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What is the difference between mathematical modeling and statistical modeling?

I only know that a mathematical model is deterministic while a statistical model is stochastic.

Is that all to answer the question?

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    $\begingroup$ I'm not so sure there is a difference. The terms are used interchangeably. $\endgroup$ Feb 22, 2015 at 11:05
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    $\begingroup$ The difference is what @tree said. One is deterministic and one is stochastic. If someone use them interchangeably, in my opinion he is wrong $\endgroup$
    – niandra82
    Feb 22, 2015 at 11:20
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    $\begingroup$ If anyone wanted to define mathematical modelling to include statistical modelling as a special case, that's fine by me. It's perhaps more common to define statistical modelling in contrast to statistical modelling. But quite where to draw the line is difficult, yet also not important, except for administrators, librarians and other people obliged to do so as part of their duties. Work on stochastic differential equations can straddle any boundary, for example. $\endgroup$
    – Nick Cox
    Feb 22, 2015 at 12:04
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    $\begingroup$ @NickCox the line would probably best be drawn at the incorporation of uncertainty. Purely "mathematical" models do not incorporate uncertainty; "probabilistic" models incorporate fundamental uncertainty in a data-generating process; "statistical" models incorporate uncertainty induced by sampling from a population. $\endgroup$ Feb 22, 2015 at 16:18
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    $\begingroup$ @ssdecontrol Good luck in sustaining that probabilistic/statistical distinction with colleagues and students, as few will promise to echo your personal division! Where do stochastic differential equations lie? More crucially, where would you place "exploratory data analysis" in Tukey's sense? $\endgroup$
    – Nick Cox
    Feb 22, 2015 at 16:22

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In my mind, statistical modelling is a special case of probabilistic modelling, which is a special case of mathematical modelling.. but I don't usually bother to distinguish them and think often the difference is more of cultures. Things I associate more with statistical modelling are replication, the special role of intuition, and data exploration.

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  • $\begingroup$ My thoughts exactly. $\text{statistical} \subset \text{mathematical}$. Although I wonder if there are some statistical models that one could argue are non-probabilistic (although it seems there is always a probabilistic interpretation even when models are not intended to be probabilistic). $\endgroup$ Feb 22, 2015 at 16:16
  • $\begingroup$ What about alternative ways to model "uncertainty", in some sense? Fuzzy sets, DEmpster-Shafer theory en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory ... $\endgroup$ Aug 7, 2017 at 13:20
  • $\begingroup$ hi @kjetilbhalvorsen, I don't know about Dempster-Shafer theory but after an extra 2.5 years of hands on modelling experience my views have been refined and I hope to revise my answer above accordingly soon :) $\endgroup$ Aug 9, 2017 at 10:19
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What is the difference between mathematics and statistics? !

Statistics is itself a branch of mathematics where most of the time we deal with mean, median, mode etc, although they require mathematical computation as well.

Same way in machine learning statistical models has most of the computation related to mean, median, quantiles etc. Ex- Linear Regression, Logistic Regression.

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    $\begingroup$ I don't agree that statistics is a branch of mathematics. Mathematics provides tools for statistics. $\endgroup$ Jan 25, 2020 at 21:23
  • $\begingroup$ If anyone knows maths of universe he must be knowing statistics, that's why statistics is branch of maths $\endgroup$ Jan 27, 2020 at 7:30

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