# How to calculate accuracy of the sample approximate of expected value

I have independent random values $x_1 < ... < x_n$ with the same distribution function. I want to find expected value of this distribution by my . Of course it will be $X = \frac{1}{n} \sum_i x_i$

Probability $P(|X - E(x)| < \epsilon) > 1- \delta$

The question is how to find the number $n$ to be accurate about $\epsilon$ and $\delta$ (they equal 0.01)? I want to find FULL proof for that accuracy. And I don't want to calculate dispersion, because it will cause same difficulties.

I had 2 variants now - Chebyshev Inequality and Hoeffding's Inequality.

First is to complicated - it needs $n> 1000000$

Second needs a precise interval for $x$ . I can estimate it theoretically, but It becomes too big, much bigger than on practice. Maybe there is some variant of HI where bounds for $x$ are changed to $x_1$ and $x_n$ ?

I also looked at Central limit theorem, but I didn't find the proof for accuracy for my $X$.

Thanks, Vasily.

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Stated that way, there is no answer to your question. Consider a distribution that gives some finite, but small, probability $\eta$ to a huge value $v$, while with probability $1-\eta$ the value is near $0$. If $v$ is large enough, the mean will be more than $1$.
But with probability $(1-\eta)^n$ the value $v$ won't occur in a sample of size $n$. So an estimate that is based only on the sample values will probably be inaccurate by more than $1$ for such a distribution.