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Inspired from this answer, I have an question about Milikan's oil drop experiment:

if we have a set of data, say $q_1, q_2, ..., q_n$, since those $g_i$'s have some error bars, we cannot directly calculate the greatest common divisor of those values because they are not exact and it will probably lead to the result $1$, so we to find a value, say $e$, s.t for all $i$, $q_i \approx e \cdot k$, where $k$ is an integer, so without trial-and error (which would be really hard to do without computers) or any other assitant of a computer, how can we find such a value $e$ ?

Edit:

Considering the fact that I have asked the question How did Milikan know that oil drops would acquire only few electron charges?, the answer given to this question is, kind of, out of range.

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    $\begingroup$ I'm voting to close this question as off-topic because it is asking about statistical analysis and belongs on Cross-Validated SE. $\endgroup$ – sammy gerbil Feb 6 '18 at 20:56
  • $\begingroup$ @sammygerbil Then don't downvote, just flat to be migrated, Jesus! $\endgroup$ – onurcanbektas Feb 7 '18 at 4:08
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    $\begingroup$ Have a look web.pa.msu.edu/courses/2003spring/PHY192/… on the how. The milikan oil drop charge measurement is a standard physics lab experiment . I vaguely remember doing it back in 1959 using a microscope. $\endgroup$ – anna v Feb 7 '18 at 4:57
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    $\begingroup$ There was no computer at the lab in 1959 either. We reproduced the experiment. Pencil and paper and brain power. :) In page 10 in my link above there is a current plot measured at a student lab. in page 6 the experimental method is clearly described. $\endgroup$ – anna v Feb 7 '18 at 5:09
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    $\begingroup$ There are some basic tips on p7 here. It suggests using the smallest difference as 1st estimate of $e$. This gives the closest multiple$n$ of $e$ for each data point. Then use Linear Regression $(q=en)$ to find the value of $e$ which minimises sum of squared errors (SSE). Tweaking the biggest multiples might further reduce SSE. There is also a formula for Standard Error within Linear Regression. $\endgroup$ – sammy gerbil Feb 7 '18 at 5:51
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Here is an example of what one gets in measuring the charge with a Millikan oil drop experiment, designed for physics students at university level.

miloil

Just 12 measurements of the charge carried by the drops are shown, already fallen into groups, as seen in this plot. When the number of measurements becomes large enough a histogram versus charge will give separate peaks .

One does not need computers, as this experiment was standard in US universities even in the 1950's, and errors and error propagation can be done by hand and brain calculations. I remember there were drops zooming fast down and drops falling slowly enough to be timed.

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