Electron charge from Millikan's oil drop experiment
Millikan's oil drop experiment first and foremost serves to establish that electron charge is quantized. However, as you say in and of itself it does not exclusively specify the charge $e$, as it could be $e/2$ as you mention. However, the simplest interpretation of the data is to say that the charges are quantized with charge $e$; if it were $e/2$ instead, you'd have to tell me why should I expect every droplet to have an even number of electrons. Essentially, you'd need some kind of conspiracy going on, some extra physics to observe such phenomena.
However, there are also other ways to measure $e$, such as measuring the shot noise in a current-carrying wire. Such experiments give us even more confidence that the electron charge is $e$. In fact, in the fractional quantum hall effect, if one tries to do the same shot-noise measurement to experimentally verify $e$, one gets a rational fraction such as $e/3$ instead. This phenomenon actually led to the Nobel Prize in 1998 because of the non-trivial physics that was discovered. All in all, because of this and more, the value of $e$ is more or less an established fact (the evidence is more than the oil-drop experiment).
Keep in mind that in doing this experiment one observers drops of many different charges, because the triboelectric effect that causes the drops to be charged in the first place is stochastic.
That is, you see values that would eventually be identified as $1e$, $2e$, $3e$, and on up to as high a value as you are able to measure with the apparatus.
Moreover, in Millikan's real experiment (rather than the simplified version presented in many basic treatments) you watch a drop while for long enough to record one or more instances of the drop's charge being reduced (an effect of cosmic radiation), so you can observe the steps down toward neutral.
In any case, when you have enough data it becomes clear that you have discrete levels, and that the spacing is the same as the lowest level. At that point the least hypothesis is pretty unavoidable.
Gonna be my first answer on this site, since I had this question right from school till college until I read the actual paper by Millikan from my library archive.
You're absolutely correct. For all we know 1.6 can be the wrong number, and instead the actual value of e is 1.6/n where n is some integer. After all, concluding what Millikan concluded requires the extra assumption that there exists at least one droplet which lost exactly one electron. However, hear me out-
The version presented in books is a very abridged one. Millikan watched the droplets for prolonged period of time, where he would zap the droplets by X-ray, and keep experimenting on it till it lost all charge, then zap it again randomly. And he observed MANY such drops.
The drops always lost charges in simple whole number ratios. First and foremost, this proved that charge is quantized.
Here's the interesting part- The common denominator the droplets were losing charge in, also happened to be the lowest measured charge in all the trials.
Let's say that e is actually e/2. This would require you to explain why, in a random process, are electrons always lost in even numbers. Now let's say e is e/3. Here, electrons can be lost in both even and odd numbers (3, 6, 9 etc), but then, it'd require you to explain why they're lost in triplets, and so on.
The above argument would break down for, say, e= e/million. Because then, the errors in measurement would far exceed our resolution to distinguish between 1 million vs 1 million and 1 electrons. This is a plausible scenario.
But there's a thing in Science called Occam's razor. If two theories explain the same thing perfectly, you choose the less complicated one. For all we know, a neutron is actually made up of two neutrons of half the accepted mass of a neutron. But the thing is, it doesn't matter. If all observable experiments can make do by considering the simplest hypothesis, that hypothesis stands until it breaks down in some other experiment
e/m ratio and mass of electrons can be measured independently. They, and every other experiment, agree on the charge found by Millikan (save for some error). However, we need to be very careful not to include experiments which already have the charge of electron secretly put in there (like those involving an oscilloscope)."First principle is you must not fool yourself".