# Measuring a mean variance for some number of objects observed per trial for multiple trials

I'm running a bunch of trials, $T$, and the outcome of each trial is some number of objects $k_i$ for $i = [1, T]$. I would like to say something about the average "spread" in terms of the number of objects observed each trial, $k_i$, something like a mean variance. However, since $k_i$ can be equal to one, and I'm not sure that the variance of a single object is well-defined, I am unsure how to proceed.

What is a good metric for measuring the average "spread" of my $k_i$? Is a variance of one object, a "variance of zero", well defined?

Let me provide an example: Say I perform an experiment, and the output of that experiment is some number of cells, $k_i = (1, 2, ...)$. I perform $T = 2$ experiments. One experiment outputs $k_1 = 1$ cell, and the other experiment outputs $k_2 = 2$ cells. What is my mean variance for an experiment (obviously two trials isn't enough)?

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I cannot make sense of this. Could you elaborate? – leonbloy Jan 7 '13 at 17:35
@leonbloy I have provided an example of what I mean. Please let me know if this clarifies things for you? – Iota Jan 7 '13 at 17:40
Not much. Let's see. Let's call the experiment a bunch of $T$ trials, each trial results in a positive integer $k_i$. If you want to compute the sample variance (I guess you mean that by "mean variance") for an experiment, then you have the usual formulas (google "sample variance"). The only doubt could arise when the number of trials is $T=1$ (here the sample variance is undefined). But this has nothing to do with $k_i$ being one. – leonbloy Jan 7 '13 at 18:33
In your example, in which you have 2 trials and results $\{1,2\}$, the sample mean is $3/2$ and the sample variance is (using the $n-1$ estimator) is $[(1-3/2)^2 +(2-3/2)^2 ]/ 1$. Where is the problem? – leonbloy Jan 7 '13 at 18:35
The only way this question makes sense to me is if you're measuring some property of each observed object and want to say something about the variance of that property... which may differ from trial to trial. Is there some other measured value besides just the number of object? – mjqxxxx Jan 8 '13 at 7:05