[R] on the growth of standard error

[Bert Gunter]

+ (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 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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