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When making box plots to represent data, is it logical to set our axis scale to logarithmic? Or does that defeat the whole point of a box plot?

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The main thing is that the ends and the middle band of the box are preserved by a logarithmic transformation, because they are the middle three quartiles, which depend only on order. So it sounds like a reasonable thing to do. The only thing can change is which data points are considered outliers if you use a 1.5 IQR criterion for the ends of the whiskers — but depending on your data this may actually work better in the log scale: see a two-dimensional example in Figure 3 of Rousseeuw et al.'s "The Bagplot: A Bivariate Boxplot". (Disclaimer: I am not a statistician.)

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