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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?

Moshe
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Red Banana
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    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. – cardinal Jul 10 '12 at 16:24
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    (+1) @cardinal amazing but not wrong :) – Stéphane Laurent Jul 10 '12 at 17:29

4 Answers4

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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.

user603
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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.

Peter Flom
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    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. – Macro Jul 10 '12 at 11:18
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    (+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. – cardinal Jul 10 '12 at 11:38
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    @macro, Some universities have separate statistics departments, some don't. But, even in the ones that do, mathematical statistics looks like math. – Peter Flom Jul 11 '12 at 10:53
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    @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. – Macro Jul 11 '12 at 14:33
  • @macro I wasn't trying to rebut you, I was just adding another point – Peter Flom Jul 11 '12 at 20:22
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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.

Itamar
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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).

Digio
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  • 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. – Nick Cox Nov 18 '15 at 18:21
  • 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. – Nick Cox Nov 18 '15 at 18:21
  • 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. – Digio Nov 19 '15 at 08:49
  • 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. – Nick Cox Nov 19 '15 at 08:55
  • 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. – Digio Nov 19 '15 at 10:15
  • A source, which shares my view would be ‘Mathematical statistics and data analysis’ by J. A. Rice. Also, I don’t see how D. R. Cox’s use of the term “applied statistics” is in conflict with what I’m saying. The thing is that, academically speaking, there’s not such thing as “applied statistics” as a discipline of its own. It’s meant as a “specialisation” for someone who’s already fluent in the theory of mathematical statistics. The fact that some of the empirical methods of applied statistics can be used by people who are not statisticians in the context of data analysis does not change that. – Digio Nov 19 '15 at 10:30
  • Neither of us is going to persuade the other to change their mind, but the discussion might have some small interest for others. But, FWIW, I have read all three editions of Rice's book and am even acknowledged in it, and I disagree again. Indeed the conjunction in his title Mathematical statistics and data analysis to me manifestly does not imply that data analysis is a subset of mathematical statistics, nor do the contents support that view. But I did not state, do not imply and have never thought that applied statistics is a discipline on its own, so there is no disagreement there. – Nick Cox Nov 19 '15 at 10:35
  • I trust you are not implying, by the way, that those formally trained in mathematical statistics are somehow intrinsically superior to those who are not. I am certainly not a statistician in any formal or certificated sense, so these hints at class distinctions can be sensitive for me, and more importantly others too. A background in some science can also inform data analysis in a way that pure mathematics does not. – Nick Cox Nov 19 '15 at 10:41
  • By citing Rice I was meaning to support my claim that generic/core statistics (first half of the book) can be synonymous to ‘mathematical statistics’, which is what this question was about in the first place. The book also implies that empirical data analysis is not synonymous to applied statistics. This is not meant to be pejorative in any way. A biologist who uses hypothesis testing to prove the efficacy of a certain medicine is not a statistician (applied or otherwise) and has no interest in being one. This doesn’t change the fact that the methods he uses are mathematically valid. – Digio Nov 19 '15 at 10:58
  • "but many principles such as strategy for model building are also based on empirical evidence or other imperatives" <- BTW, that is not true. What statisticians call "methods for model-building" has a very deep theoretical background in things such as computational complexity theory and algorithm theory, both studied in discrete mathematics and logic (the latter being a subfield of pure math)... – Digio Apr 04 '16 at 10:29
  • I just don't recognise the picture of statistics you paint and find your assertions implausible and unsubstantiated. For example, how do generalised linear models, or any other standard tool you care to use as an example, arise out of, or are even illuminated by, computational complexity theory? I am content to let my earlier comments. stand as expression of dissent and puzzlement. – Nick Cox Apr 04 '16 at 10:38
  • Let's take forward regression and variable selection as an example, the aspect of linear regression that most statisticians would view as empirical methods. Variable selection is mathematically a NP-hard combinatorial problem (can be reduced to the max-cut problem), which a deterministic algorithm would solve in exponential time. Forward selection and backward elimination are nothing more than a sort of generilised Hill Climbing, a local search heuristic "solving" a NP-hard problem in polynomial time. This makes little sense in the world of statistics, but it's not an empirical approach. – Digio Apr 04 '16 at 12:49
  • That helps me to see what you're getting at, but nevertheless it leaves what I consider my most important single point (second comment above) quite untouched. I don't see that the relevance and indeed importance of these computing issues undermines that in any sense. Indeed, these examples make your initial emphasis on pure mathematics as a basis for statistics seem more peculiar. Sure, applied mathematics depends on pure mathematics, but I don't think of computational complexity theory as pure mathematics. If you want to argue back, I am happy to let you have the last word. – Nick Cox Apr 04 '16 at 13:04
  • My point all along has been that every aspect of statistics is mathematical in nature, be it pure, applied, discrete, computational, etc. To be honest, it's not important. Thank you for replying to an old discussion just for the sake of arguing. – Digio Apr 04 '16 at 13:33
  • It is important to me (not a matter of life or death, but still important) that statements here are clear and correct. Thanks for your summary. So long as any kind of consideration of data and their analysis is included in your very broad definition of mathematics, then I can agree. – Nick Cox Apr 04 '16 at 13:39