Why is the inverse of an average of numbers not the same as the average of the inverse of those same numbers? I have a set of numbers (in my case: mean retention time (MRT) in the stomach (h)) of which I want to calculate the average gastric passage rate (/h). Gastric passage rate = 1/MRT.
My question is why 'the average of the calculated gastric passage rates of those numbers' is not the same as 'the calculated gastric passage rate of the averaged MRTs'. The next question is: what is the right way?
So for example:
$x = 5; 10; 4; 2.$ Average $= 5.25 h \Rightarrow 1/5.25 = 0.19$/h
$1/x = 0.2; 0.1; 0.25; 0.5.$ Average $= 0.26$/h
So should I first take the average of the MRTs and then take the inverse for calculating the gastric passage rate (first way) or should I first take the inverse of all numbers for calculating the gastric passage rates and then take the average of that number (second way). 
Thanks in advance!

Given some positive values $x_1$, ..., $x_n$, their average is
$$\frac{x_1+\cdots+x_n}{n}=\frac{1}{n}\sum_{i=1}^nx_i,$$
and its inverse is
$$\frac{1}{\frac{1}{n}\sum_{i=1}^nx_i}=\frac{n}{\sum_{i=1}^nx_i}.\tag{1}$$
On the other hand, the average of the inverses is
$$\frac{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}{n}=\frac{1}{n}\sum_{i=1}^n\frac{1}{x_i}\tag{2}.$$
In general, these two expressions are not the same, as your calculations also show.

I wrote an incorrect and confusing bit here, after reading the question properly (and giving it some thought) I fully support @Ethan Bolkers answer.
