# Sampling distributions of sample means

Given a population from which we draw repeated samples of fixed size, say, 'n', my questions is what size is considerable? Is there any lower bound on sample size? Also how many such samples do we have to select? Is there any lower bound on the number of samples so that it would follow central limit theorem?

• "considerable" in what sense? "Is there any lower bound on sample size?" -- to achieve what, exactly> – Glen_b -Reinstate Monica Apr 17 '15 at 11:17

To get lower bound for 'n', there are 2 methods.... you just need tolerance level.

the two methods are:

Let $\bar x=sample~mean,~\mu=population~mean,~\bar x-\mu=\epsilon(tolerance~level),~S=Sample~Standard~Deviation,~n=number~of~samples,~\alpha=error~level$

1. Absolute error method:-$$\epsilon\sim N(0,\frac{S}{\sqrt{n}})$$ So you can find lower bound for n,$$n\geq\frac{Z_{\alpha/2}*S^2}{\epsilon^2}$$

2. Relative error method:- $$n\geq\frac{Z_{\alpha/2}*S^2}{\epsilon^2\mu^2}$$(if you don't know $\mu$, then take $\bar x)$

It depends on the distribution and its moments, of course. Without this information, you can see how much the jack-knife estimation (leave-one-out) of the first moment fluctuates, for a decreasing sample size. And, of course, the CLT applies to the sum of an infinite number of RVs

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