# Average of ratios - I don't get it

I'm calculating ratios from paired samples and there is a point I don't understand. Supposed that I measured the values of 2 paired samples: A and B, and then I calculate the ratios from those values.

Ratio 1: $\frac{A}{B} = 0.5$

Ratio 2: $\frac{A}{B} = 2.0$

Normally one would calculate the average $ratio = \frac{(Ratio 1 + Ratio 2)}{2} = 1.25$. Then the conclusion would be: the value of A is 1.25 times higher than that of B.

But, the ratios can be understood as:

Ratio 1 = 0.5 --> the value of B is double the value of A

Ratio 2 = 2 --> the value of A is double the value of B

Then, the average ratio of A and B should be equal 1.

Does that make sense to you? Where is the flaw?

Thanks all,

• There are different concepts of mean, and it is upto you to decide which is useful. Your idea is known as geometric mean
– Guy
May 28, 2017 at 9:47
• If you think the ratios $0.5$ and $2.0$ are in a sense equal and opposite, you might consider using geometric means or logarithms May 28, 2017 at 22:08

Your ratios imply $A=0.5B$ and $A=2.0B$.

These multiplicative relationships are compatible with geometric sequences and the geometric mean which is $\sqrt{0.5 \times 2}=1$

The 'flaw' is using the arithmetic mean $\frac{0.5+2}{2}$