As read on Wikipedia, the binomial distribution $B(n, p)$ is approximately normal with mean $np$ and variance $np(1−p)$ for large $n$ and for $p$ not too close to zero or one. Why ? Why this condition on $p$ ?
I know that the sum of Bernoulli distributions gives a binomial distribution, but this is not enough to apply the central limit theorem. Indeed the central limit theorem involves the mean of some variables not only the sum of these variables. Can somebody show me at which point one obtains a mean of variables ?