I'm developing a tool to conduct expert elicitations of scientists. As answers to questions are speculative, we like experts to help us understand the range of answers they find plausible, to get some idea of uncertainty.

For one type of question, we have them input a lower-bound, upper-bound, 25th percentile, 50th percentile and median (essentially the "5 number summary").

In testing we found that some scientists find this data intuitive, and others find it easier to visualize in the form of a PDF. I'm trying to figure out a straightforward algorithm to provide a PDF-style visualization of this "5 number summary". I've considered a "Skew Normal" distribution, would this be a good choice?

Put another way: given a quartiles+median, what's the most straightforward function to render a PDF for a distribution with those characteristics?

Any suggestions? I know this is sloppy at best, but the answers to these questions are speculative themselves, and we'll get better answers if we can communicate both ways.

  • $\begingroup$ Take a look at this paper: ba.stat.cmu.edu/journal/2007/vol02/issue01/… $\endgroup$
    – Zen
    Apr 10, 2012 at 2:28
  • 1
    $\begingroup$ A normal distribution does not have lower or upper bounds. The simplest solution would be to have a PDF made up of four uniform distributions but you may think it does not illustrate the point. $\endgroup$
    – Henry
    Apr 10, 2012 at 6:55
  • $\begingroup$ I second Henry's suggestion: construct a histogram. If you want the histogram to appear more "pdf-ish" (i.e., continuous over the support) then use triangles on the edges and trapezoids in the middle (just preserve total volume). $\endgroup$
    – jmtroos
    Apr 10, 2012 at 16:29
  • $\begingroup$ @Zen I couldn't get the link to work $\endgroup$
    – Seth
    Apr 16, 2012 at 19:58

1 Answer 1


Choosing the family of distributions is not straightforward, and strongly depends on a-priori knowledge and assumptions. Is the value always positive? is it restricted to a range? is it bimodal? etc You could consider using several families, all of which give a pdf with the given properties, and showing the variety of underlying possibilities.

For the technical aspect, an implementation for a few families can be found at John D Cook's blog, who also considers this question for the purposes of eliciting priors for Bayesian analysis.

  • $\begingroup$ Since we're presenting this as "feedback" in a UI, we can set the assumptions we like, and perhaps have an "outlet valve" where they can explain how the PDF misses their intuition. Cook's blog is very helpful and pragmatic, thank you for directing me toward that! $\endgroup$
    – Seth
    Apr 16, 2012 at 20:01

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