Why was not mean included in the five-number summary, when it was first conceived? What was the motivation of choosing sample minimum & maximum, lower & upper quartile and median?


1 Answer 1


The five-number summary was, I believe, introduced by John W. Tukey about 1970. The point was that once you have ordered the data (e.g. using a stem-and-leaf plot), then those summaries could be produced by at most counting and averaging pairs of values. The context was pencil and paper methods for tens or (say) a few hundred values.

Now it is, as we all know, immensely more likely that people have their data on a computer and may even be unused to mechanical arithmetic such as adding numbers and dividing by 2. But there is usually no difficulty in calculating a mean. Whether a mean is a useful summary is open to discussion, but we can always have a look.

The five-number summary idea lives on in the form of box plots. Arguably, box plots have even been oversold, as when box plots without means or SDs are presented as cognate to analysis of variance. More on that in this thread

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    $\begingroup$ You're right, but I think you have omitted to mention a crucial consideration: that the median and hinges ("quartiles") are resistant to outliers. A most basic description of a batch of data will tell us a typical value (median), indicate how much it is spread above and below that value (the hinges), and give us a sense of just how far out the data really do go (the extremes). The mean and standard deviation do not serve the first two purposes for general data exploration because they are too heavily influenced by even one outlier. $\endgroup$
    – whuber
    Commented Apr 24, 2014 at 18:05
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    $\begingroup$ @whuber Quite right; and to think that I've spent a few decades emphasising precisely that in introductory courses, but omitted to mention it this time.... $\endgroup$
    – Nick Cox
    Commented Apr 24, 2014 at 18:18
  • $\begingroup$ @whuber I see it now. It's a matter of resistancy to outlier. Am I right? $\endgroup$
    – hans-t
    Commented Apr 25, 2014 at 4:31

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