In my opinion, any scientific discipline utilizing quantitative methodologies as part of their scientific analysis and inference, is a domain of applied mathematics.

Statistics is not a branch of applied mathematics, but rather a separate field like physics.

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Ismail Abbas M. Abbas Hello Ismail. To follow-up on your response: In your opinion, what is required for a topic to be considered a branch of applied mathematics? What is the criteria? Thanks.

Rainer Duesing I'd be curious to hear your thoughts on this matter, Rainer

Blaine Tomkins to be honest, I did not really think about the topic and just used statistics as a tool, until I heard a talk of a mathematician. He wanted to talk about statistics (and Bayes vs. frequentism) and to prepare himself he looked into several mathematics textbooks. To his surprise, none of the textbooks covered statistics, only a few mentions and some remarks that there is "something else called statistics".

So, from this perspective, it seems as if statistics is a separate field, which incorporates mathematics but also finds new insights and contributes to it.

On the other hand, what else as an applied form of mathematics should it be? Where are the boundaries and what are the definition that define what to include and what not to. I thinks this is very hard to distinguish.

I would say that even some physicists would say that theoretical physics is nothing more than applied mathematics.

My question: is it important how we call it?

Rainer Duesing Many good points, and a good question. In addition to genuine curiosity, the reason I asked this question is my institution is currently revising it's curriculum and whether statistics is considered a branch of mathematics is one of the central issues.

Specifically, the answer to this question could determine whether 'statistics' satisfies the 'math' requirement. It could also determine which departments (and which faculty) teach statistics and which do not.

Ok, your question has a real world application, very interesting. Maybe, your institution should ask itself, for which purpose and for whom you want to teach statistics and how you will see the "landscape" in 10 years or so.

To be more specific: I am "only" a psychologist, who is highly interested in statistics and I took a deepdive into the field. Many of my colleagues didnt do it and they seek for help, if they want to puplish a paper. They tell me that they are also interested in statistics, but they do not have the time to be up to date with current analyses strategies. So maybe, in 10 years or so, we need highly trained statisticians, with a solid mathematical background, who will support all publications and do all the work and heavy lifting. If that is envisioned, then I would think that it should be part of mathematics, to have also a fundamental education in this field.

If on the other hand the goal is to educate applied researchers in statistics, so that they can do analyses widely on their own, maybe it should be a separate branch with a different kind of people as teachers. Applied statistician, which are from the field itself may have a better insight in the specific needs of researchers, which may be no so much the case, if the statisticians will do all of the work, as in my first example.

My personal experience: my first statistics professor was a mathematician. He was very good in the things he did, but his teaching didactics was poor, because saw the audience only from his perspective. But most of us were not trained in mathematics and therefore, it was quite hard to follow his ideas. Maybe a more applied statitician would have been better to teach statistical concepts to psychology students.

How my two cents help a little bit.

Richard Cember Well observed. That is precisely the situation we're faced with. The majority of liberal arts colleges include 'statistics' among the courses that satisfy the general education math requirement. By not doing so, we would be an outlier.

Though the second issue you mentioned (an interdepartmental competition over resources) is the primary driving force behind this decision, I fear. As such, arguing over semantics is pointless.

Be that as it may, I am still intrigued by the question from an academic perspective; namely, how to classify statistics.

Returning to this question, it's interesting that probability theory is unanimously considered a topic that falls under applied mathematics, and - in a sense - statistics ultimately boils down to probability. Any work with probability requires a variable (i.e., probability of what ?) and the only difference, as I see it, between a study of probability theory proper and statistics is the nature of the variable. In probability theory (or a study of axioms of probability) the variable is typically abstract p(A) whereas in statistics, the variable is usually something empirical p(height)

However, regardless of whether a variable is (A) or (Height), the axioms and theories of probability DO NOT change. If a probability axiom is true of A it MUST also be true of height. The only real difference as I see it is the nature of the variable. In probability theory, probabilities reflect some abstract event or outcome considered a priori. In statistics, probabilities are informed by data. The underlying logic and principles, however, are the same. Thus, I conclude that if probability theory is a branch of applied mathematics, so too is statistics.

....Furthermore, statistics should not be compared to fields like physics. Physics is restricted to a particular domain of variables, or content. For example, physics does not concern itself with the study of language.

Statistics is not restricted in this way. Statistics is not bound to a particular domain of content or study like physics. It applies wherever there is data - be it the study of physical properties, language, or anything else. Whenever a variable can be measured, statistics applies.

I would disagree with your second answer. Statistics is restricted to a particular domain of probability, i.e. where variables are empirical and not abstract. Statistics is not concerned with the study of language or language itself, but with data and probability of language studies. Therefore, I is fair to compare physics and statistics.

Just to directly pick up your former points.

Hmmm...I'll need to think on this awhile

I think we agree in our characterization of statistics as a 'tool'. And for this reason, I see it as different from physics. I'll try to explain:

It's true that in the vast majority of cases, the variables in statistics are empirical rather than abstract - but not always. For instance, researchers sometimes generate simulated data for some random variable in order to assess the "behavior" of a particular model. In these rare instances, the variable doesn't reflect anything in the empirical world - it's simply an abstract placeholder in the same way that 'A' is a placeholder in p(A). While these are relatively rare, I would still consider these exercises as falling under 'statistics' since there is still data present. It just happens to be simulated data.

I suppose my point is statistics is fundamentally a 'method' (tool) used to learn about something else (that is not statistics). There's really no way to study statistics in itself the way you can with branches of pure mathematics (e.g., algebra, geometry, calculus). These branches examine how principles of math work within it's own internal logic. In other words, statistics has no inherent content. It depends on data. If there is no data, there can be no statistics. The same way that probability theory depends on a variable (probability of WHAT?). Without an outcome, event, or variable, one cannot examine probability.

The situation is different with physics. Physics is not a tool or method that can only be used to study something else (that is not physics). There are variables studied in physics (e.g., force, energy, matter) that are specific to the field of physics. This is what I mean by 'content' above. Statistics has no variables of this kind (i.e., that are specific to itself). It must find it's variables elsewhere, making it fundamentally a tool. It's a hammer in search of nails.

In a way, I would agree with you Blaine Tomkins , and in another way I wouldn't. I think it boils down to how you would define any natural sciences. In my personal opinion, there is no such thing as "force" or "energy", but physics is a well elaborated tool to describe phenomena, give it names, formalize it and make predictions. And maybe this is the point where we disagree. Physics "only" describes and simplifies nature, which is latent and not directly visible to us. Physics is "just" an abstraction of reality.

Similarly, in statistics, you can "discover" how theoretical models behave under different circumstances, like in theoretical physics, free of any content or application. But you can also apply it to specific data, like in experimental physics. There is a distinction between experimental and theoretical physics.

But I think this problem/discussion is rather philosophical and does not fit to your OP.