Quick questions on Normal approximation of the binomial distribution: continuity, validity and z-scores

The approximation is not a coincidence: as the number of trials grows, the binomial bars settle into the shape of a bell curve with the same mean and standard deviation. These panels overlay the matching normal $N(np,\, np(1 - p))$ on the bars of $B(n, 0.5)$ for growing $n$.

For a normally distributed $X$ with mean $\mu$ and standard deviation $\sigma$, the standardised value is

$P(X \le k)$ uses $k + 0.5$ in the numerator (extend the interval by half on the upper side). $P(X \ge k)$ uses $k - 0.5$ (extend the interval by half on the lower side).

If $n p$ or $n(1 - p)$ is less than $5$, the normal approximation is poor.