What's the difference of mathematical statistics and statistics?

I've read this:

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.

And this:

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

So what would be the difference beetween them? I can understand that the processes of collection may not be mathematical, but I guess that organization, analysis and interpretation are, am I missing something?

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    $\begingroup$ Slightly tongue-in-cheek (and to steal a [modified] line from someone else): I would say that a mathematical statistician is someone that the mathematicians consider a statistician and that the statisticians consider a mathematician. $\endgroup$ – cardinal Jul 10 '12 at 16:24
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    $\begingroup$ (+1) @cardinal amazing but not wrong :) $\endgroup$ – Stéphane Laurent Jul 10 '12 at 17:29

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.

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    $\begingroup$ I'd like to think of myself as the kind of statistician who has a problem of originating with (1), goes on to play around with (2) to find a way to solve it and then uses (3) to show that the solution is valid. :) $\endgroup$ – MånsT Jul 10 '12 at 12:11
  • $\begingroup$ @MånsT: yea, that should be something like "those whose research question emanate from..." $\endgroup$ – user603 Jul 11 '12 at 11:04
  • $\begingroup$ Very nice answer!!! $\endgroup$ – Jase Dec 14 '12 at 10:38
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    $\begingroup$ I do not understand this answer: is it a joke?! $\endgroup$ – Xi'an Nov 18 '15 at 17:46
  • $\begingroup$ what about type (2)? $\endgroup$ – Haitao Du May 5 '17 at 14:37

Mathematical statistics concentrates on theorems and proofs and mathematical rigor, like other branches of math. It tends to be studied in math departments, and mathematical statisticians often try to derive new theorems.

"Statistics" includes mathematical statistics, but the other parts of the field tend to concentrate on more practical problems of data analysis and so on.

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    $\begingroup$ This is a good answer (+1) but I'm not sure I agree with the statement "It tends to be studied in math departments". In my department (a stat dept) in grad school there was a lot of mathematical statistics research. In the math department there was plenty of probability/analysis research being done but none that I would call "mathematical statistics". Perhaps my University was not the norm. $\endgroup$ – Macro Jul 10 '12 at 11:18
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    $\begingroup$ (+1 to Peter and Macro): It doesn't sound like your university is outside the norm, @Macro. There are also plenty of people outside these two departments that engage in aspects of mathematical statistics research, including in various engineering, economics, finance, computer science, genetics and medicine depts. $\endgroup$ – cardinal Jul 10 '12 at 11:38
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    $\begingroup$ @macro, Some universities have separate statistics departments, some don't. But, even in the ones that do, mathematical statistics looks like math. $\endgroup$ – Peter Flom Jul 11 '12 at 10:53
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    $\begingroup$ @Peter, I wasn't saying whether or not it looked like math. I was just saying that, in my experience, "It tends to be studied in math departments" is not the case, and it appears Cardinal has a similar impression. $\endgroup$ – Macro Jul 11 '12 at 14:33
  • $\begingroup$ @macro I wasn't trying to rebut you, I was just adding another point $\endgroup$ – Peter Flom Jul 11 '12 at 20:22

The boundaries are always very blurry but I would say that mathematical statistics is more focused on the mathematical foundations of statistics, whereas statistics in general is more driven by the data and its analysis.

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There is no difference. The science of Statistics as it is taught in academic institutions throughout the world is basically short for "Mathematical Statistics". This is divided into "Applied (mathematical) Statistics" and "Theoretical (mathematical) Statistics". In both cases, Statistics is a subfield of math (or applied math if you will) while all its principles and theorems are derived from pure math.

"Non-mathematical" statistics, for lack of a better term, would be (for me) something like the percentage of ball possession of a football team after a game, i.e. the act to register and report some real-world statistic(s).

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  • $\begingroup$ It takes a lot of evidence to be clear on what is true throughout the world: but I just note that I frequently see distinctions between mathematical or theoretical statistics on the one hand and applied statistics on the other. To regard applied statistics as a subset of mathematical statistics just doesn't match the way the terms are used in my experience. $\endgroup$ – Nick Cox Nov 18 '15 at 18:21
  • $\begingroup$ I also would deny that all the principles of statistics come from pure mathematics. The argument can be protected by defining principle in a sufficiently narrow way, but many principles such as strategy for model building are also based on empirical evidence or other imperatives. Evidence on how procedures work with real data influences how they are (recommended to be) used, which is not directly deducible from pure mathematics. $\endgroup$ – Nick Cox Nov 18 '15 at 18:21
  • $\begingroup$ Applied Statistics has no common definition among academic institutions and methods of teaching can differ significantly. However, in all occasions it consists strictly of applying of the mathematical principles established in theoretical statistics. This doesn't mean that the person learning/applying those methods is necessarily a mathematician (or even a Statistician in some cases). But it also doesn't make the scientific discipline any less mathematical. $\endgroup$ – Digio Nov 19 '15 at 8:49
  • $\begingroup$ I would have to differ on "strictly". My own namesake Sir David Cox has written books with titles like theoretical statistics and applied statistics. Much of the content of the latter is not deducible from the former. Your comment doesn't really address my earlier points. $\endgroup$ – Nick Cox Nov 19 '15 at 8:55
  • $\begingroup$ To address your earlier points, it all depends on someone's perception of where does Statistics start and end. We could argue forever on where does Statistics end and machine learning or data analysis begin, but I think we all agree on where does Statistics begin, and it’s with pure math. In that sense, mathematical Statistics is for me synonymous to "core Statistics", which can be focused on theory or application. Empirical methods such as model-building that you perceive as ‘applied statistics’ are, for me, part of ‘data analysis’ or ‘data science’ and not Statistics per se. $\endgroup$ – Digio Nov 19 '15 at 10:15

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