# Boxplots with lognormally distributed data

### Context

Environmental data (e.g., pollutant concentrations in water, soil, air) are often lognormally distributed. Even when they are not, we tend to assume that they are (for better or worse).

Because of this, 99.5% of the time that I create boxplots, they are presented with a log-scaled concentration axis. Here's an example:

It occurs to me that computing the lower fence as $Q_1 - (1.5 \times \mathrm{IQR})$ may not be the best way when the data are presumed to be lognormal.

### (Restated) Question

Given that, for me, the main value of defining the fences as with $1.5 \times \mathrm{IQR}$ is detecting potential outliers, should I define those fences and outliers in log-space, or keep everything in arithmetic space?

I'm currently leaning towards log-transforming the data, but have concerns that this may cause undue confusion or even not be an acceptable practice.

### Similar questions

The accepted answer to this question: Is there a boxplot variant for Poisson distributed data? suggests simply transforming the data -- in that case by taking the square root. I'm specifically curious if the fences should be computed in log-space and then converted back to arithmetic space.

• I believe you refer to my answer, but you do not quite correctly characterize it. It suggests re-expressing the data and redrawing the boxplot based on the re-expressed data. That means that you write down the logarithms of the data and proceed with the usual computations based on the logs. Although the medians and hinges will be the logs of the original medians and hinges, the step (which determines the fences) will change. That is different than merely drawing the original boxplot on a logarithmic scale. – whuber Aug 21 '14 at 1:05
• @whuber understood. I'll amend my question accordingly. – Paul H Aug 21 '14 at 1:08
• @whuber I my only concern is that if I log-transform the data and plot those results, I'll label the y-axis "log of Zinc concentrations" and it'll be absolutely clear what's going on. But if I compute the boxplot values on the logs of the data, and then transform back, the figure could need some heavy caveats to adequately convey the whole process. I'm not sure how even a fairly sophisticated audience would feel about that. Would it open the analysis up to criticism? Is it just so unconventional that it'll detract from the analysis? – Paul H Aug 21 '14 at 4:32
• You show all the data (good idea), so the box plots on your display only act to provide summaries. Drawing whiskers based on 1.5 IQR has minimal extra diagnostic value given all the detail in the tails. For these and other reasons I favour drawing whiskers to selected quantiles (e.g. 1% and 99%). That is easy to explain and marches well with monotonic transforms. Of course, you should always explain what you do. Conversely, it is striking how many researchers use just one of several conventions in preparing box plots, but don't explain which in their reports. – Nick Cox Aug 21 '14 at 8:39
• The specific question here is: should be the 1.5 IQR rule be applied on the original scale or on a transformed scale? As @whuber comments, you should apply that rule on the scale used to draw the box plots. If a transformation seems natural or appropriate, calculations should be on that scale. Above all, don't mix scales (e.g. calculate whiskers based on 1.5 IQR on the raw scale, then log transform to get a new graph). – Nick Cox Aug 21 '14 at 17:17