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Does it ever make sense to compare the variance for two sets of data, neither of which are even approximately normal (e.g. bimodal)?

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Yes it makes sense. The variance of the data approximates the variance of the theoretical distribution from which the data is generated (if the data are i.i.d replicates of this distribution). But is it useful/relevant ? That depends on the context and your goal. – Stéphane Laurent Aug 1 '12 at 13:10
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If your distribution is bimodal, as you indicate, measures like mean and variance can be hard to interpret. For example, large variance could mean the two modes are far from each other or it could mean that each mode, separately, has a lot of variation around it (or both). As a summary statistic, it lacks clear meaning. It may be that the bimodality arises from aggregating two different distributions. If that is the case, you may want to disaggregate (e.g. by using a mixture model) before you look at things like variance. – Macro Aug 1 '12 at 13:47
Vegard, the comments and reply show that this question in its present form may be too speculative for this site: it asks us to imagine hypothetical situations. Could you please make this more concrete by telling us what practical problem you are facing? – whuber♦ Aug 1 '12 at 16:00
My concrete practical problem is simply whether to report variance or not. If you think the question is speculative, I think you're reading too much into it. I am simply asking for examples of scenarios where variance is useful in practice even though the data is non-normal. – Vegard Aug 1 '12 at 17:28
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One thing to note, is that if you already have a family of distributions in mind, the parameters of that distribution will give rise to a set of sufficient statistics (assuming some parametric family). If the set of sufficient statistics for this family coincide with the mean and variance, then your tests between distributions will most likely be at least partially based on the sample quantiles. This would be especially evident under the exponential family of distributions where UMPU tests are determined by functions of the sufficient statistics. – Jonathan Lisic Aug 1 '12 at 19:21
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2 Answers

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Does it ever make sense? I don't know. It depends on what you are trying to do. But you say you aren't trying to do anything, so there's no good answer.

I can't think of a case where the variance of a bimodal distribution makes much sense. But you say (e.g.) bimodal. If the data are non-normal, but not all that non-normal, variance can make sense. My general rule is that, if the mean makes sense, the variance makes sense. If the median makes sense then either the mean absolute deviation or interquartile range makes sense. For bimodal distributions, neither the mean nor the median makes all that much sense. A density plot is a good method here, or a table of percentiles.

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Not to quibble, but the variance of a binary variable always has a useful interpretation :-). Its mean definitely makes sense. Ergo, by your syllogism, its variance makes sense. Most binary data are bimodal, practically by definition... . I wonder, then, whether indeed there might be some other multimodal distributions or datasets where the variance is a useful statistic. – whuber♦ Aug 1 '12 at 21:11
OK, you're right. But I think calling these statistics "mean" and "variance" for binary distributions can lead to confusion. Should we not use proportion? – Peter Flom Aug 1 '12 at 21:21
Well, with a 0-1 coding, mean and proportion are (of course) equal. Since the variance contains no more information than either, you could argue that reporting the variance is redundant, which might resolve my concern. I'm struggling to imagine a bimodal situation where the variance is either a natural or helpful statistic. But then again, the question asks whether the variance is useful for comparing two datasets. Maybe we have wandered off that topic a little bit. Perhaps the way out lies in considering how one would go about comparing two bimodal datasets in the first place. – whuber♦ Aug 1 '12 at 21:26
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I think the question of how to compare two bimodal distributions depends on what you are interested in comparing, but I would go for graphical methods here. Overlaid density plots, quantile-quantile plots, perhaps parallel box-plots. – Peter Flom Aug 2 '12 at 10:04
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We frequently use scaling relationships between central moments to ascertain the most likely underlying generative process of biological data. I studied the dynamics of the HIV promoter and it produces very non-Gaussian distributions that are wide and highly skewed. The Var~Mean relationship was the first step in determining candidate stochastic models. Moment analysis is never an end in itself, without interpretation they're of limited use.

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