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.
Solved – meant by the interquartile range of ordinal data
descriptive statisticsinterquartileordinal-datarange
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From definition, this defines the range witch holds 75-25=50 per cent of all measured values.
: (median-24/2,median+24/2). Median should be written somewhere near this IQR.
The above was false of course, it seems I was still sleeping when writing this; sorry for confusion. It is true that IQR is width of a range which holds 50% of data, but it is not centered in median -- one needs to know both Q1 and Q3 to localize this range.
In general IQR can be seen as a nonparametric (=when we don't assume that the distribution is Gaussian) equivalent to standard deviation -- both measure spread of the data. (Equivalent not equal, for SD, (mean-$\sigma$,mean+$\sigma$) holds 68.2% of perfectly normally distributed data).
EDIT: As for example, this is how it looks on normal data; red lines show $\pm 1\sigma$, the range showed by the box on box plot shows IQR, the histogram shows the data itself:
you can see both show spread pretty good; $\pm 1\sigma$ range holds 68.3% of data (as expected). Now for non-normal data
the SD spread is widened due to long, asymmetric tail and $\pm 1\sigma$ holds 90.5% of data! (IQR holds 50% in both cases by definition)
If you want to describe the difference in distribution, perhaps something like a Q-Q plot, or a pair of kernel densities would be common tools, though I assume you have quite discrete looking distributions, in which case that may make something like a pair of barcharts/histograms or a pair of ECDFs better (if possible on a single display)
If you do go with looking at the difference in ECDF, the two-sample Kolmogorov-Smirnov statistic would be the ideal "measure" of significance to go with that, since it's based on the biggest (vertical) difference in ECDF.
If you compare barplots(/histograms) for a discrete distribution, you might like to look at the kinds of measures of discrepancy indicated by smooth tests of goodness of fit (Neyman-Barton type tests), but where the statistic is adapted to two samples. This would in effect correspond to partitioning a chi-square into components associated with orthogonal polynomials, the first of which would correspond to location (a linear term), the second to variance (a quadratic term), and so on (typically you'd look out to order 4 or 6). Books and numerous articles by Rayner and Best (and some others) cover some of that territory.
Best Answer
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).