I've just started a beginner's statistics course, and they've presented us with the variance of the sample mean as:

$$ \operatorname{Var} \left[ \bar X_m \right] = \frac{N-n}{N-1} \frac {\sigma^2} n$$

where, as I understand it, $\bar X_m$ is the mean of $m$ random variables, each of size $n$, from a population of $N.$

But, in looking this up, I've also seen this formula: $\operatorname{Var} [\bar X_m] = \frac {\sigma^2} n$

I take it both are right under different circumstances, and I've tried looking it up, but can't seem to find when to use one and when the other?

Any insight would be greatly appreciated, thanks so much.

  • 1
    $\begingroup$ The first (displayed) formula is for sampling from a 'finite population' of size $N$ without replacement. The second is for sampling with replacement or from an essentially infinite population. The first is most often used when the sample size $n$ is more than 10% of the population size $N.$ (If $n$ is much smaller than $N,$ then $\frac{N-n}{N-1} \approx 1$ and the two formulas are about the same.) $\endgroup$ – BruceET Dec 7 '17 at 18:18

$\operatorname{Var} \left[ \bar X_m \right] = \dfrac{N-n}{N-1} \dfrac {\sigma^2} n$ is correct when sampling without replacement.

$\operatorname{Var} \left[\bar X_m\right] = \dfrac {\sigma^2} n$ is correct when sampling with replacement, so that the observations are independent of each other.


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