[R] on the growth of standard error

[Jeff Newmiller]

stats.stackexchange.com

On August 21, 2020 1:25:06 PM PDT, Wayne Harris via R-help <r-help using 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?
>
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-- 
Sent from my phone. Please excuse my brevity.