stats.stackexchange.com On August 21, 2020 1:25:06 PM PDT, Wayne Harris via R-help <r-help@r-project.org> wrote: > >I'm intested in understanding why the standard error grows with respect >to the square root of the sample size. For instance, using an honest >coin and flipping it L times, the expected number of HEADS is half and >we may define the error (relative to the expected number) to be > > e = H - L/2, > >where H is the number of heads that we really obtained. The absolute >value of e grows as L grows, but by how much? It seems statistical >theory claims it grow by an order of the square root of L. > >To try to make things clearer to me, I decided to play a game. Players >A, B compete to see who gets closer to the error in the number of HEADS >in random samples selected by of an honest coin. Both players know the >error should follow some square root of L, but B guesses 1/3 sqrt(L) >while A guesses 1/2 sqrt(L) and it seems A is usually better. > >It seems statistical theory says the constant should be the standard >deviation of the phenomenon. I may not have the proper terminology >here. The standard deviation for the phenomenon of flipping an honest >coin can be taken to be sqrt[((-1/2)^2 + (1/2)^2)/2] = 1/2 by defining >that TAILS are zero and HEADS are one. (So that's why A is doing >better.) > >The standard deviation giving the best constant seems clear because >errors are normally distributed and that is intuitive. So the standard >deviation gives a measure of how samples might vary, so we can use it >to >estimate how far a guess will be from the expected value. > >But standard deviation is only one measure. I could use the absolute >deviation too, couldn't I? The absolute deviation of an honest coin >turns out to be 1/2 too, so by luck that's the same answer. Maybe I'd >need a different example to inspect a particular case of which measure >would turn out to be better. > >Anyhow, it's not clear to me why standard deviation is really the best >guess (if it is that at all) for the constant and it's even less clear >to me why error grows with respect to the square root of the number of >coin flips, that is, of the sample size. > >I would like to have an intuitive understanding of this, but if that's >too hard, I would at least like to see some mathematical argument on an >interesting book, which you might point me out to. > >Thank you! > >PS. Is this off-topic? I'm not aware of any newsgroup on statistics at >the moment. Please point me to the adequate place if that's >applicable? > >______________________________________________ >R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see >https://stat.ethz.ch/mailman/listinfo/r-help >PLEASE do read the posting guide >http://www.R-project.org/posting-guide.html >and provide commented, minimal, self-contained, reproducible code.
-- Sent from my phone. Please excuse my brevity. ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
