Scott Goelzer asked if his students will be okay when they get to
college if he doesn't teach significant-figure rules. He also asked
about rounding when doing textbook problems.
I can provide one data point, an opinion, and a recommendation.
Data Point... in the Bluffton University Science Department we do not
teach/use significant-figure rules. For lab work, the expectation is
that the student either formally works through, or at least gives some
thought to, the uncertainty and/or what makes sense for reporting.
Whether the uncertainty is worked through in detail, or some minimal
thought is given to the number reported, depends on the class.
Organic chemistry labs often involve a synthesis, and the student
reports a percent yield. It doesn't make sense for a student to report
a yield as 27.123 %. This is not so much a matter of breaking sig-fig
rules as it is a lack of appreciation of why we calculate a percent
yield and how that number might be used. Thus, if a student reports
27.123 % she is not likely to get criticized for sig-fig violation;
rather, she is likely to be asked why she thought a yield should be
reported with that rounding as opposed to reporting 27%. Having learned
sig-fig rules in high school is of no use here because it is not a
matter of how many digits can be justified from the data. Rather, it is
a matter of tailoring the number to the type of data being communicated.
Indeed, if an analytical balance was used to weigh reactants and
products, the 5-digit number might be consistent with sig-fig rules, but
hardly justifiable as a worthwhile contribution to a percent-yield
communication.
On the other hand, in calculus-based-physics labs I typically want to
see something like 3.57 +- 0.12 or 3.57(12). That is, I am looking for
a more formal error analysis. Having learned sig-fig rules in high
school offers no benefit to this type of estimate/calculation.
An opinion... If you simply ignore sig-fig rules your students will be
okay with most profs at most colleges. If you replace sig-fig rules
with some level of discussion about estimating uncertainties and perhaps
some discussion about what is needed to communicate the science you are
trying to communicate, the students will be ahead of those who just
learned sig-fig rules.
A recommendation... For problem sets and exams I tell students the
following.
(1) In almost all cases, please report your answers rounded to three
significant digits (*). This is generally sufficient to notice anything
that a particular problem is designed to teach you. By automatically
rounding to the three digits, you aren't wasting any of your time
worrying about sig-fig stuff, and you aren't wasting any grader time to
read extra digits or think whether your 1-sig-digit number is close
enough to indicate you worked the problem correctly.
(*) Note... this is the only way in which I use the words significant
digits. I do expect that students know that 0.001 is only one
significant digit rather than three. Thus, if the calculator shows
0.0012345, I want them to write down 0.00123 rather than 0.001. The
rationale is partly ease in grading as well as unshackling the student
from sig-fig rules. But also note that 3 sig-figs would typically be
sufficient for students to notice what they need to notice in problems
where the intent is for the student to note differences or similarities
in results.
(2) Try to perform the calculation in your calculator or using Excel
such that you don't have to write down then reenter any intermediate
numbers. That way you don't have to worry about excess round-off
errors. If you do have to write down intermediate numbers, use two more
digits than what you'll round to. These are called guard digits. Thus,
if you are going to round to three digits, write down intermediate
answers to five digits.
(3) Occasionally a problem is designed to show something that requires
more than three significant digits. A good clue for when this is the
case is to notice that the problem's data are supplied with more than 3
significant digits. However, the best way to realize this is to
understand the problem and work it with sufficient digits to see the
point being made about the problem. An example of this would be if you
are adding the masses of protons, neutrons, and electrons in an atom to
show how this differs from the isotopic mass for that atom. The intent
is to demonstrate the "mass defect."
Final note... If students don't follow this advice on problem sets and
exams, I do not take off points, but I continue to ask them why they are
reporting answers the way they are. The goal is not compliance with any
sig-fig rules. The goal is to get them to think about what they are
trying to communicate with their numbers.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu
.