1. The problem statement, all variables and given/known data
I am reading a definition a quantile plot and a quantile and I am getting a little confused. It says:

Ok, I get that. Then it says:

Get that too; but then it says

I don't get that part. What exactly is this 'order of observations?'

I am interpreting it as the following:
If i = 4, we are finding the fraction f_{4}, which is the fraction of data values that are less than or equal to the value of the 4^{th} observation.

The 4^{th} smallest observation. The text is talking about order statistics.

When you collect data, your measurements will be rather random. Suppose you perform some experiment in which you expect to be measuring the same random process over and over. You might well think something fishy if your first measurement is -0.43, the next one -0.21, the next one 0.16, 0.28, 0.45, and so on. You wouldn't think anything is fishy if you took your randomly-collected data and sorted it from smallest to largest and saw -0.43, -0.21, 0.16, ...

That sorted data tells a whole lot about the underlying distribution, even when you haven't the foggiest idea about the nature of the random process. When you do have some idea about that process it also tells quite a bit. Here you are assuming a normal distribution. The quantiles generated by that statistic, (i-3/8)/(N+1/4), are approximately unbiased if the underlying random process is normal. This is the type 9 quantile formula. There are, not surprisingly, 8 others.

OK, I follow you, I think. This is the first week of class and he started us on this random section in the 8th chapter in the text. So I am assuming that the derivation of the formula above might be outside of my scope of understanding for the time being?