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A colleague brought to my attention documentation for a standardised questionnaire which recommended using the median and interquartile range to describe summary variables created by combining responses to multiple questions. In most instances these summary variables had 20+ possible values (e.g. the sum of responses to 5 questions each on a 0-4 scale resulting in a 0-20 scale).

Strangely, this advice also applied to single item summaries of yes/no questions - i.e. the new summary variable is just the response to a yes/no question. Clearly, for a binary variable such as this the median + IQR gives much less information than reporting the proportion of 'yes' and 'no' responses. My question is - how many levels would a variable require for the median + IQR to be an informative summary of the central tendency and spread of that variable?

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    $\begingroup$ I believe the answer might depend on what you believe to be "meaningful." In the sense of potentially providing information about the data, the median and IQR are indeed meaningful for binary variables: when the IQR is zero, you know that at least three-quarters of the data equal the median. It is unclear in what sense such information would be "practically meaningless" to you. Perhaps, then, you could edit the question to provide some guidance considering how we are to understand "meaningful"? $\endgroup$
    – whuber ♦
    Sep 10, 2014 at 15:07
  • $\begingroup$ A very good point - I will edit the question and try to avoid the word 'meaningless'. $\endgroup$
    – ChrisP
    Sep 10, 2014 at 15:16

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In general, quantiles are mainly useful for continuous variables. To answer your specific question, the distribution would need to be nearly continuous in the neighborhoods of the quantiles of interest for those quantiles to be useful. Another way of saying this is that if the 0.15-0.35 quantiles are not the same, the 0.25 quantile is likely to be meaningful. You are likely to need more than 20 unique values of the variable to make this happen.

For discrete variables I tend to use the mean as a measure of central tendency but have trouble selecting a dispersion measure other than perhaps Gini's mean difference.

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  • $\begingroup$ You seem to be saying that the decision to report medians+IQR should be led (partly) by whether the discrete variable has sufficient levels that you might consider it continuous it could be considered continuous - is that right? If so, there seems to be a large gulf between variables with few values (2 or 3, so reporting proportions would be sensible) up to discrete variables with 30+ levels for which the median+IQR might be appropriate (although mean + Gini's mean difference is a new alternative for me). What is the best way to describe discrete variables which sit in this middle ground? $\endgroup$
    – ChrisP
    Sep 11, 2014 at 15:53
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    $\begingroup$ A good question. Probably a histogram. Regarding your first question, the discrete variable has to not only have a lot of levels overall but to have a lot of levels in the vicinity of the quantile you are estimating. $\endgroup$
    – Frank Harrell
    Sep 11, 2014 at 18:15

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