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Suppose I wish to know the variance within a sample or of the population from which it is drawn. However, I do not have true measurements for most of my "observations". Think of them as like wooden blocks or books on a shelf. It is easy to line them up from biggest to smallest, even though we have not taken accurate individual measurements of any of them.

Now suppose I do take accurate measurements of the tallest and shortest. I will refer to the difference between these measured values as the range. My question is, if I am really interested in the variance, how good a proxy is the range for the variance? How much of the sample information have I lost by precise examination of only two observations? How should I estimate the variance if given only the range? If I have prior information providing an accurate sample mean, will that enable me to significantly improve my variance estimate?

If accurate individual observations are difficult or expensive, but I am basing some medical or financial judgement on the variance estimate with a potential for extremely negative outcomes if my estimate deviates from the true variance by a large amount, is there some way to get at least a rough idea of how likely that is?

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Unfortunately, unless you have some reasonable preliminary idea of the "shape" of the distribution, there is not a very close relationship between the variance and the range. In particular, given a fixed variance, the range of a distribution is heavily influenced by the kurtosis of the distribution (fatness of the tails), so a leptokurtic distribution will tend to have a larger range than a platykurtic distribution. This effect can be quite extreme, so the range on its own is not very helpful in estimating the variance.

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  • $\begingroup$ Thanks Ben! That is very helpful. $\endgroup$
    – andrewH
    Dec 24, 2022 at 9:18

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