Distribution of means of sets of random numbers If I make sets of $n$ random bits (numbers which are either $0$ or $1$) and take their means, what function $f_n(\bar{x})$ would be the probability distribution of the means?
Also, if I make sets of $n$ random reals between $0$ and $1$, what function $g_n(s)$ would give me the probability distributions of their standard deviations.
These would be useful to check if a result in an experiment is meaningful or if there just weren't enough tests done to show that the results weren't being random. 
 A: If you are dealing with large $n,$ then the Central Limit Theorem may
make it possible to use the normal distribution to get some useful
results. 
Suppose the individual bits have $P(X = 1) = p$ and $P(X = 0) = q = 1-p.$
Let $S_n = \sum_{i=1}^n X_i.$ Then for large $n,$ we have $S_n \sim Norm(np, \sqrt{npq}),$ the normal distribution with mean $np,$ variance $npq,$
and standard deviation $\sqrt{npq}.$ It follows that 
$\bar X = S_n/n \sim Norm(p, \sqrt{pq/n}).$
I'm wondering if by 'random' you might mean 'fair' in the sense that
$p = q = 1/2,$ in addition to meaning that the bits are independent of
one another. In that case $\bar X \sim Norm(.5, .5/\sqrt{n}).$
For example, if $n = 2500,$ then $\bar X \sim Norm(.5, .01).$
This would mean that the average of 2500 fair bits should be in
the interval $0.5 \pm .02$ (within 2 standard deviations of the mean)
about 95% of the time.
As to your second question, you seem to be asking about values of $X$
that are uniformly distributed in $(0,1),$ and I don't immediately
see the connection to the first question.
At this point, I would like to know approximately what size $n$ you
have in mind (several hundred is large enough to use the normal distribution), whether you mean $p = 1/2$ (otherwise we need to talk about more general procedures
to estimate $p$), and whether my discussion
seems to be going in a direction that might be useful.
The figure below shows the distribution of averages, each of 2500 'fair' bits, based on a million simulated samples with $n = 2500.$ Vertical bars at $.5 \pm .02$ enclose about 95%
of the averages. The density curve of $Norm(.5, .01)$ is shown in dark blue.

