Stratified sampling I am learning about stratified sampling, and I have the following question: can stratified sampling achieve more precision than SRS if data is plentiful and the samples are big enough?
For example, it seems to me (I'm learning!) that if there isn't much data, and I want to make sure I sample e.g. men and women equally, I might want to partition the data into men and women and sample $n$ of each, lest SRS would unluckily pick many more women than men.
But if I have plenty of data, and I sample "enough", I could reduce the chance of not sampling men and women equally. Am I missing something? Does stratified sampling make sense only when the data/sample sizes are small?
 A: No, stratified sampling still makes sense with large sample sizes. It is true that the absolute benefit decreases, but the relative benefit as a fraction of the total uncertainty doesn't.
Suppose the population has an equal proportion of men and women. If you sample $n$ people, you will get on average $n/2$ men, but with a standard deviation of $\sqrt{n/4}$. So if there's a difference of $\delta$ between the mean in men and the mean in  women, your estimated mean will be off by something  like
$$\delta\frac{\sqrt{n/4}}{n}=\frac{\delta}{2\sqrt{n}}$$
The overall standard error of the mean also scales as $1/\sqrt{n}$, so the fraction of the standard error due to sample imbalance in gender doesn't go down.
In theory, stratification is at worst harmless, as long as you have at least two observations per stratum so you can compute variances.  In practice, you might want more than that so you don't lose a stratum to missing data.  Also, it might be operationally harder to collect a heavily stratified sample, especially if you don't have a sampling frame that includes the stratification variable.
It is quite common for the first stage of a multistage sample to have just two sampling units (clusters) per stratum.
