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Stratified sampling is a probability sampling method while quota sampling is a non-probability sampling method.
Stratified sampling includes sub-dividing the sample into mutually exclusive and exhaustive groups. A simple random sample is then chosen independently from each group. This way, the probability of each element in a given group being selected is equal.
Quota sampling, on the other hand, is a type of judgement sampling. In quota sampling, quotas are set up according to some characteristics. Within a given quota, the interviewer has to interview a certain number of people. The interviewer is free to choose anyone he wants. So personal judgement and bias may come in. However, this is the most common type of non-random sampling.
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