It is commonly said that the interquartile range (IQR) is suitable to describe ordinal-, interval- and ratio-level data (one of many examples found on the Internet). But calculating the IQR includes finding the difference between two values, and that requires the interval level of measurement. Likewise for the range. I could understand if the IQR and range were to be reported as "from x to y", but I have not seen that as a definition.

  • $\begingroup$ As long as the data has an ordering you can determine if an observation is at the 25th percentile and the 75th percentile. If no observation is exactly at the required percentile you can pick one of the two that it is between. This of course wouldn't be exact and interpolation is not possible. This provides the endpoints bt cannot determine a length. $\endgroup$ – Michael R. Chernick Apr 8 '18 at 20:18

That is a good observation, so to use interquartile range for ordinal data it certainly must be redefined. If $Q_1, Q_3$ are respectively first and third quartile, so the usual interquartile range is $R=Q_3 - Q_1$, for ordinal data I would define it as the interval $$ R^* = [Q_1, Q_3] $$ (for ordinal data where there are often few unique values, it is important to use the closed interval, including the endpoints, there for $[]$)

(I do not have any references for this definition, but I cannot see some other way to make meaning).

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  • $\begingroup$ Thanks for using the word "interval". I have now been able to find the following reference "Remark 3.2.1 Note that the interquartile range is defined as the interval [x0.25, x0.75] in some literature. However, in line with most of the statistical literature, we define the interquartile range to be a measure of dispersion, i.e. the difference between x0.75 and x0.25." Heumann et al, Introduction to Statistics and Data Analysis. Also, Cumming, Introduction to The New Statistics, defines IQR (and range) as the interval between two values. I now see that interval is not a synonym for distance. $\endgroup$ – jon1000 Apr 9 '18 at 16:32

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