# How to teach sensible elementary statistics to lower-division undergraduates?

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Kevin Carlson writes:

Though my graduate education is in mathematics, I teach elementary statistics to lower-division undergraduates.

The traditional elementary statistics curriculum culminates in confidence intervals and hypothesis tests. Most students can learn to perform these tests, but few understand them. It seems to me that there’s a great opportunity to reform the elementary curriculum along Bayesian lines, but I also see no texts that attempt to bring Bayesian techniques below the prerequisite level of calculus and linear algebra. Do you think it’s currently possible to teach elementary stats in a Bayesian way? If not now, what might need to happen before this became possible?

My reply:

I do think there’s a better way to teach introductory statistics but I’m not quite there yet. I think we’d want to do it using simulation, but inference is a sticking point.

To start with, let’s consider three levels of intro stat:

1. The most basic, “stats for poets” class that provides an overview but few skills and no derivations. Currently this seems to usually be taught as a baby version of a theoretical statistics class, and that doesn’t make sense. Instead I’m thinking of a course where each week is a different application area (economics, psychology, political science, medicine, sports, etc.) and then the concepts get introduced in the context of applications. Methods would focus on graphics and simulation.

2. The statistics course that would be taken by students in social science or biology. Details would depend on the subject area, but key methods would be comparisons/regression/anova, simple design of experiments and bias adjustment, and, again, simulation and graphics. The challenge here is that we’d want some inference (estimates and standard errors, and, at the theoretical level, discussions of bias and variance) but this all relies on concepts such as expectation, variance, and some version of Bayesian inference, and all of these can only be taught at a shallow level.

3. A statistics class with mathematical derivations. For this you should be able to teach the material any way you want, but in practice these classes have a pretty shallow mathematical level and give pseudo-proofs of the key results. I don’t think there’s any way to teach statistics rigorously in one semester from scratch. You really need that one semester on probability theory first.

Option #2 above is closest to what I teach, and it’s what Jennifer and Aki and I do in our forthcoming book, Regression and Other Stories. We do lots of computing, and we keep the math to a minimum. Bayes is presented as a way of propagating error in predictions, and a way to include prior information in an analysis. We don’t do any integrals.

I’m not yet sure how to do the intro stat course. Regression and Other Stories starts from scratch, but the students who take that class have already taken introductory statistics somewhere else.

For that first course, I think we need to teach the methods and the concepts, without pretending to have the derivations. Students who want the derivations can go back and learn probability theory and theoretical statistics.

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