I have a bit of confusion on this topic. If you were for instance approximating a binomial distribution as normal (Given by N(np, npq)), when would you just use the standard $\bar x-\mu$/$\sigma$ equation and when would you have to divide the bottom also by the square root of the sample size?
As an example, assume a distribution B(100,0.4). It would be ~N(40,24). Would you then proceed to use the formula (using the obtained $\sqrt{24}$ standard deviation using the $\sqrt{npq}$ formula) with the $\sigma$ divided by $\sqrt n$ or just divide by $\sigma$ assuming we just wanted to find the Z score of an arbitrary $\bar x$?
Comment (continued): Here is a plot of $\mathsf{Binom}(n=100,\,p=.4)$ [vertical bars] along with the density function of $\mathsf{Norm}(\mu = 40, \sigma = \sqrt{24})$ [blue curve].