Arithmetic Mean of Random Variables

$Z$ denotes a random variable with $m=EZ$ and $/sigma^{2}=Var(Z)=E(Z-m)^{2}.$ Sample values $z_{1},/ldots,z_{n}$ are interpreted as $n$ independent realizations of $Z$. They also can be interpreted as realizations of $n$ independent random variables $Z_{1},/ldots,Z_{n},$ which all have the same distribution as $Z$. This $Z_{i}, /; i=1,/ldots,n,$ are called sample variables. The arithmetic mean

then is a random variable, with a given distribution, of which we would like to compute the expectation and variance.

From this we need two properties of linear combinations of independent random variables:

It follows immediately for the arithmetic mean

and

The corresponding deviation or standard error is

and we summarize:

The arithmetic mean shows the same expectation as the random variable $Z_{i}$ and a standard deviation (or error), that reduces that of $Z_{i}$ by a factor $/sqrt{n}$.

From this, for example, if we assume an approximate normal distribution of the data, we can give immediately an important confidence interval for the theoretical mean value $m$ of the distribution (Hartung et al., 1984[9]): with about 95% probability the interval

covers the value $m$, where $/bar{z}$ respectively $s$ denote again the estimated (sample) values of $m$ and $/sigma$.