Question

When would researchers want to investigate an empirical distribution and when would researchers want to apply a normal distribution?

Answer 1

Empirical Distribution: To describe a sample of observations of a given variable researchers want to investigate an empirical distribution. Its value at a given point is equal to the proportion of observations from the sample that are less than or equal to that point. You don't need to find a theoretical distribution that fits your data (this can be really tricky!). Also it might give you an idea of what theoretical distribution fits your data if you need a theoretical distributions for forecasting or testing or whatever.

Normal Distribution: When there is large number of independent random variables, irrespective of the original distribution of these variables, their normalized sum tends towards a Normal distribution. For example, the distribution of total distance covered in an random walk tends towards a Normal probability distribution. Unlike many other distribution that changes their nature on transformation, a Normal tends to remain a Normal.