Empirical Rule: Where Does it Come From? Is there any compound proof of the empirical rule? Where do the 68 - 95 - 99.7 percentages come from? Is it simply just an observed probability for most normal distributions?
 A: But the empirical rule is just a more specific statement about a very general fact about CDFs. For every distribution, cumulative distribution function is defined as $F_X(x) = \mathbb{P}(X \le x)$. If you want to know the probability that a sample is in an interval $(a,b]$, then you can use the difference of CDFs: $$\mathbb{P}(X \in (a,b])=F_X(b) - F_X(a).$$ This fact is important because it's true for any probability distribution, for any interval. In the special case that $F_X$ is a normal CDF, then we can show that the empirical rule that you've presented just reproduces this identity.
For the more specific case of a normal distribution with mean $\mu$ and standard deviation $\sigma$, we can write
$$\begin{align}
\mathbb{P}(X \in (\mu-\sigma,\mu+\sigma]) &= F_X(\mu+\sigma) - F_X(\mu-\sigma)\\
&= 0.8413447\dots -0.1586553\dots \\
&=  0.6826895\dots
\end{align}$$
and likewise we can write down and evaluate similar expressions for $(\mu-2\sigma,\mu+2\sigma]$ and $(\mu-3\sigma,\mu+3\sigma]$, or indeed any number of standard deviations $k\ge0$.
