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I see that all my understanding of statistics (& so of probability which is a branch of statistics) from high school came from the mathematics textbook and it all appears too mathematical to be accepted as a branch of mathematics but why then it isn't considered a branch of mathematics?

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The first two answers I've got are contradicting each other one is claiming that it (statistic) falls under the domain of measure theory which is a branch of mathematics. So, it's entirely mathematical. The other one is saying that they are different.

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  • $\begingroup$ Maybe because it is a combination of combinatorics/probability theory and other things that come from other branches of mathematics. It then teaches how to apply this in 'statistics', which no longer really makes it a math of its own. Pun intended. $\endgroup$ – Simply Beautiful Art Oct 16 '16 at 1:43
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    $\begingroup$ It sounds to me like what people are trying to say is this: Probability theory is a mathematical subject, and statistics is the application of the mathematical subject to real world experiments and polls. Statistics, therefore, places itself outside the realm of pure mathematics and into the realm of applied mathematics. I'm not sure I buy that definition - I just thought that maybe other people's thoughts here weren't articulated clearly enough, so I figured I'd rephrase it. $\endgroup$ – Dustan Levenstein Oct 16 '16 at 2:02
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    $\begingroup$ FWIW, the meaning of the words that I know is what @DustanLevenstein explained. In particular, probability is definitely not "a branch of statistics", statistics is not "a branch of measure theory" and is certainly not "entirely mathematical". $\endgroup$ – Did Oct 16 '16 at 15:02
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You won't get any consensus in answers here because probability and statistics can be both theoretical and applied. Here, roughly speaking, is how I think about these things; I'm posting this to clarify the confusion mentioned in the comments and in the edited question, not to answer the historical question of why statistics departments are often separate from math departments. (Note, however, that MIT, for example, has no statistics department. All probability and statistics courses are in the math department, or done within the various scientific or engineering departments.)

Theoretical statistics (also called mathematical statistics) -- e.g. what can be found in the book by Schervish -- is probability theory applied to particular theoretical problems: inference and estimation. You could construe this as a branch of probability theory, but in practice the theory of statistical inference usually follows more foundational courses in probability theory and the theory of stochastic processes. Theoretical statistics deals with the sampling distributions, interval estimation, hypothesis testing, alternative modes of inference (Bayesian, non-parametric), etc.

Statistics also has an applied component, of course. Applied statisticians work with actual data sets and use the theory done by theoretical statisticians to draw conclusions about particular empirical problems. Theoretical statisticians hardly ever look at data; they prove mathematical theorems.

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Statistics is a branch of measure theory and it is entirely mathematical. Introductory statistics and probability are usually taught heuristically because the theorems and proofs of statistics are often more sophisticated than the average lower level undergrad student can handle comfortably.

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Just sharing the story from my teacher.

He said that long ago, math and stats were different apart. Stats was just chart, mean, median, modus, standard deviation, and some other methods to simplify and generalize governmental numbers, and considered as a branch of governmental science.

After that, Blaise Pascal started the earliest Probability Theory, a branch of math. But still, math and stats were different.

However, people started to look for some arguments that would make statistical methods reasonable scientifically. And since math is the science about truth, reason, and logics (as long as we assume the postulates we need are true), and because drawing sample is actually a random experiment (in prob.theory point of view). Then, Mathematical Statistics born as the 'bridge' to connect stats and math (especially Probability Theory).

That is why, stats is not considered as a brach of math, but Mathematical Statistics is.

I don't know the validity of this story. This is just a sharing.

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  • $\begingroup$ After that, Blaise Pascal started... That's a funny story for sure. Can only wonder what that time was before Blaise Pascal when stats were considered a branch of governmental science. $\endgroup$ – dxiv Oct 16 '16 at 1:56
  • $\begingroup$ hahaha, well, that is why I wrote I don't know about the validity :-D but I like this story, and right, it is funny $\endgroup$ – Rizky Reza Fujisaki Oct 16 '16 at 2:01
  • $\begingroup$ So, probability theory lies under the domain of mathematical statistic? $\endgroup$ – ankit Oct 16 '16 at 2:22
  • $\begingroup$ based on those story, mathematical stats lies under prob theory $\endgroup$ – Rizky Reza Fujisaki Oct 16 '16 at 4:06

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