## What Should Be in the Introductory (Short) Course in Statistics for Non-Mathematicians?

Murray Aitkin is a professorial fellow with the department of mathematics and statistics at the University of Melbourne. A Fellow of the ASA, he has been associate editor of many journals, including the

Journal of the Royal Statistical Society Series B,Biometrics, Journal of Educational Statistics,andBiometrics.

Most study programs in the social sciences have an introductory statistics course at some early point. The content of this course, and who should teach it, have been a sore point with both social scientists and statisticians for many decades.

I have been deeply concerned about these courses since my appointment to Macquarie University in Sydney in 1969. The university’s common course at that time, which was taken by a large majority of first-year students, had been designed by a committee of foundation professors. It will be no surprise that the early drafts of the proposals required far more time than was available, since the course was two hours/week for one semester or one hr/week for a full year.

The final agreed-to version went from nothing to linear regression in the 24 lectures (there were no tutorial or practical classes for the 1,200 students). No mathematics could be assumed, and no theory of anything was given—just methods. The main statistical principle, if there was one, was the Central Limit Theorem justifying the normal distribution of the sample mean, leading to the t-test, the centerpiece of the course.

The course was perhaps no worse than other such course, but it was disliked by students, and they learned little from it. Many took away a heartfelt dislike of statistics (and sometimes statisticians) and a determination to avoid any more of it, or them, if possible. This made the task of Macquarie statisticians much harder in second- and higher-level courses.

I attribute much of the low esteem in which statistics and statisticians are held directly to such first courses. Why are these courses so bad, and how can they be improved?

The first important point is that a first-year student does not have a program so narrow that he or she cannot be exposed to real-world statistical issues that do not directly relate to his or her major field of study (which may, in any case, change during the student’s university experience). On the contrary, it is very important to demonstrate that statistical issues come up in all fields of study and are met in daily life, regardless of the student’s major field.

The second point is that the main statistical issue to be raised should not be how to formulate a statistical hypothesis and do the t-test; this should be the subject of a second or later-year course. The main issues should be those of design, experimental and survey, and how to be able to assess whether the design used allows believable conclusions to be drawn from the study. Central to this is the sample design: Is the sample from which conclusions are being drawn representative of the population to which the inference is to be applied? Randomness, and how to achieve it, play a critical role in this. This approach is now widely used, for example in the books by David Moore.

At a late stage in my career, I returned from a leave of absence to take over a new small (20-lecture) introductory course designed for arts and social science students in the technical university.

The course was developed in an attempt to increase student numbers, by then the critical determinant of departmental success at the university.

The proposer of the course had the concept of students being required to read the daily papers and bring to the class stories involving, or which should have involved, statistics, which would then be used by the lecturer to motivate and develop the subject.

While pondering this concept, I was told the course would not continue after this first year, as the engineering faculty was restructuring its program and the possibility of a small course in statistics would be eliminated. No one cared, or would care, what was in the course.

This released me from the newspaper story idea and I knew what to do. I had two strands in the course. One was the use of a small population database (the StatLab database from Hodges, Krech, and Crutchfield’s 1974 book) for students to investigate with random dice sampling used for the estimation of population quantities. The other strand was a series of set-piece lectures on design issues and their importance: bias in observational studies (Moore’s book has great examples), the Women and Love surveys of Shere Hite, “testing” students for ESP with the 25-card ESP deck, randomization in clinical trials. For this, I used an Australian trial of depepsen for duodenal ulcers, finishing with the Nobel Prize in Medicine in 2005 awarded jointly to the Australian doctors Barry Marshall and Robin Warren for their discovery of the bacterium *Helicobacter.*

I wrote a small course book for the course and gave this out to students; the lectures were essentially tutorials on the book content.

To minimize the amount of probability theory needed, I restricted inferential questions to proportions, developing the likelihood function for the ESP experiment with *n*=25 and *p*=1/5. The form of the course was frequentist, but if I were to give it again, it would be Bayesian, which would be simpler and allow me to include the prosecutor’s fallacy and other court examples of misuse of probability.

I had the greatest professional satisfaction in giving this course, and the students enjoyed it; the only criticism came from an international student who would have liked it to be more mathematical. I thought later that the course might be of value in other universities and submitted the book to several publishers. I did not expect the angry attacks on it from reviewers. It was judged totally unsuitable for any introductory course, as it did not even have the t-test!

Anyone interested can download it from my website.

Vincent Granvillesaid:My data science book won’t feature the t-test either, nor any statistical test for that matter, and very little mathematics. There will be other stuff, such as how to build simple confidence intervals without any statistical models.