# Stats is not maths?

Is stats maths or not?

Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.

I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles.

• Compulsory XKCD reference: xkcd.com/435 . Anyway, does it really matter? – nico Dec 4 '13 at 21:17
• (i) How would we quantify such things? It's not like it's been the subject of a survey! (ii) The calculations almost always involve numbers, but what makes it statistics, in my mind, is usually not in the calculations. (iii) When I did my undergraduate statistics major, it wasn't in the mathematics department. The place I did my PhD at - under two fairly well known statisticians - wasn't a maths department either. (iv) I don't think it's a joke. It relates to a very important idea - that what makes statistics "statistics" is more about a way of reasoning about particular types of problems. – Glen_b Dec 4 '13 at 21:36
• I feel obligated to give a short answer, as I am former pure mathematician (PhD and 3.5 years of postdoc in some kind algebra), and now an applied statistician... well, the kind of stats you learn for applied stats, like "when do I use a $t$-test" or what not, for a mathematician, looks like a recipe book, not like maths. But, for example, van der Vaart’s Asymptotic Statistics is definitely a math book... There are plenty of intermediate levels – some of them not well populated, I think there are not enough books explaining stats with lots of real examples and all the mathematical details. – Elvis Dec 4 '13 at 22:19
• I don't know what to make of the statement, "the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles", I'm not even sure if it can really even be right or wrong. It mostly seems to proceed from confused premises. – gung Dec 4 '13 at 22:25
• @Guy By analogy, we could characterize chemistry (another "mathematical discipline") as asymptotic distribution theory and C* algebras. Doing so is nominally accurate but so completely misses the essence of what chemistry is and its aims that no chemist would recognize it. Similarly, contrast your characterization with what leading professional societies say statistics is: they are worlds apart. "The science of learning from data, and of measuring, controlling, and communicating uncertainty." Not one mention of probability there. – whuber Dec 5 '13 at 7:52

Mathematics deals with idealized abstractions that (almost always) have absolute solutions, or the fact that no such solution exists can generally be described fully. It is the science of discovering complex but necessary consequences from simple axioms.

Statistics uses math, but it is not math. It's educated guesswork. It's gambling.

Statistics does not deal with idealized abstractions (although it does use some as tools), it deals with real world phenomena. Statistical tools often make simplifying assumptions to reduce the messy real world data to something that fits into the problem domain of a solved mathematical abstraction. This allows us to make educated guesses, but that's really all that statistics is: the art of making very well informed guesses.

Consider hypothesis testing with p-values. Let's say we are testing some hypothesis with significance $\alpha = 0.01$, and after gathering data we find a p-value of $0.001$. So we reject the null hypothesis in favor of an alternative hypothesis.

But what is this p-value really? What is the significance? Our test statistic was developed such that it conformed to a particular distribution, probably student's t. Under the null hypothesis, the percentile of our observed test statistic is the p-value. In other words, the p-value gives the probability that we would get a value as far from the expectation of the distribution (or farther) as the observed test statistic. The signficance level is a fairly arbitrary rule-of-thumb cutoff: setting it to $0.01$ is equivalent to saying, "it's acceptable if 1 in 100 repetitions of this experiment suggest that we reject the null, even if the null is in fact true."

The p-value gives us the probability that we observe the data at hand given that the null is true (or rather, getting a bit more technical, that we observe data under the null hypothesis that gives us at least as extreme a value of the tested statistic as that which we found). If we're going to reject the null, then we want this probability to be small, to approach zero. In our specific example, we found that the probability of observing the data we gathered if the null hypothesis were true was just $0.1\%$, so we rejected the null. This was an educated guess. We never really know for sure that the null hypothesis is false using these methods, we just develop a measurement for how strongly our evidence supports the alternative.

Did we use math to calculate the p-value? Sure. But math did not give us our conclusion. Based on the evidence, we formed an educated opinion, but it's still a gamble. We've found these tools to be extremely effective over the last 100 years, but the people of the future may wonder in horror at the fragility of our methods.

• The p-value is not the probability that we are wrong when we reject the null hypothesis, as that also depends on H1 which does not enter into the calculation of the p-value (well illustrated by i.stack.imgur.com/tStr4.png - the probability that H0 is wrong and that the sun has exploded is rather less than p = 1/36). – Dikran Marsupial Dec 6 '13 at 9:05
• Could you suggest a better simple language interpretation of the p-value? "The probability that we observe the data at hand given the null is true" perhaps? I've already delved much deeper in the p-value example than I was intending to. My intention was to make a point about statistics, not provide a tutorial on interpreting p-values. I don't want to get too derailed. Thanks for pointing that out, in any event. – David Marx Dec 6 '13 at 17:29
• The p-value is the probability of a result at least as extreme as that observed if the null hypothesis is true. The point that the link between the plausibility of the null hypothesis and the p-value being largely subjective, rather than a logical necessity, is a good point though (+1). I have been wondering lately whether frequentist hypothesis testing is any less subjective than the Bayesian approach, where at least the subjectivity is made more explicit. – Dikran Marsupial Dec 6 '13 at 18:21
• It's not clear to me how your p-value interpretation/definition differs from the alternative I offered in my last comment. There's certainly a degree of subjectivity in frequentist hypothesis testing, but it's the same kind of subjectivity that gets invoked when interpreting a Bayes Factor. And it's not like significance level isn't communicated (i.e. the subjectivity is made explicit here too), it's just often chosen based on convention, whereas there's usually more thought put into choosing (informative) Bayesian priors. – David Marx Dec 7 '13 at 3:43
• @David: The "at least as extreme" makes a great difference - the probability of the observed value under the null is not in general the p-value, even for discrete test statistics where it makes sense. I know it's tangential to the point you were making, but if Wikipedia can get it right, we should be able to on Cross Validated. – Scortchi Dec 7 '13 at 18:53

Tongue firmly in cheek:

Einstein apparently wrote

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

so statistics is the branch of maths that describes reality. ;o)

I'd say statistics is a branch of mathematics in the same way that logic is a branch of mathematics. It certainly includes an element of philosophy, but I don't think it is the only branch of mathematics where that is the case (see e.g. Morris Kline, "Mathematics - The Loss of Certainty", Oxford University Press, 1980).

• Is Logic a branch of Mathematics? Including three-valued logics & modal logics, or just first-order predicate calculus? Are all formal sciences somehow Mathematics? – Scortchi Dec 6 '13 at 17:33
• I would view the study of any system for manipulating symbols according to a set of rules (e.g. formal languages) to be a variety of mathematics, so yes, I suppose I probably would. The trouble with labels is that they are not always fully descriptive of everything to which they are applied (I wouldn't say I was exactly a mathematician, a statistician or a computer scientist, but I have some aspects of all three). Similarly the same thing can often be placed in more than one hierarchy, so perhaps there isn't a unique solution to the question! – Dikran Marsupial Dec 6 '13 at 19:10
• By your argument statistics, as a description of reality, comprises geometry and quantum field theory, too, but it does not include hypothesis testing (because most hypotheses are contra-factual--they are intended to be falsified--and therefore plainly do not "describe reality"). – whuber Dec 6 '13 at 20:51
• The Einstein quote was the tongue in cheek bit, and wasn't meant to be taken seriously; I'm pretty sure it isn't quite what Einstein actually had in mind! – Dikran Marsupial Dec 6 '13 at 21:31

Well if you say that "something like statistics, where you can't build everything on basic axioms" then you should probably read about Kolmogorov's axiomatic theory of probability. Kolmogorov defines probability in an abstract and axiomatic way as you can see in this pdf on page 42 or here at the bottom of page 1 and next pages.

Just to give you a flavour of his abstract definitions, he defines a random variable as a 'measurable' function as explained in a more 'intuitive' way here : If a random variable is a function, then how do we define a function of a random variable

With a very limited number of axioms and using results from (again maths) measure theory he can define concepts are random variables, distributions, conditional probability, ... in an abstract way and derive all well known results like the law of large numbers, ... from this set of axioms. I advise you to give it a try and you will be surprised about the mathematical beauty of it.

For an explanation on p-values I refer to: Misunderstanding a P-value?

• Isn't there still an important distinction, though, between Probability Theory (Maths) & its application to problems of inference (Statistics)? The Bayesian & frequentist approaches show the same mathematical apparatus (typically, or almost) used with quite different concepts of probability. – Scortchi Sep 18 '15 at 11:05
• @Scortchi: I am not sure whether the concepts of probability are different for frequentists and Bayesians; see stats.stackexchange.com/questions/230415/… – user83346 Aug 23 '16 at 6:13
• I don't see any disagreement between my comment & your answer to Is there any mathematical basis for the Bayesian vs frequentist debate?. By "mathematical apparatus" I mean what follows from Kolmogorov's axioms; by "concepts" I mean the interpretations as limiting frequency, degree of belief, &c. – Scortchi Aug 24 '16 at 8:58

I have no rigorous or philosophical basis for answering this, but I've heard the "stats is not math" complaint often from people, usually physics types. I think people want guarantees certainty from their math, and statistics (usually) offers only probabilistic conclusions with associated p values. Actually, this is exactly what I love about stats. We live in a fundamentally uncertain world, and we do the best we can to understand it. And we do a great job, all things considered.

Maybe its because I'm a plebe and haven't taken any advanced mathematical courses, but I don't see why statistics isn't mathematics. The arguments here and on a duplicate question seem to argue two primary points as to why statistics isn't mathematics*.

1. It isn't exact/certain, and as such relies on assumptions.
2. It applies math to problems and anytime you apply math it is no longer math.

### Isn't exact and uses assumptions

Assumptions/approximations are useful for lots of math.

The properties of a triangle that I learned about in grade school I believe are considered true math, even though they don't hold true in non-Elucidean geometry. So clearly an admission of the limits, or stated another way "assuming XYZ the following is valid", to a branch of math doesn't disqualify the branch from being "true" math.

Calculus I'm certain would be considered a pure form of math, but limits are the core tool we built it on. We can keep calculating up to the limit, just as we can keep making a sample size larger, but neither give increased insight past a certain threshold.

### Once you apply math it isn't math

The obvious contradiction here is we use math to prove mathematical theorems, and no one argues that proving mathematical theorems isn't math.

The next statement might be that thing x isn't math if you use math to get a result. That doesn't make any sense either.

The statement I would agree with is that when you use the results of a calculation to make a decision then the decision isn't math. That doesn't mean that the analysis leading up to the decision isn't math.

I think when we use statistical analysis all the math performed is real math. It is only once we hand the results to someone for interpretation does statistics exit mathematics. As such statistics and statisticians are doing real mathematics and are real mathematicians. It is the interpretation done by the business and/or the translation of the results to the business by the statistician that isn't math.

### From the comments:

whuber said:

If you were to replace "statistics" by "chemistry," "economics," "engineering," or any other field that employs mathematics (such as home economics), it appears none of your argument would change.

I think the key difference between "chemistry", "engineering", and "balancing my checkbook" is that those fields just use existing mathematical concepts. It is my understanding that statisticians like Guass expanded the body of mathematical concepts. I believe (this might be blatantly wrong) that in order to earn a PhD in statistics you have to contribute, in some way, to expanding the body of mathematical concepts. Chemistry/Engineering PhD candidates don't have that requirement to my knowledge.

The distinction that statistics contributes to the body of mathematical concepts is what sets it apart from the other fields that merely use mathematical concepts.

*: The notable exception is this answer that effectively states the boundaries are artificial due to various social reasons. I think that is the only true answer, but where is the fun in that? ;)

• If you were to replace "statistics" by "chemistry," "economics," "engineering," or any other field that employs mathematics (such as home economics), it appears none of your argument would change. As such it seems to be without any substance. – whuber Sep 17 '15 at 4:52
• Statistics PhDs do not have to "contribute to the body of mathematical concepts." Most stats PhDs are awarded for contributions to statistical methodology and statistical theory. (Few mathematicians, if any, pay attention to the statistical literature. It just isn't a good source of new or fruitful mathematical ideas in general. I am not referring to literature in probability theory here.) Moreover, chemists, engineers, physicists, etc. often do create (or, usually, re-create) mathematical ideas in their work; that does not automatically turn their fields into branches of mathematics. – whuber Sep 17 '15 at 17:56
• @whuber That is very interesting. It appears as if I don't have a leg to stand on. – Erik Sep 17 '15 at 18:20
• For the record, I have not downvoted your contribution. This is a sensitive topic for many--for example, many college math departments are still trying to treat statisticians as mathematicians, to the detriment of both--and so it's likely to elicit some strong reactions. – whuber Sep 17 '15 at 20:16
• @whuber I'm tough enough to stand a few down-votes regardless. :) I think you were respectful at all times, so don't worry about that. Besides voting is anonymous for a reason. No need to go on the record. – Erik Sep 17 '15 at 20:26

Statistical tests, models, and inference tools are formulated in the language of mathematics, and statisticians have mathematically proven thick books of very important and interesting results about them. In many cases, the proofs provide compelling evidence that the statistical tools in question are reliable and/or powerful.

Statistics and its community may not be "pure" enough for mathematicians of a certain taste, but it is definitely invested in math extremely deeply, and theoretical statistics is just as much a branch of mathematics as theoretical physics or theoretical computer science.

• Hi Paul, as you say, stats is full of nice theorems and proofs (+1), there is even an axiomatic theory of probability, developed by Kolmogorov, as I explain in my answer. – user83346 Sep 17 '15 at 16:43

The "difference" relies on: Inductive reasoning vs. Deductive reasoning vs. Inference. For instance, no mathematical theorem can tell what distribution or prior you can use for your data/model.

By the way, Bayesian statistics is an axiomatised area.

• Mathematics need inductive reasonning too... – Elvis Dec 5 '13 at 15:28
• @Elvis Yes, that's why my example ... I am sure you know there is no general answer to this question ... I have edited the answer, for your pleasure ... – Compay Segundo Dec 5 '13 at 15:31
• I really don't get your point. – Elvis Dec 5 '13 at 17:00
• @CompaySegundo: I am not sure you have a valid point here, at least, it's not clearly stated. – Quora Feans Dec 5 '13 at 18:16
• @QuoraFea Probably I am just too drunk ... – Compay Segundo Dec 5 '13 at 19:08

This may be a very unpopular opinion, but given the history and formulation of concepts of statistics (and probability theory), I consider statistics to be a subbranch of physics.

Indeed, Gauss initially formalized the least squares regression model in astronomical predictions. The majority of contributions to statistics before Fisher were from Physicists (or highly applied mathematicians whose work would be called Physics by today's standards): Lyapunov, De Moivre, Gauss, and one or more of the Bernoullis.

The overarching principle is the characterization of errors and seeming randomness propagated from an infinite number of unmeasured sources of variation. As experiments became harder to control, experimental errors needed to be formally described and accounted for to calibrate the preponderance of experimental evidence against the proposed mathematical model. Later, as particle physics delved into quantum physics, formalizing particles as random distributions gave a much more concise language to describe the seemingly uncontrollable randomness with photons and electrons.

The properties of estimators such as their mean (center of mass) and standard deviation (second moment of deviations) are very intuitive to physicists. The majority of limit theorems can be loosely connected to Murphy's law, i.e. that the limiting normal distribution is maximum entropy.

So statistics is a subbranch of physics.

• This thesis is as implausible as it is illogical. As Stephen Stigler points out in his books, psychologists, economists, and most other social scientists did not adopt the physicists' methods for up to another century due to real doubts about their applicability and their interpretation. That is prima facie evidence that statistics is far more than a branch of physics. Other disciplines, ranging from engineering through biology, also employ physical methods and physical theories, but that does not make them branches of physics either--at least not in any meaningful or insightful way. – whuber Dec 6 '13 at 20:47
• Didn't the Bernoulli's interest in probability stem from gambling rather than physics? – Dikran Marsupial Dec 6 '13 at 21:36
• @whuber As with my field, biostatistics, I'm keenly aware that these applied sciences existed in various forms prior to their distinct identification as a field of science. I believe these fields, though, were formally preceded by the field of statistics itself. This of course is not the case for physics. The one central theme in these applied sciences the formulation of a process as a model relating some predictor to a response. Perhaps the language of statistics was in part born out of the need to generalize such concepts as to apply to these fields. – AdamO Dec 6 '13 at 21:59
• You're thinking of Jacobus Bernoulli, posthumous author of ars conjectandi (ed. Nicholaus Bernoulli, 1713). Probably the last people who seemed to be motivated by gambling problems were Pascal and Fermat in 1654, but even then it appears they were using certain gambling problems (the "problem of the points") only as a motivational example and not as the focus of their investigation. (Modern scholarship actually traces the problem of the points to Islamic contract law c. 1200.) The last mathematician of note who truly was motivated by gambling probably was Cardano (1501-1576). – whuber Dec 6 '13 at 22:08
• Diaconis the magician? I wouldn't conflate gambling with showmanship! You have a point, but you could push back a little better by suggesting that many "investors" are actually gamblers, whence many theoreticians in mathematical finance might truly be motivated by that form of gambling. Just a thought... Anyway, it is clear that by the time Huygens published his little treatise in 1657 that people were creating a theory of probability (and statistics) for reasons much more profound and far-reaching than doing better at the gambling tables. – whuber Dec 6 '13 at 22:18