My take on significant digits

Last summer, my wife’s friend told me that she was planning to sunbathe on the rooftop patio of her condo. I told her to apply extra sunscreen — she’ll be closer to the sun. Her reply was an honest-to-goodness, “Really?”.1

You and I know that my suggestion was ridiculous. What difference does 150 m make when the sun is about 150 million km, or 1.5 x 1011 m away?2

This raises a question about people’s understanding of confidence level and significant digits — and the need to help our students with this important concept.

I teach significant figures in grade 11 Chemistry but not at the beginning of the course. I wait until students need to perform calculations based on measured quantities. My opener involves two activities. The first looks at the precision of different items of glassware.3 The second asks students to calculate the thickness (mm) of a (ca 10 cm by 10 cm) sheet of Al foil, using a mm ruler, a centigram balance and a periodic table that includes density values. They soon figure things out and write their answers on the board.

A post-activity discussion makes it clear (I hope) that an answer for the thickness of a piece of foil — determined with a plastic ruler and a two decimal place balance — accurate to 12 significant figures — is fantasy.

I state the bottom line: All measured quantities must have one estimated, or uncertain, digit. For an analogue measuring device, this means estimating between the smallest scale divisions. For electronic devices, I simply tell students to assume that the right-most digit is estimated. Good enough for high school.

Sig figs have rules — I get it: one for multiplication and division of measured quantities, and one for addition and subtraction. But beyond — or beneath these rules, depending on your point of view — we have what finance people call “the underlying”.

Which we call confidence level — the foundation of the whole sig fig thing: the “weakest link”.

We tell our students that counted quantities have infinite sig figs. This makes sense if one is counting students or beakers. But estimates, such as “There were 105,000 spectators at the Winter Classic” hardly have infinite precision. Further, the counting of droplets in a micro-titration does not amount to infinite sig figs — each droplet does not have exactly the same volume.
For multiplication and division of measured quantities, we have the general rule that the lowest quantity of sig figs “in” determines the quantity of sig figs “out”. This is an easy sell but needs to be paid attention to…

The addition and subtraction of measured quantities boils down to lining up the place values — and keeping an eye on what’s estimated. At the end of a calculation, make sure the answer has one uncertain digit.

The more we focus on this day-to-day, the easier and more intuitive it gets. Thinking about sig figs (and units!) before solving a problem can be more than helpful.

And here’s the kicker: Understanding sig figs on a conceptual level can make one’s life easier(!) Shortcuts can, in many cases, be taken — because they don’t matter.

For example, students can determine the molar volume of hydrogen gas (at ca SATP) in a quick, low precision experiment, based on the reaction represented by:

1 Mg(s)  +  2 HCl(aq)  →  1 MgCl2(aq)  +  1 H2(g)

A weighed amount (ca 0.20 g) of clean Mg ribbon is treated with excess 1 mol/L HCl(aq). Collect the hydrogen gas by downward displacement of water in a 250 mL Erlenmeyer flask, as illustrated above. Mark the level of gas collected with a grease pencil; use a 100 mL volumetric flask (and water) to determine the volume of H2(g).

If this sounds imprecise — it is. Using the molar ratio between Mg and H2 (spoiler alert: stoichiometry) the class average will be 25 L/mol, totally precise to 2 sig figs. The imprecision of the experiment allows it to be done “quick and dirty” — no correcting for the partial pressure of water vapour or conditions that aren’t quite SATP.

In the post-lab discussion, you’ll see that many students will have reported the volume of H2(g) precise to the 10th of a mL, which pays homage to the “letter of the law” for handling measured quantities, but completely ignores the confidence level inherent in the data collection. Aside from nailing the molar volume, the discussion on significant figures is solid gold!

In AP Chemistry, we do a spectrophotometric determination of the Ka (and pKa) for bromocresol green, an acid-base indicator and a weak acid itself. (We’ll publish this in a subsequent issue.)

As an undergraduate, we did this to a high precision level, using pipets and volumetric flasks and controlling the Earth’s magnetic field (kidding!!). Needless to say, our answer was precise (and accurate!) to several significant figures. At Crescent School, this determination is limited to one sig fig.

And why is that?

Because the concentration of our stock solution of bromocresol green is 0.04% (m/v) — one sig fig!!!! No need for any high precision glassware; we measure volume by the “drop” and use the “poor man’s” volumetric flask — a graduated cylinder.

How is our result, you ask? The pKa is spot on — to one sig fig. By this time, students have come to realize how a knowledge of sig figs can make one’s life easier. So carry on proselytizing on the topic of significant figures; it’s a fight worth fighting.
They’ll love you… eventually.

Notes and references

1. Jean Hein, the dedicated editor of this publication, uses the example of the change in one’s body mass, post-hair cut. We’d have to weigh ourselves to the gram (!), for Pete’s sake.
2. That said, the World Health Organization reports that UV radiation increases 10 — 12% for every 1000 m increase in altitude. www.who.int/uv/uv_and_health/en/
3. J. Shields, Chem 13 News, November 2017, uwaterloo.ca/chem13-news-magazine/november-2017/feature/close-enough-said-no-chemist-ever — this is an excellent activity, well worth the time.

[If you would like a package with Michael’s PowerPoint, review and activities for significant digits, email Jean Hein, jhein@uwaterloo.ca.]