Absolute Bounds on the Mean and Standard Deviation of Transformed Data for Constant-Sign-Derivative Transformations

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Abstract
Absolute bounds (or inequalities) on statistical quantities are often a desirable feature of statistical packages since, as contrasted with estimates of those same quantities, they can avoid distributional assumptions and can often be calculated very fast. We investigate bounds on the mean and standard deviation of transformed data values, given only a few statistics (e.g., mean, standard deviation, minimum, maximum, and median) on the original data values. Our work applies to transformation functions with constant-sign derivatives (e.g., logarithm, antilog, square root, and reciprocal). We can often get surprisingly tight bounds with simple closed-form expressions, so that confidence intervals are unnecessary. Most of the results of this paper seem to be new, though they are straightforward to derive by geometrical arguments and analytical optimization methods.

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