I don't quite get why you can't find the average (mean) time of wake up here. I wasn't referring to range as in math range >.<
Ok, basically, what we have observed here is that there are a group of people who wake up extremely early and a group of people who wake up extremely late, both with relatively small standard deviations. This suggests a type of distribution which is known as a Bimodal distribution, it would look similar to two normal distributions joined together.
This would look something like this:
Say that we refer to this distribution as the distribution of the continuous random arable X, bimodally distributed with Y (people waking up early) having the probability of alpha and Z (people waking up late) having the probability of 1-alpha. Y and Z, hence, are unimodal continuous random variables, where 0<alpha<1, this is known as the mixture co-efficient.
Thus, when you want to find the mean, you will find the mean of the individual unimodal continuous distributions, otherwise your mean will make no sense. Your mean will neither belong to the group waking up early or the group waking up late. You can't force a bimodal distribution into a unimodal one.
To answer your second question, it does not matter, range, in plain English means from A to B. If you only have one value (which is your so called mean), there is no range to begin with, thus, how can you say that your wake up time falls within any sort of range. If I were to tell you, "I have a range of pens, from black to blue", it means I have many pens, I cannot have a range of pens if I only have one pen, just like you cannot have a range of values, if you only have one value. Thus, your wake up time cannot be within that non-existent range.
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