The classical [de Moivre-Laplace theorem](http://en.wikipedia.org/wiki/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?



Quanitative de Moivre–Laplace theorem (reference request)
Firstly, I think by "qualitative" you mean "quantitative". Secondly, while there is a huge literature on the quantitative versions of the central limit theorem, the canonical results can be found in Feller's Vol 2. For the center of the distribution there is the Berry-Esseen theorem, for the tails there is the large deviations theory, the introduction to which is also covered by Feller.


**EDIT** If you really care about the specific approximation of the binomial by the normal (or *vice versa*) you are just talking about the higher terms in the Stirling approximation to the factorial (and hence to the binomial coefficients). You can read all about it in, eg, Graham/Knuth/Patashnik's Concrete Math.


