normal approximation to the binomial distributions

Let $X$ be binomial with parameters $n$ and $p$. For large $n$, $X$ is approximately normal with mean $mu = np$ and variance $sigma^2 = np(1-p)$.

$$P[X=x]=frac{n !}{x !(n-x) !} p^{x} q^{n-x} approx frac{1}{sqrt{n p q} sqrt{2 pi}} e^{-dfrac{(x-n p)^{2}}{2 n p q}}$$

The approximation is good if $p$ is close to 0.5 and $n>10$. Otherwise, we require that

$$n p>5 quad text { if } p leq 1 / 2 quad text { or } quad n(1-p)>5 quad text { if } p>1 / 2$$

we can approximate the sum over all $x leq y$ by integrating to y + 1/2:

$$P[X leq y]=sum_{x=0}^{y} left( begin{array}{l}{n} \ {x}end{array}right) p^{x}(1-p)^{n-x} approx Phileft(frac{y+1 / 2-n p}{sqrt{n p(1-p)}}right)$$

This additional term 1/2 is known as the half-unit correction for the normal approximation to the cumulative binomial distribution function.