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Statistics and Mathematics

  • 1.  Statistics and Mathematics

    Posted 06-29-2022 15:53
    Hello all,

    I haven't given this much thought, but with a current collaboration, I can use your knowledge on the topic.

    Is the field of statistics a branch of mathematics, science or something else all together.  The always reliable WWW is all over the place on this topic saying:
     - Statistics = branch of mathematics
     - It is a separate discipline and not a branch of mathematics.
     - Statistics is a branch of applied mathematics.
     - Statistics is a mathematical science.
     - Statistics is an applied science.
     - Many universities have their Department of Mathematics and Statistics

    Some solid wisdom on the topic would be appreciated.

    Michael J. Anderson

  • 2.  RE: Statistics and Mathematics

    Posted 06-29-2022 16:18
    Hi Michael,
    Way back in the day when I was an undergrad the maths were considered humanities and my math and stats classes were under the humanities dept.

    I guess i just made it more confusing ;)

    Elaine Eisenbeisz
    Owner and Principal Statistician
    Omega Statistics

  • 3.  RE: Statistics and Mathematics

    Posted 06-29-2022 16:37
    Hi Michael,

    Speaking as a recovering quantum field theorist, I believethat statistics is a mathematical science. Statistics, like physics, chemistry, biological sciences, etc., does not focus on mathematics as such. Mathematics is a means, not an end for these disciplines. They simply use it to address topics that are NOT mathematics, but concern different areas of human knowledge.

    That's my two cents.

    Nelson Lipshutz
    President, Regulatory Research Corporation

  • 4.  RE: Statistics and Mathematics

    Posted 06-30-2022 11:45
    I, too, love this question!
    Many statisticians I know (including myself) have a bachelors degree in mathematics.
    Three things led me to get a higher degree in biostatistics (pre-WWW).
    1. As a sophomore or junior in college, a Physics lab that requested I "fit a line" to the data. (my math background knew there was a better way than eyeballing it, as many in the class did, but I had not had "statistics training" to do it - so I independently looked up how and fit the line using OLS by hand (took a good hour or more on my rusty bought in 1986 TI "scientific" calculator - for those who remember "scientific" meant it had square root, exponent, and log functions!)).
    2. As a junior I was looking through a book for "graduate programs in arts and sciences" - and I saw biostatistics, which seemed like a natural combination of science projects that I did in grade school and high school and all the "higher math" that I learned in college.
    3. As a senior, I took Probability and Statistics in the Mathematics department. It was mostly mathematical, but there was something "different" - we had to estimate an MLE. Somehow I knew taking the log would make things easier, but many other math majors in the class did not think of that. It seemed that there was a slightly different intuition in that class than in most of my other courses.

    I think it would be really hard to be a theoretical statistician without the (Calculus and Linear Algebra based) math background.
    I know MANY excellent applied statisticians that do not have the (Calculus and Linear Algebra based) math background.
    So, my take:
    Perhaps we should consider the definition of Mathematics.
    Science direct posits that "[M]athematics... is the science of the forms of intuition (space and time which are conditions under which all objects of experience are made known to us)."

    If one considers the larger definition of mathematics, it is hard to say that statistics is not a branch of mathematics.
    However, if by "mathematics" one simply thinks of the usual math curriculum, then I think it is more a Venn-diagram situation where there is overlap, but mutually exclusive areas.  

    Mary Kwasny

  • 5.  RE: Statistics and Mathematics

    Posted 06-29-2022 18:21
    Hi Michael,

    In my opinions statistics is not mathematics, because all data and all random variables have distributions. Most (if not all) statistical ideas are based on reality and on randomness. Mathematics is absolute in many ways. Mathematics also uses a lot of imagination (think of the properties of symmetry and all possible structures that can come from this). So, philosophically, statistics and mathematics are very different.

    Now, statistics uses many mathematical results in proofs, but so does physics. This does not make physics an area of mathematics. The same is true for statistics.

    However, people who love mathematics can also learn statistics, but they may not love it.

    Alexandra Kapatou
    Hurst Senior Professorial Lecturer

  • 6.  RE: Statistics and Mathematics

    Posted 06-29-2022 19:01

    This is a fun and interesting topic. The following may be out of left field, but i have found that when I have hired graduate students for statistical support, it did not work so well when I hired graduate students from the mathematics department. The students from the statistics or biostatistics department usually worked out quite well. I don't think the difference was that the students with a statistics background had learned the things that I needed. No, in learning new material the mathematicians did not "get it" as quickly as the statisticians did. Statistical work involves a different way of thinking than mathematical work. Neither one is better or worse than the other, they are just different. They are quite different fields because of the different kind of imagination and way of thinking involved. There is my 2.04 cents worth on the topic.



    Nayak Polissar
    Principal Statistician

  • 7.  RE: Statistics and Mathematics

    Posted 06-30-2022 08:33
    I agree.  This IS a fun topic.  We could all use a little fun in our lives right now.  I agree that math is needed for statistics, as well as other sciences such as chemistry, engineering, physics, etc.  One could make the argument that deeper knowledge of math is more important for physics than statistics.  Where practicing statisticians differ is in that one little term hanging off the end of every model, + e.  As a practicing statistician, it is more important to appreciate, understanding and recognize variability than it is to know advanced calculus.  What are all the potential sources of variability?  Can I recognize them?  Can I account for them?  Can I reduce them?  Are they special cause, bias or seemingly random?  Even with 'big' data, variability does not go away.  It only appears to go away and many bad decisions are being made assuming that the root causes are known when they may only be artifacts of bias.  Without variability, statistics does just become math, but variability takes statistics into a very different arena.  Have a great day everyone.

    Philip R. Scinto
    Senior Fellow
    The Lubrizol Corporation

  • 8.  RE: Statistics and Mathematics

    Posted 07-01-2022 11:13

    This is an interesting topic. When I introduce statistics to students for the first time, I try to use the following example to make them see the fundamental difference between math and stats:

    Suppose someone asks you when should she leave for the airport if the airport is 20 miles away. What would you suggest?


    The math answer is something like: if she drives at 60 miles/hour she would take 20 mins, so she can leave with 20 mins in hand.

    The stats answer is something more practical: if she has 45 mins in hand we are fairly sure she would reach there in time. The "45" here is an estimate. It may be calculated from existing data (from multiple past trips to the airport – i.e. via a frequentist approach), or a mix of hunch/experience and data (i.e. via more of a Bayesian integration) using some kind of a model. The "fairly sure" part as we know can also be quantified and presented as a confidence/credible interval.


    What it essentially indicates is that math is deductive logic – given certain axioms the answers just follow, whereas stats is a part of inductive logic – here the answer is derived from only a sample of data and generalized (and is inherently prone to error). Of course as others have already mentioned stats uses a heavy dose of math since there are numerous deductions that are just an integral part of it, but philosophically it is quite different from math.




  • 9.  RE: Statistics and Mathematics

    Posted 06-30-2022 13:41
    I will share a story that relates to this line of discussion only because it is, in my opinion, fascinating.  

    I worked at the Department of Mathematics and Statistics at Wright State University (WSU) for 35 years.  In the mid-1980s the great David Blackwell came to WSU to visit the African-American Student Union ("Bolinga Center").  The Bolinga Center generously allowed our department to borrow Dr. Blackwell for a couple of hours for a brief visit and colloquium.  Shortly before his talk, during general banter, he said "I find that statisticians understand mathematics talks better than mathematicians understand statistics talks."  I'm afraid the mathematicians in the department were privately upset by this statement.  

    I suppose one can argue the veracity of such a statement endlessly, but that a person of such stature should make it is, as I mentioned before, fascinating.  

    Harry Khamis
    Professor & Director Emeritus
    Department of Mathematics & Statistics
    Statistical Consulting Center
    Wright State University
    Dayton, OH

  • 10.  RE: Statistics and Mathematics

    Posted 06-30-2022 17:11
    There are so many branches of mathematics. Statistics seems to be widely considered to be one of the branches.

    From Britannica:

    Among the principal branches of mathematics are algebraanalysisarithmeticcombinatoricsEuclidean and non-Euclidean geometries, game theorynumber theorynumerical analysisoptimizationprobabilityset theorystatisticstopology, and trigonometry.

    Elizabeth Newton
    Newton Statistical Consulting.

  • 11.  RE: Statistics and Mathematics

    Posted 06-30-2022 19:19
    That is interesting, Elizabeth, but since we escaped from the British in 1776, I am not sure I am ready to have the British (Encyclopædia Britannica) define what I do. Also, their sales plummeted in the internet era, and I am not sure that they are considered to be the authority that they used to be. (Side note: when I was a kid, I loved looking through Encyclopedia Britannica.)

    If we want to aggregate even more, we are all just scientists: physics, chemistry, biology, statistics, geology, mathematics, etc.

    LIke the Declaration of Independence, I think that we can declare that the differences are enough that we are on our own boat. Definitely mathematics is a parent, but we have grown up and become our own thing. Just as the tension between Britain and America was intense long ago, and now we are friends (pretty much), so the days of us being just a part of Math Departments (which some statisticians found hard) are mostly over, but we all get along.

    Finally, this process of defining new entities is not over. There are now Departments of Biostatistics. And Statistics has also calved "Data Science," and Data Scientists are now happily defining themselves as different.

    On we go.

    Best wishes,


    Nayak Polissar
    Principal Statistician

  • 12.  RE: Statistics and Mathematics

    Posted 06-30-2022 20:19
    Britannica is not alone.

    This site identifies 33 branches of mathematics. Here probability, statistics and game theory are part of applied mathematics.

    This site, on the other hand, identifies 10 branches of mathematics, with statistics the 10th (and"easiest"):

    This site identifies only 4 branches of mathematics,
    "Algebra, Geometry, Calculus and Statistics & Probability are considered to be the 4 main branches of Mathematics".

    And from the ultimate resource Wikipedia, https://en.wikipedia.org/wiki/Mathematics
    "Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics".

    Elizabeth Newton, Ph.D.
    Newton Statistical Consulting

  • 13.  RE: Statistics and Mathematics

    Posted 07-01-2022 18:46
    HI Elizabeth, it is very impressive on how many sources you found that lump us under mathematics. However, even if  15,000 sources lump us under mathematics, I still feel like a statistician and not a mathematician.

    Thanks for finding all those interesting sources.

    Humans, gorillas, chimpanzees, baboons, etc., all seem to have come from the same place, but they all are recognized by their separate names and qualities.

    Best wishes,


    Nayak Polissar
    Principal Statistician