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I know that stratified random sampling is almost always preferred over simple random sampling, but I have also read that the Variance of sample mean (x-bar) from stratified random sampling could sometimes be larger than the Variance of x-bar under simple random sample. In those cases, SRS might a better method compared to stratified random sampling.

Logically, it seems to make sense, but I'm wondering if there's a more mathematical way to explain this, and maybe a (simple) real life example?

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    $\begingroup$ stratified random sampling is almost always preferred over simple random sampling who said that? Do you have any sources for these and other claims? $\endgroup$ – user2974951 Jul 17 '19 at 11:06
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Stratified random sampling (STRS) is often preferred to simple random sampling (SRS) because it usually lead to better precision, if strata are chosed wisely. However, sometimes STRS may return higher variance with comparison to SRS. There is a mathematical explanation and it depends on the variance between and within the strata. You may refer to Sampling Techniques (Cochran, 1977) for more details.

Anyway, the STRS is not only chosen to reduce the variance, but also because it lets you represents all the strata in the population. For example, you may want to stratify over the gender, to be sure to include both males and females in your sample, or you could stratify on age classes, etc...

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