The observation in 4.3 Approximating Sample Proportion Distribution is a well-known effect. Statisticians has investigated such phenomenon and came up with a theorem for it, known as the Central Limit Theorem.

The Central Limit Theorem (CLT) states that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

To explain in math, suppose that X=\frac{1}{n}\left(X_{1}+X_{2}+X_{3}+\ldots+X_{n}\right), then we can assume X \sim \color{red}N\color{black}\left(\mu, \frac{\sigma^{2}}{n}\right). The most important part here is N. as for the mean of variance of X, it can be calculated without the usage of any approximation/assumption (shown below).