Simple question regarding ratio and average

Question

I have 2 lists of numbers (with an equal number of numbers in each). Each number is then divided by the number of which it is paired with (by index), and a ratio is received.

I then want to calculate the global average ratio, so I try 2 ways:

1. Sum all numbers in each column, and divide one with the other.
2. Make an average of all ratios I have received from the calculation above.

Both give me different results.

How could that be? And what would be considered the true global ratio average?

Example lists:

1  : 1 = 1
2  : 1 = 2
3  : 2 = 1.5
4  : 2 = 2
-----------
         1.625 average
10 : 6 = 1.667

Answer 1

It is more instructive to regard just the case of lists of length 2:

You want to have

$$\frac{a}{c}+\frac{b}{d} = \frac{a+b}{c+d}.$$

But this is almost never true, there is a reason for the more complicated way to add fractions.

With 27 numbers, it does not get better.

Edit:

To answer your second question: It depends on your situation what you want to calculate.

The first calculation in your list gives you the ratio of the averages, the second calculation gives you the average of the ratios.

If, for example, you know of every person in a country how much they earn and how much they spend on food, then the first calculation will tell you what percentage of the total income in the country is spent on food. The second calculation will tell you the average percentage people spend on food.

The result will be quite different, because the few very rich people will influence the second result much less than the first. If you want to know how food price rises impact people the second calculation will be more important, if you want to know how it impacts the part of the economy that produces and sells food, the first calculation will be more interesting.

Answer 2

Asking which is "the true global ratio average" brings you very close to Simpson's paradox where, for example, treatment A can be better for men than treatment B, and also better for women, but worse overall. There is an active literature on how to deal with the paradox when it arises.

Answer 3

The answer to your first question is the observation that $\frac{a}{b} + \frac{c}{d}$ is in general not equal to $\frac{a+c}{b+d}$. In other words, the average of ratios is not necessarily equal to the ratio of sums.

As regards the true global ratio average is concerned- I think the answer is context specific. A meaningful answer can only be given if you could shed some light on the nature of the numbers you have.