Standard Deviation for Sample Proportions

For the normal approximation to hold, both $np$ and $n(1-p)$ should be at least 10.

Increasing the sample size decreases the standard error of the sample proportion.

The standard deviation of the sampling distribution for sample proportions is independent of the sample size.

The standard deviation for the sample proportion is given by $\sqrt{\frac{p(1-p)}{n}}$, assuming a sufficiently large sample.