# Quora Why don't I get any dissenting votes?

## Statistics is not math?

Is statistics math or not?

Given that these are numbers that are mostly taught by math departments and that you get math credits for, I wonder if people are only half joking when they say it as they say when they say it , be it a small part of math or just applied math.

I wonder if something like statistics, where you can't base everything on basic axioms, can be viewed as mathematics. For example value, a concept that was created to understand data but is not a logical consequence of more fundamental principles. P

Reply:

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

Statistics uses math, but it's not math. It's an educated guess. It's playing.

Statistics is not concerned with idealized abstractions (although it uses some as tools), but with real world phenomena. Statistical tools often make simplifying assumptions in order to reduce the confusing data of the real world to something that fits into the problem domain of a solved mathematical abstraction. That way we can make educated guesses, but that's actually all that statistics are all about: the art of making educated guesses.

Consider hypothesis tests with p-values. For example, suppose we are testing a hypothesis with a significance of, and after collecting data, we find a p-value of. So we reject the null hypothesis in favor of an alternative hypothesis 0.001α = 0.010.001

But what is this p-value really? What is the meaning? Our test statistic is designed to match a certain distribution, probably a student's t. Under the null hypothesis, the percentile of our observed test statistic is the p-value. In other words, the p-value indicates the probability that we will get a value as far from (or further) away from the expectation of the distribution as the observed test statistic. The significance level is a pretty arbitrary rule of thumb: if you set it to, you can say, "It is acceptable for 1 in 100 iterations of this experiment to indicate that we reject zero, even if the zero is actually true." 0.01

The p-value gives us the probability that we are observing the data at hand, provided that the zero is true (or rather, a bit more technical, that we are observing data under the null hypothesis, which gives us an at least as extreme value as the one tested Statistics than the ones we found). If we reject zero, we want that probability to be small and approach zero. In our particular example, we found that if the null hypothesis is true, the probability of observing the data we are collecting is only true, and we rejected the null. *This was an educated guess. *We never have *really* know for sure that the null hypothesis is wrong, using these methods we are developing a measure of how strongly our evidence supports the alternative. 0.1%

Did we use math to calculate the p-value? For sure. But math didn't give us our conclusion. We formed an educated opinion based on the evidence, but it's still a gamble. We've found these tools to be extremely effective over the past 100 years, but people in the future may be horrified at the fragility of our methods.

Tongue firmly in the cheek:

Apparently Einstein wrote

In so far as the laws of mathematics relate to reality, they are not certain; and as far as they are certain they do not relate to reality.

So statistics is the branch of mathematics that describes reality. ;O)

I would say statistics is a branch of mathematics, just as logic is a branch of mathematics. There is certainly an element of philosophy in it, but I don't think it is the only branch of mathematics in which it does (see, for example, Morris Kline, "Mathematics - The Loss of Certainty," Oxford University Press, 1980).

Well when you say "so *something like statistics, where you can't base everything on basic axioms* "then you should probably read about Kolmogorov's axiomatic probability theory. Kolmogorov defines probability in an abstract and axiomatic way, as you can see in this PDF on page 42 or here below on page 1 and the following pages.

To give you a taste of his abstract definitions, he defines a random variable as a measurable function. This is explained “more intuitively” here: If a random variable is a function, how do we define a function of a? random variable

With a very limited number of axioms and using results from (again mathematical) measure theory, he can abstractly define concepts such as random variables, distributions, conditional probability, etc. and derive all known results such as the law of large numbers. ... from this set of axioms. I advise you to give it a try and you will be amazed at the mathematical beauty.

An explanation of p-values can be found under: Misunderstandings with a p-value?

I don't have a rigorous or philosophical basis to answer this, but I've heard that the "statistics is not math" complaint often comes from people, usually physicists. I think people want certainty about their math, and statistics (usually) only offer probabilistic inferences with associated p-values. That's what I love about statistics. We live in a fundamentally insecure world and do our best to understand it. And we're doing a great job all in all.

Maybe it's because I'm a plebeian and haven't taken advanced math classes, but I don't understand why statistics isn't math. The arguments here and on a double question seem to support two main reasons why statistics are not math ^{*} .

- It is not accurate / safe and so is based on assumptions.
- It applies math to problems and whenever you do math, it's not math anymore.

### Is not accurate and uses assumptions

Assumptions / approximations are useful to many mathematicians.

The properties of a triangle that I learned about in elementary school are, in my opinion, considered real math, even if they don't apply in non-Elucidic geometry. So to allow the limits or otherwise state "provided that XYZ the following holds true" does not disqualify a branch of mathematics from being "true" mathematics.

I'm sure calculus is a pure form of math, but boundaries are the core tool on which we built it. We can continue to calculate to the limit, just as we can make a sample larger and larger, but we also cannot provide greater insight beyond a certain threshold.

### Once you start doing math, it's not math

The obvious contradiction is that we use math to prove math theorems and no one argues that proving math theorems is not math.

The next statement could be that if you are using math to get a result, then it is not math. That doesn't make any sense either.

The statement that I would agree with is that when you use the results of a calculation to make a decision, **the decision is not mathematical** . *That is not to say that the analysis that leads to the decision is not a mathematical one* .

I think when we use statistical analysis, all of the math that is done is real math. **Only when we give the results to someone for interpretation do statistics leave mathematics. ** As such, statistics and statisticians do real math and are real mathematicians. It is the company's interpretation and / or the statistician's translation of the results into the company that is not mathematical.

### From the comments:

Whuber said:

Replacing "statistics" with "chemistry," "economics," "engineering", or any other subject that uses math (such as home economics) doesn't seem to change your argument.

I think the main difference between "chemistry," "engineering," and "balancing my checkbook" is that in those fields **only existing** math concepts are used. I understand that statisticians like Guass **expanded** the body of math concepts. I am of the opinion ^{(that may be obviously wrong),} that in order to get a PhD in statistics one must, in some way, help expand the mathematical concepts. To the best of my knowledge, chemistry / engineering doctoral students do not have this requirement.

The distinction that *statistics contributes to this to the body of math terms* is what it sets apart from the sets

*other fields that only*.

**use**math concepts*: The notable exception is this answer, which is effectively stating that the boundaries are artificial for various social reasons. I think that's the only real answer, but what's the fun in that? ;)

Statistical tests, models, and inference tools are written in the language of mathematics, and statisticians have mathematically proven thick books with very important and interesting results about them. In many cases, the evidence provides 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 they are definitely extremely deeply invested in mathematics, and theoretical statistics is as much a branch of mathematics as theoretical physics or theoretical computer science.

The "difference" is based on: Inductive thinking *vs.* Deductive Thinking *vs.* Inference. For example, no mathematical theorem can tell what distribution or priority you can use for your data / model.

Bayesian statistics are, by the way, an axiomatized area.

This may be a very unpopular opinion, but given the history and formulation of concepts in statistics (and probability theory), I see statistics as a sub-branch of *physics* .

In fact, Gauss first formalized the least squares regression model in astronomical predictions. Most of the contributions to statistics before Fisher were made by 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 overriding principle is the characterization of errors and apparent randomness that spread from an infinite number of unmeasured sources of variation. Because experiments were harder to control, experimental errors had to be formally described and accounted for in order to calibrate the preponderance of experimental evidence against the proposed mathematical model. Later when particle physics dealt with *Quantum physics* dealt with, the formalization of particles as random distributions provided a more precise description of the seemingly uncontrollable randomness with photons and electrons.

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

Statistics is therefore a sub-area of physics.

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