Well first I want to say that I have a very weak background in stats/probability but I still hope that the question I want to share with you makes sense.
I have a set of binary (coming from a set of simulations of physics system, each returning me the value of boolean 0 or 1). This set as size around 40k.
Trivial calculation give me a mean of 0.084 and a standard deviation of 0.27. I'd like to put an uncertainty on this estimation of the mean but it shouldn't be the standard deviation (telling me actually how much the value fluctuates around it). I want something telling that if I pick random samples of size y on my original data set and calculate their mean it will usually fall around this 0.084 .
It does seem much more relevant in my head, but when trying to apply it I can't figure out what should be this value y of my samples size (and it will sure influence the result), how many of these samples should I take, and should I allow or forbid to pick twice the same value from the data set?
Thank you in advance for your help/enlightening.
Let $$\bar x ={ {\sum{x_i}} \over n},$$ whre each $x_i$ is a Bernoulli (0 or 1) random variable.
Then $$\sigma^2_{\bar x}={1 \over n^2} \left( \sigma^2_{x_1}+\sigma^2_{x_2} + \cdots+\sigma^2_{x_n} \right) = {1 \over n^2} (np(1-p))={{p(1-p)} \over n}$$
Use your estimate of $p$ (which is $\bar x$) in this equation to give you an estimate of the variance of the sample mean. Take the square root to get your estimate of the standard deviation of $\bar x.$