# How should I characterize the “typical” variance of a large collection of variables?

For an experiment I'm analyzing, I took many measurements of many different values. (To make it concrete, let's pretend I measured the height of people from many different age groups -- I have the height of ten 20-year-olds, ten 21-year-olds, ten 22-year-olds, etc.)

I'd like to demonstrate that the variance of all my readings, per variable, is pretty low. (In my example, I'd like to show that people of any given age are all about the same height. That is, say 20-year-olds are all close to 5'4" with low variance and 21-year-olds are all close to 5'6". The particular heights don't matter; just that people of any given age are similar in height to other people of that age. That's obviously not true, but let's pretend it is.)

What's the best metric to report? I could, for example, take the arithmetic mean of the variances -- or maybe the harmonic mean or geometric mean would be better? Or perhaps I should be averaging the standard deviations or the standard errors?

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I think that your pretend example doesn't match your real case. In the pretend example, if you want to show that people of a given age are about the same height, you can just report the variance of height for that age. But if you'd like to show that ALL the people of a given age are about the same height, I'd report the range of values. But then you start talking about measures that combine the variances or something across different groups. So apparently you want to say something not about people of a particular age, but about people of different ages. But what do you want to show? –  Peter Flom Sep 9 '11 at 20:48
Yes; I should have made this clearer. I'll try to clarify the wording above. I'm interested in showing that people of ANY given age are, regardless of the particular age, around the same height. –  adrian Sep 9 '11 at 20:56