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 ?

  • $\begingroup$ You can calculate it from the individual sample information. $\endgroup$
    – Glen_b
    Commented Jul 28, 2014 at 8:44

2 Answers 2


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.

Things get more complicated if you assume difference variances in each classroom but the basic ideas remain the same. You should look into pooled analysis for multiple samplings.



Do you mean: (1) how do you calculate the variance of the means of the means empirically, or (2) how do you estimate that variance based on your data together with statistical theory? To use the first approach, you would have to repeat your data collection multiple times. To use the second approach, you can use this formula: the variance of the mean is equal to the variance of the variable divided by N. (Check some statistics textbooks on "standard error of the mean". As used here, "standard error" is a synonym for standard deviation.) In your case, your observation involves means rather than individual scores, but the formula is still appropriate. Clearly the second approach involves less work than the first.


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