The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution:

$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi npq}}e^{-(k-np)^2 / (2npq)}.$$

My question is: are there more precise, quantitative versions of this theorem in the literature? Are there good estimates how to measure the error? I am unfortunately not familiar with the subject but need a result of this type.

Of course there is always the option of going through existing proofs and checking the details, and turning them from "soft" to "hard", but I suspect this has to be already done. And maybe this is not optimal, maybe there are good accessible ways.

Can someone point me a good reference in this direction?

whichnormal distribution. – Will Jagy Sep 29 '11 at 20:22