An interesting thing happened this week in my labs. We do a neat little exercise that treats pennies as isotopes. Pennies minted before 1982 have a different mass than pennies minted after 1982—like all things “of the future,” pennies got lighter in 1982. Students are provided with opaque film canisters of fifteen pennies and asked to determine the “isotopic abundance” of pre- and post-1982 pennies in their canisters. The canisters are glued shut, but standard pennies and empty canisters are available for weighing.
Student: Do my calculations look good?
mevans: Looks great.
The student had calculated something like 10.1025 pre-1982 pennies and 4.8975 post-1982 pennies. Variance in the masses of the canisters and pennies causes some non-ideality.
Student: How many significant figures should I include?
Ouch. This question, to me, was evidence that the point of significant figures was lost on this student. Of course, I thought, just round to whole numbers, since fractional numbers of pennies are nonsensical.
mevans: Does a fraction of a penny make sense when counting them?
mevans: Then throw all the digits after the decimal away. Boom! Done.
Humans have a disturbing natural attraction to numbers, even when said numbers are nonsensical, so small as to be meaningless, or outright lies that ignore statistics (as when a calculation based on measured, uncertain values is reported to too many decimal places). Throwing numbers away is cognitively hard! Deep down, we know that a number with more significant digits is more precise, and we cling to those digits even if the precision is imaginary or nonsensical. A big part of science education is training the mind to overcome this deception and deal with numbers in a healthy way.
The saga continued. Students got the sensible idea to report isotopic abundances as percentages of pennies in the entire sample. New set of numbers, same set of issues with significant figures!
Student: How many significant figures should I include in the percentage?
Things suddenly got interesting! I have to admit that this question caught me off guard. The calculation is simple enough: 10 / 15 * 100. Dogmatic application of the “rules” for significant figures would produce the number “67%.” Yet, the exact ratio of pennies is known, since we know that there are fifteen pennies and—relying on the idea that fractional pennies are nonsensical—there is no uncertainty in the numbers of pennies. There is no uncertainty in the percentage at all, so it’s fine to report the percentage as “66.6 repeating.” Hm, perhaps there is more to significant figures than meets the eye!
I’m fascinated by the link between significant figures and scientific misconduct. I think it’s rarely appreciated, but significant figures really are an issue of scientific misconduct. Reporting too many digits in a number is tantamount to lying about the precision of one’s instruments and ignoring (willfully or not) the impact of uncertainty on reported values. How do you get students—and other number-obsessed humans in the general public—to appreciate the contingency of scientific quantities?
I can’t resist one more fun fact about significant figures to finish this post. A value calculated from a logarithm (say an energy calculated from –RT ln K) has only as many significant figures as digits after the decimal place. Why? Think of the logarithm as a stand-in for a power of 10 (never mind the conversion of ln to log for a second). The integer part of a logarithm, then, is just a simpler way of writing “times ten to the power of…” It’s just the exponent part of a number in scientific notation—a placeholder that is never significant! The decimal portion of a logarithm, on the other hand, actually represents a number with meaning. Hence, only the numbers following the decimal point are significant in a logarithm-based value. Slick, eh?