Editing Albert A. Michelson (section)

=== Application of basic statistical principles in Michelson's study of speed of light ===
During June and early July 1879, Michelson refined experimental arrangements from those developed by [[Hippolyte Fizeau]] and [[Léon Foucault]]. The experimental setup was as follows: Light generated from a source is directed towards a rotating mirror through a slit on a fixed plate; the rotating mirror reflects the incoming light and at a certain angle, towards the direction where another fixed flat mirror is placed whose surface is perpendicular to the incoming ray of light; the rotating mirror should have rotated by an angle&nbsp;''α'' by the time the ray of light travels back and is reflected again towards the fixed plate (the distance between the fixed mirror and the rotating one is recorded as&nbsp;''D''); a displacement from the slit is detected on the plate which measures&nbsp;''d''; the distance from the rotating mirror to the fixed plate is designated as the radius ''r'' while the number of revolutions per second of the mirror is recorded as ''ω''. In this way, {{nowrap|1=tan(2''α'') = ''d''/''r''}}; {{nowrap|1=Δ''t'' = (''α''/2{{pi}})/''ω''}}; speed of light can be derived as&nbsp;{{nowrap|1=''c'' = 2''D''/Δ''t''}}.

While at plain sight, four measured quantities are involved: distance ''D'', radius ''r'', displacement ''d'' and rotating mirror revolution per second&nbsp;''ω'', which seems simple; yet based on the limitation of the measurement technology at that time, great efforts were made by Michelson to reduce [[Observational error|systematic errors]] and apply subsequent corrections. For instance, he adopted a steel measuring tape with a said length of 100 feet and he intended to measure tens of times across the distance; still, he measured its length against a copy of the official standard yard to find out it as 100.006 feet, thus eliminating a systematic error, albeit small.

Aside from the efforts to reduce as much as possible the systematic errors, repeated measurements were performed at multiple levels to obtain more accurate results. As R.J.&nbsp;MacKay and R.W.&nbsp;Oldford remarked in their article,<ref>{{Cite journal|last1=Oldford|first1=R. W.|last2=MacKay|first2=R. J.|date=August 2000|title=Light|journal=Statistical Science|volume=15|issue=3|pages=254–278|doi=10.1214/ss/1009212817|issn=0883-4237|doi-access=free}}</ref> 'It is clear that Michelson appreciated the power of averaging to reduce variability in measurement', it is clear that Michelson had in mind the property that averages vary less which should be formally described as: the standard deviation of the average of ''n'' independent random variables is less than that of a single random variable by a factor of the square root of&nbsp;''n''. To realize that, he also strived to have each measurement not influencing each other, thus being mutually [[independent random variables]].

A [[statistical model]] for repeated measurements with the assumption of independence or identical distributions is unrealistic. In the case of light speed study, each measurement is approached as the sum of quantity of interest and measurement error. In the absence of systematic error, the measurement error of speed of light can be modeled by a random sample from a distribution with unknown expectation and finite variance; thus, the speed of light is represented by the expectation of the model distribution and the ultimate goal is to estimate the expectation of the model distribution on the acquired dataset. The law of large numbers suggests to estimate the expectation by the sample mean.<ref>{{Cite book|title=A Modern Introduction to Probability and Statistics: Understanding Why and How|year=2005|isbn=978-1852338961|pages=248|last1=Dekking|first1=F. M.|last2=Kraaikamp|first2=C.|last3=Lopuhaä|first3=H. P.|last4=Meester|first4=L. E.|publisher=Springer }}</ref>