your costs. For example, if you have people door-knocking to collect responses, it'll cost far more for them to travel around farms than in the city. If you don't stratify by country/city then you might happen to select a lot of farms in your sample and go bankrupt attempting to survey all of them.

*The variability of responses in some sections of your population might be different. You could even have very extreme cases of this. For (a contrived) example, you might have a list of all campus colleges and the number of students staying in each one. Let's say you're trying to estimate the number of males and females staying on campus. If on your list you have an indicator of which colleges are single-gender and which are mixed, then you only need to survey one student from each single-gender college to work out which gender everyone in that college is. So you can have extremely small samples for the single-gender colleges, and larger samples for the mixed colleges.

