TITLE: Greatest common denominator of measurements
QUESTION [6 upvotes]: In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ can be represented as a multiple of a fixed elementary charge $e$ like $$q_i = n_i e.$$
Now I do not know $e$ in advance. Is there some method to get $(e, \Delta e)$ out of the sequence of measurements?
I found this question on StackOverflow, but it does not have any answers that solve the problem in a stable way.

REPLY [3 votes]: This is how you would do it if there were no experimental error:  Let $q_0$ be your smallest measured value for the charge and enumerate the other charges $q_1,\dotsc,q_n$.  By assumption, $q_k=en_k$ for some positive real $e$ and $n_k$ an integer.  Thus, $\frac{q_k}{q_0}$ is rational for all $k$.  Write all of these rationals is reduced form and let $\ell$ be the least common multiple of all the denominators.  Take $e=q_0/\ell$.  Then, by construction, $q_k/e$ is an integer for all $k$ and $e$ is the smallest positive real number such that this is true.
The problem with this, however, is that, when you calculate $\frac{q_k}{q_0}$, you are only getting an estimate, and so $\ell$ is going to be completely off.  In practice, I think you just have to hope that $\frac{q_k}{q_0}$ is (approximately) an integer for all $k$, and if that's not the case, go back to the lab and take more data until this is the case.
Of course, as pointed out, if $e$ works, so will $e/2$, and $e/3$, etc.  The best you will ever be able to do here is calculate the probability that you measured your value of $e$ (call it $e_0$) was indeed the correct value of $e$ under the assumption the true value of $e$ is $e_0/2$, $e_0/3$, etc.