I recently spoke to a large qualitative analysis company who were working on in-depth analysis of the potential customers of train company. I asked them how they chose the people to include in their sample, they said, that they picked them at random from list of people in the region, but they also said that had predefined criteria, that the sample needed have equal distribution of both genders as in the region, and similarly for ethnicity, income, education and so on. I asked them whether they believed that the list they picked from was biased, but this was not the reason, it was rather to make the sample more random. Does this make sense, or do they risk interfering with the randomness, and hence make it less random? Their sample was around 100 people, but I am interested in an answer also for very few people.
The procedure you describe here is known as quota sampling. This is a non-ramdom sampling method. The procedure is frequently used in situations when there is no sampling frame available. In such a situation it provides a convenient and cheap way to draw a sample.
Stratified sampling is different. In stratified sampling the population is divided into a certain number of non-overlapping sub-populations, and each population is sampled independently. Or to put it differently, the sampling rate varies between the sub-populations.
Example: To draw a simple random sample from a group, you apply a uniform sampling rate to a population, say 10%. In stratified sampling you split the populations into two groups, say rural and urban. For each group you apply a different sampling rate, e.g. 5% for the urban population and 20% for the rural population.
It might be easy to make the mistake (if you are careless) and interpret it as stratified sampling instead of quota sampling. As whuber pointed out in the comments below, stratified sampling requires the the partitioning of the population into groups BEFORE sampling.
"This is known as stratified sampling. It does not make it "less random" than simple random sampling.
For smaller samples, it is actually more appropriate. With simple random sampling, your samples are more likely to over-represent a category of the population."