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There are three types of statisticians;

1. those that (prefer to) work with real data,
2. those that (prefer to) work with simulated data,
3. those that (prefer to) work with the symbol $X$.

math stat types would be (3). Typically, type (1) statisticians have some prefix attached to make clear the source of the data they work with (biostatistics, econometrics, psychometrics,....) because these fields have implicit shared assumptions about the data they use and some commonly accepted ordering of the plausibility of these assumptions.