More on sig. figs.

Andrew asks us sig-fig-haters, “How would you want students to report this measurement?”

” I measured the length and width of a sheet of printer paper in centimeters. I came up with l=27.95 cm and w=21.60 cm. Each of the measurements I believed to be within ±0.05 cm. If I want to find the area, what value should I report? l×w=603.72 cm2 without regard to the number of figures being reported.”

Ignoring units for a moment, I’d be happy with any of following, plus some more:

604 ± 2
603.7  ± 1.9
“A little more than 600 cm , give or take a few”
“Most likely somewhere in between 601 and 605.”

So, beyond all of that, to me, it’s important to distinguish among three different things:

(1) The habits of mind we are trying to cultivate (e.g., here a sensitivity toward describing measurement in terms of distribution in which the actual value is likely to reside, and an understanding of why we might care to do so)

(2) A particular strategy or set of strategies we want students to feel confident using when determining those distributions (e.g., crank three times, monte carlo, calculus methods, etc.)

(3) The particular standards or conventions for reporting measurment and uncertainty when publishing for an broader audience (e.g., error bars, sig figs, confidence intervals, etc).

My thinking: If you are conflating these three, you are going to run into trouble. Of course, I want students to learn #1. But I don’t think you can learn that in the abstract. So we’ll have to talk about #1 in relation to a variety of #2’s. But doing #2’s of course doesn’t imply that they are supporting #1. Students can learn to do carry out the procedures of #2 without any idea of #1.  #3 is where I think I should be really careful. Me personally, I’d hold off on demanding particular standards until students are publishing their work for an audience greater than the teacher. Different communities, even different journals, have different standards for reporting measurements and their uncertainty. Those conventions serve a purpose, but I want students to make contact with the necessity for convention when they are communicating their research to an audience (even if just peers in class), not just completing exercises for homework or on exams.