calculation of ratio with random error

I have trouble solving a problem assigned to us for class and I dont even know how to start. The problem goes as follows: We look at a guitar string and its frets (numerated from fret 0 (the whole string) to fret 12 (half). The string is 600 mm long. Now the frets are arranged such that the ratio from fret n / fret n+1 = 2^(1/12). (The following setup was created by me, it might therefore be wrong.) So the position of fret n (denoted by a_0):

$\displaystyle a_n = 600 \mdot (2^{-\frac{1}{12}})^n$

Here's the catch. Now in an experiment we measure the position of all frets. We will by by that have a random measurement error and get:

$\displaystyle a'_n = a_n + \epsilon_n$ where $\displaystyle \epsilon_n \in [-0.5,0.5]$

Now I am supposed to explain why the average of all the ratios:

$\displaystyle \frac{1}{12}\sum_0^{11}\frac{a'_n}{a'_{n+1}}$

improves my accuracy by a factor 3-4. I get the intition but to state numbers, I tried to solve this filling in what I know but even MATLAB only gives me unreadable results on this. Furthermore I am expected to explain why this only derives from $\displaystyle 2^{-1/12}$ by no more than 0.0000004.

Any ideas? Thanks a lot, I am lost on this.