Confidence in sample mean, given sample variance?

Question

Let's say that I have an large population of data, but that I have a sample mean and sample variance calculated from a subset of that data.

Can I use my sample variance (or standard deviation) to know how confident I should be in my sample mean being close to the population mean?

It seems like I should because a low variance seems to indicate that there isn't very far that the mean could move, but on the other hand, taking more samples isn't going to make the variance approach zero.

Is there some other calculation I should be using for getting a confidence amount in my sample mean?

Answer 1

if you look carefully at the construction of a confidence interval:

$$ \mu +/- Z_{\alpha} * \sqrt{VAR(\mu)} $$

$$ VAR[\mu] = VAR[\sum{x_i}/n] = n * VAR[ {x_i}/n] = VAR[X] / n $$

so you just have to divide the variation of the sample by the number of observations that were made.

hope this helps!