How to calculate the variance for mean of means?

I have been reading up on how to calculate variance for different situations and the following has me confused…..

If I take a number of samples (e.g. choose 5 kids in a class and ask them how many pets they have), each time I go into a class and choose 5 kids I am going to get a different mean number of pets, and then each sample will have its own variance.

If I then took the mean of all my samples and added them up and then divided them by the number of the sample I had I'd get the mean of the means.

The part I don't understand is how do I calculate the variance of the mean of means ?

Do I subtract the each mean from the mean of means, square the answer and then divide by n ?

• You can calculate it from the individual sample information. – Glen_b -Reinstate Monica Jul 28 '14 at 8:44

As a general statistical advice, it is best to formulate your questions and it will be very easy to see the answer. Suppose $x_{i,j} \sim N(0,\sigma^2)$ where $i=1 \ldots I$ is your classroom and $j = 1 \ldots J$ is the samples in each class. Let's assume $I=2$ and $J=5$. For each $i$, $\bar{x}_i$, the mean of classroom samples, follows a normal distribution with smaller variances i.e $\bar{x}_i \sim N(0,\sigma^2/5)$ for all $i$. Now the mean of $\bar{x}_i$, which is represented as $\bar{x}$, follows another normal distribution $N(0,\sigma^2/5/2) = N(0,\sigma^2/10)$. $\sigma$ can be estimated directly from your data and variance of mean of means follows from the formula.