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How would you motivate an undergraduate student to take an Introduction to Mathematical Statistics course, if they know that much of their time will be spent proving results they've already seen in Introduction to Applied Statistics? That is, much (though certainly not all!) of the course is "just" about proving mathematically that standard methods for p-values and confidence intervals (z and t tests and intervals; F and chi-square tests) work as advertised.

Context: I teach at a US small liberal arts college. Here is the course description for our version of this class:

Topics in Statistical Inference Building on their background in probability theory, students explore inferential methods in statistics and learn how to evaluate different estimation techniques and hypothesis-testing methods. Students learn techniques for modeling the response of a continuous random variable using information from several variables using regression modeling. Topics include maximum likelihood and other methods estimation, sample properties of estimators, including sufficiency, consistency, and relative efficiency, Rao-Blackwell theorem, tests of hypotheses, confidence, and resampling techniques.

We have used textbooks such as Hogg, McKean, & Craig (table of contents) or Wackerly, Mendenhall, & Scheaffer (table of contents).

Our statistics majors are required to take Math Stats, but I try to give better motivation than just "it's required." And occasionally other students ask me why they should consider taking this course too. What are some ways I could convince them to see the value of taking Math Stats?

I'll post my own answer with what I currently tell them, but I expect that the community here has other great suggestions.

civilstat
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    I'm not convinced that this is on-topic, but I know why it has been posted and am giving a big +1 for that reason. – Dave Feb 14 '23 at 16:50
  • @Dave Thanks :) Do you have suggestions for making it more on-topic? I assumed that if questions like https://stats.stackexchange.com/q/3262/17414 are OK, then something like this would be fine too. I do genuinely want to hear others' answers. But I understand and will not object if the consensus is that this question is out of place here. – civilstat Feb 14 '23 at 17:00
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    This is a good question. But such kind of questions are always open ended for the answer would be more opinion inclined rather than a robust post. So while I upvoted it, I voted to close it too on the same grounds. – User1865345 Feb 14 '23 at 17:35
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    Why don't you talk to some of your students and see what motivates them? Then, you can try to frame the course as tending to some of those interests. – Demetri Pananos Feb 14 '23 at 17:39
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    One would imagine that the value of such a course would vary according to student backgrounds, interests, and objectives. I'm not sure there will be one motivation common to them all apart from recognizing the department views the subject as important enough to require it. Perhaps instead of describing the course so technically (why should we care about relative efficiency, for instance?), perhaps it would help to give an account of the deep concepts and powerful techniques students will learn. Why are you excited about these topics? – whuber Feb 14 '23 at 17:40
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    @whuber Thanks, and I agree. The technical description above is from the course catalog, and it's quoted here only so that readers of the question will know what the course covers. It's certainly not the language I use when I try to pitch the course to students! – civilstat Feb 14 '23 at 18:00
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    A very common question I get from people a couple of years after they do a basic/applied stats course is "I really want to understand where these statistical procedures come from, what resources are there?". If you have some way of conveying that desire to people who haven't yet asked that question, that might help. (I'm not sure how to achieve it -- I was typically asking that question right at the start of learning anything that didn't start with explaining it.) – Glen_b Feb 14 '23 at 21:28
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    @Glen_b Thanks -- that reminds me of this approach to lesson planning: https://blog.mrmeyer.com/2015/if-math-is-the-aspirin-then-how-do-you-create-the-headache/ One approach may be to find a catchy example that can't be solved using Intro Applied Stats methods (ideally a case where those methods break down), but can be derived using Intro Math Stats tools. Or perhaps to ask students, "If you hadn't learned method XYZ in Intro Applied Stat, how else would you have tried to solve it? Now, how can we decide if your solution is any better than the standard one?" – civilstat Feb 14 '23 at 22:24
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    @DemetriPananos in addition to this suggestion, providing something like a career panel would be a good opportunity to meet at both ends. One of the biggest anxieties of committing to an academic program is being unsure what your day-to-day work will be like. I definitely saw some "interesting perspectives" from statisticians within "certain organizations" - although I was a few weeks out from my thesis defense, so it was too late to consider plumbing or carpentry. – AdamO Feb 14 '23 at 23:07
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    @civilstat the directory of mathematical headaches (linked at the bottom of that blog post) has a handy example - that of fitting lines; it's easy enough to come up with a variety of simple slope estimators for example (average pairwise slope, median pairwise slope, slope from least squares, slope from least absolute deviations, Tukey three-group line ...), and one can indeed ask motivating questions like the ones you pose at the end there about those guesses or any other suggestions. – Glen_b Feb 15 '23 at 00:49
  • The exercise at the link was of course to do a 'by eye' line to generate the headache, but to compare performance, there's a need for formally defined estimators (to compare across more than one data set). Aside from the ones mentioned above there's a host of other possibilities but some may not be a good fit for an elementary class . . . – Glen_b Feb 15 '23 at 01:40
  • @Glen_b Great example! Would you post these ideas as an answer so they don't get lost in the comments? – civilstat Feb 15 '23 at 03:24
  • rename the class to "Mathematics for Data Science" – safex Feb 15 '23 at 06:58
  • @safex My sense is that the title "Math for Data Science" is already being used for remedial courses that review high school algebra, or courses that provide the minimal applied linear algebra needed to run ML algorithms. That's quite different from a course in mathematical stats. But if you're just saying "give it a catchier name for better marketing," fair enough :) although perhaps "Math for AI" might be more up-to-date? https://books.google.com/ngrams/graph?content=statistics%2Cdata+science%2Cai%2Cmachine+learning&year_start=2010&year_end=2019&corpus=en-2019&smoothing=3 – civilstat Feb 15 '23 at 17:33
  • “Data Science Done Right”? @safex – Dave Feb 15 '23 at 17:53
  • The counterpoint to "Statistics Done Wrong" https://www.statisticsdonewrong.com/ – civilstat Feb 15 '23 at 19:40
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    There’s no shortage of that one. – Dave Feb 15 '23 at 19:41

3 Answers3

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This is not an answer, it's a story -

My experience matriculating from an agricultural-based community college to a large urban liberal arts college was culture shock on every front. I knew I was hungry for mathematics and, especially, statistics. In 3rd year mathematics, students are exposed to proofs, like Rudin's Analysis. It doesn't feel "gentle" - but imagine being dropped into Functional Analysis or Measure Theory without a proofs background!

In stats, I often find there's no 3rd year statistics book or course that establishes fundamental statistics. I wrote a cotaught undergraduate (4th level)/graduate level probability and statistics course from Hogg and Craig as a 3rd year student.

Hogg and Craig's book perfectly reflects the awkwardness of this discipline to establish itself as an insular and self-sufficing degree of study. The introduction, for instance, discusses the probability space as a non-negative measure on a collection of Borel sets. I remember needing wikipedia to read the first pages. Not only is that definition highly sophisticated (and I think there are more general probability spaces than are described), that result is not used anywhere else in the book except that very general results about integration are used with reckless abandon. The rest of the book is multivariable calculus. Hogg and Craig provide different treatments for continuous and discrete DFs as a result. Hogg and Craig is not a good 3rd year book at all.

I think Casella Berger is a better book for this reason. It's very focused on problems and concepts. I also think students would benefit having prior or concurrent exposure to Sheldon Ross' Probability Models which goes over probability in great detail, skipping the statistical results - the difference between these fields is as underappreciated by students as it is by us practicing professionals. Lastly, I haven't found any textbook to be of adequate quality dealing with the topic of regression, definitely the most widespread and discussed area of statistics and inference- Nachtsheim et al. is perhaps accessible at the undergraduate level, but something like a capstone project would be a better way to expose students to using the software, doing analyses, and presenting the results.

How do you motivate students to study these areas? Well, the students have to motivate themselves. You could try the "carrot" method about all the opportunities they have, and you can say that it fosters "critical and quantitative thinking", although I find training in probability theory to be more useful than statistics in this regard. You can go farther to point out that virtually every scientific discipline requires its practitioners to be critical producers of and consumers of information that is presented using statistical summaries. For students going into mathematics and statistics, you can point out that there are many exciting results just waiting for them in a 4th year course, or graduate theory - a special topics day could go over the Cantor distribution function, or you could present the Hodges' Superefficient Estimator and explain why it "breaks" the Cramer-Rao bound.

AdamO
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    This is an unorthodox post in that you share your own experience from a not so conventional ground and this makes it more insightful. +1. – User1865345 Feb 14 '23 at 17:33
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    I've not read Hogg & Craig, but I've noticed the same thing with Lindgren (can't remember which edition) & other texts - this dragging in of a lot of fancy mathematical apparatus in the first chapter, just for it to sit gathering dust until the end of the book. (Lehmann does make use of it, but I think one could manage Cox & Hinkley - or Pawitan - with little more than school maths.) – Scortchi - Reinstate Monica Feb 14 '23 at 19:08
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  • Do you hope to do research on statistical or machine-learning methods? Even if this class just proves conclusions that you've already seen, the mathematical techniques and frameworks behind these proofs are the same ones used by research statisticians to develop new cutting-edge results. This class is the first step towards making your own contributions to statistical methodology. It will also help you sanity-check a colleague's work when they say they've developed a "better" statistical method for some scenario. (How is it better: more efficient? more robust? etc. Does it actually work the way they think it does?)
  • Do you appreciate beauty in math? I find that some of the math techniques or results themselves are beautiful. For instance, you can approach several common situations from totally different angles (MLE, MOM, Bayesian posterior mode, minimizing MSE...) and they all give the same answer (sample mean)! How cool is that?
  • Do you only care about applications, not the "pure" math itself? Even so, a Math Stats course will give you a much deeper understanding of what these methods are doing. Personally, I had seen p-values and confidence intervals many times, but they did not really click for me until I took Math Stats. It will also help you learn to watch out for situations when the standard methods won't work and possibly develop your own solutions or workarounds.
  • Do you want to work with data-minded colleagues but not analyze data yourself? Even so, you'll be better prepared to follow other people's new developments in statistics, AI, and machine learning, so you can sensibly decide whether they're useful to your work. For instance, differential privacy has been a recent hot topic in my areas of work, but it's not trivial to understand what it's even trying to solve or how it proposes to do so; some Math Stat background definitely helps.
civilstat
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  • I think this attempt to distinguish math vs non-math kind of undermines what math is. I mean, what is math but a highly organized way to present and share ideas? I would argue that the ideal class at this level "unlocks" math skills to organize and present data in ways that will generate insights, curiosity, and strategies. There are aspects that are "pure" math, meaning there are new symbols aside from established arithmetic or calculus. You might say it's a way to pursue an in demand and challenging quantitative field as an alternative to pure math... but even that's iffy! – AdamO Feb 14 '23 at 18:34
  • I didn't mean to say X is math and Y isn't -- just shorthand for different students having different interests, as @DemetriPananos pointed out. I would use a different motivation to someone who says "I love proofs, but why prove these things?" than to someone who says "Proofs scare me, and I'm taking stats only so that I can apply it to do rigorous research in another field (biology, sociology, etc.)" – civilstat Feb 14 '23 at 19:26
  • @AdamO I like your suggestion that "it's a way to pursue an in demand and challenging quantitative field as an alternative to pure math." The academic job market for traditional math faculty positions seems much tougher than the market for statistics faculty positions, at least at the moment. So if a "pure-math"-minded student is open to choosing a field of mathematics based on job opportunities, then getting a PhD in mathematical statistics might open up more doors than a PhD in, say, number theory. – civilstat Feb 14 '23 at 19:33
  • You can definitely pitch it that way. What seems surprising to me is what students will do with a BS in stats. It almost seems the better degree for a graduate stats program is math more than stats. Think about measure theory! Two decades ago, there weren't accredited majors in stats, you just did math with a focus on stats. I went to a very theory heavy graduate program, and despite being a working statistician, I actually use a lot of theory. When hiring, I find a lot of graduates from less rigorous programs are a bit underprepared for industry... – AdamO Feb 14 '23 at 22:58
  • ...add to that industry options for statisticians are recently waning. I think there was a bit of a hype bubble that burst after the Google CEO said stats would be the "sexiest" profession of the decade. But the point of an academic program is to meet the needs of tomorrow... I wouldn't discourage anyone from either math or stats, and say either has a decent shot in informatics, econometrics, biostatistics, etc. – AdamO Feb 14 '23 at 22:59
  • I agree that students who want to go to grad school in stats need a strong math background -- stronger than the minimum required for a stats major alone. In the other direction, statistics can also be a useful second major for someone whose primary major is in something else, such as econ or biology. – civilstat Feb 15 '23 at 03:18
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If the undergraduates have just taken a course in Introduction to Applied Statistics, well, frankly I can't blame them for not taking the Introduction to Mathematical Statistics course. For interested students, I would think that the "theory" behind applied statistics would be at least minimally presented in the applied statistics course; enough for them to understand it in principle. And these things take a while, and some experience, to intuitively understand and be cogitated upon. Unless perhaps there was some deficiency in the first course in explaining basic concepts and they got stuck on that.

I would suggest that perhaps other courses might fit students' progression up the statistical ladder, such as regression (logistic, Poisson and others); sampling methods; a software-oriented epidemiology course, or even an applied course in genetics. These would help them to intuit statistical logic. It seems your college might be forcibly dividing intro statistics in to an "applied" camp where the basic how-to is performed, and a "mathematical" camp where more mathematical, probability-based and inter-related statistics are looked at under the hood. You could always change the name of the course to make it a more logical follow-on to the first course. My guess is that those who took the first course are actually hungry for more applications of statistics, and this second course might be better to have some real-world examples, making it "semi-applied."

With that said, I appreciate trying to teach the undergraduates more about the logic of inference and probability and so forth, beyond what they learned in the nuts-and-bolts course. But it might be (or sound) overwhelming for them. In my own personal journey I've come to appreciate statistical theory on things well after I've had multiple statistical courses on everything from design of experiments to logistic regression and probability, and a lot of real-world research. And I've found the theory to usually be much simpler than it sounds; again, partly because a certain amount of applied statistics has been put under my belt.

kjetil b halvorsen
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Jamie
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  • Thanks. Math Stat is not our 2nd course -- we do recommend that most students go on to other applied courses first, including many of those you've described. We just want Stats majors to see Math Stat at some point before they graduate. The applied-vs-mathematical separation into distinct courses is common (not just at my campus). Here, our Intro Applied Stat course also serves students from other majors (eg biology) who don't have the calculus background to follow Math Stat proofs, but still need practice applying these methods and intuition behind how they work. – civilstat Feb 14 '23 at 22:30