We have such statistical tools for the normal distribution as the empirical rule and Z-score. But in real life we frequently work with bell-shaped distributions that look kind of "normal", i.e. they are approximately normal. But how do we determine if a specific distribution is close enough to a normal distribution for use of such things as the empirical rule and Z-scores? What if, for example, the distribution is too skewed to the right/left?
I wonder if there is a formula or some kind rule of a thumb. I know only that both np>=5 and nq>=5 must be true if we want to approximate the binomial distribution through the normal distribution, but I don't know anything like this for applicability of the empirical rule and Z-score to an approximately normal distribution.
I ask it because I currently have a problem with my population, namely I want to apply the empirical rule and Z-score to it, but I'm afraid that it's too skewed.