# What is meant by the interquartile range of ordinal data?

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.

• 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. 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 $[]$)