The interquartile range is defined as the difference between the third and first quartiles:
$IQR = Q_3 - Q_1$
When studying intensive and extensive properties, only values of extensive properties can be summed, but the differences of values of intensive (or extensive) properties can usually be summed together. For example, if the outside air temperature rose by 2°C from day 1 to day 2, and by 3°C from day 2 to day 3, it rose by 5°C from day 1 to day 3.
For quartiles (or quantiles in general), this is not the case: two interquartile ranges cannot be summed together. Obviously, quantiles are summary statistics on observed variables, not directly-observed variables. Nevertheless, what kind of algebraic proof can we give to justify the fact that two interquartile ranges cannot be summed?
My intuition is that the growth of a variable across space or time is structurally different from the deltas between quantiles across a population, but I would really like to find a more robust justification for this intuition, backed by standard equations.
And to be clear, I make a distinction between expressions that are arithmetically valid but might make no sense from a statistical analysis standpoint, and expressions that not only make no statistical sense, but are first and foremost arithmetically invalid.
For example, on one hand, computing the mean of two IQRs is arithmetically valid, but whether it makes sense or not from a statistical analysis standpoint is probably open to debate. On the other hand, computing the sum of two IQRs not only makes no statistical sense, but it's also probably arithmetically invalid (this is the part that I would like to firmly establish).
How can one know that something probably makes no sense? Simply by the fact that nobody ever came up with a valid example for which it would make sense.
Now, it is obvious that the IQR is not additive, in the sense that in most cases:
$IQR(X + Y) \neq IQR(X) + IQR(Y)$
But this does not establish that two IQRs should never be summed.
Summability does not imply additivity (while additivity implies summability of course).
Or does it? In other words, can we find a summary statistic $S$ for which:
$S(X + Y) \neq S(X) + S(Y)$
$S(X) + S(Y)$ is arithmetically correct and makes statistical sense?
If the answer to this last question is negative, then we have a proper empirical definition of summability for summary statistics. If the answer is positive, then we do not, and the summability (not the additivity) of each summary statistic will need to be studied individually.