+ (in addition to Jeff's link) https://en.wikipedia.org/wiki/Binomial_distribution
Bert Gunter "The trouble with having an open mind is that people keep coming along and sticking things into it." -- Opus (aka Berkeley Breathed in his "Bloom County" comic strip ) On Sat, Aug 22, 2020 at 6:50 AM 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. > [[alternative HTML version deleted]] ______________________________________________ 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.
