August 30, 2015 Filed in: PER Corner

OK, let’s begin by admitting that we are all playing a numbers game. Or, at least, we make our students play this game where they bet their marks on correctly figuring out the last digit to write down in their answers. (The classic numbers game is an illegal betting pool where people try to guess the last few digits of some “random” number like a stock price listing.) To make it sporting, we teach our students rules for identifying the significant digits in a given number and rules for deciding how many digits to keep after a calculation. Now, you likely know what happens next. For the rest of the year we are plagued by noisome questions during lessons and tests: “How many significant digits does this have?” “Is this two or three?” “Mr. Meyer, you started with 1000 and your final answer was 17.5 m/s ...” Sound familiar?

Now, how did we come up that quantity 1.75 s in the first place? After all, my calculator, in all its obsessive compulsiveness, gave me an average time of 1.75395742 seconds. I had to decide when the significance of the digits became so small that reporting the digit no longer had any use. When collecting our data for the ball drop, we would have noticed a certain amount of spread or variation between each time measurement. If the amount of spread turned out to be fairly small (all our measurements are fairly close together), we would expect our averaged result (1.75 s) to quite closely match the true value. If the amount of spread was great, we expect a great difference between our result and the true value. One technique for finding the amount of spread or uncertainty is through the standard deviation and determining the standard error of the mean. We won’t focus on the statistical details here, since I do not use these techniques with my students, we will just skip to the end. The final result of the calculation for the uncertainty (the standard error of the mean) in the time for the ball drop is a value of 0.07 s. This tells us that there is a pretty good chance (~68%) that the true value lies with a range of values: 1.68 s to 1.82 s. A short way of writing this is: 1.75 s ± 0.07 s, or 1.75(7) s. In real science (the kind we ought to teach), only the uncertainty allows us to decide on the significance of any digits. With an improved experiment we might be able to reduce the uncertainty to 0.006 s. Then we would be justified in writing 1.754 s as our result. Given the result alone (1.75 s), we cannot decide how much uncertainty there is and how many digits to keep. A measurement such as 2000 m might indeed have 1 significant digit or it could have 4 – it depends on the uncertainty! Our traditional rules for significant digits are not only wrong; they do not even start students down the right path. We are justified in abandoning them entirely.

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While this may burst some blood vessels in statisticians, I think it is a reasonable start for my students. It allows them to begin thinking about ranges of acceptable results and it opens an important mental door for scientific decision making. It’s interesting that this is exactly the part of science that most frustrates the general public. Scientists seldom talk about absolute certainties. They say obnoxious things like, “we conclude that climate change is caused by human activity with a 95% (2σ) level of confidence.” Reality comes with a grey haze: there is no one correct answer to most scientific questions, instead there is a range. Next, I introduce a simple scientific decision-making rule: if two results overlap in their range of uncertainties, they agree (apologies to stats people!).

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With these techniques, students can make crude, but reasoned, decisions about the agreement between their prediction for the ball drop (1.71 s) and their results (1.75 s ± 0.07 s). How often have you had a perplexed student ask if their experiment was “OK” since they didn’t get the exact answer they were looking for? In the past I would have glibly responded, “oh yeah, that’s close enough,” leaving the student mystified, but now willing to blunder on. With this new framework for scientific decision making, my students have a basic, conceptually correct set of tools to help them think scientifically that are a good first-order (zeroth-order?) approximation to the proper techniques.

Now, it is not always practical, or desirable, for students to collect a set of data and find an average for each quantity they measure (we do lots of measuring). When one measurement is sufficient, we report the readability of the result from that device and use the readability as an uncertainty. The readability is our term for the reading error: a subjective estimation of the amount of uncertainty in a single measurement that depends how well the experimenter can use the device. For example, if a student can reliably estimate to half a millimetre on a typical ruler, she could report a readability of ± 0.05 cm.

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- The uncertainty in a quantity is the only way to determine the appropriate number of digits to record

- When multiple measurements of one quantity are made, the uncertainty is estimated from the spread of the data: σ = (highest datum – lowest datum)/2.

- When a single measurement is made, the uncertainty is the readability of the measuring device for that student

- Two quantities agree with one another if the range of their uncertainties overlaps

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All useful physics calculations are (or should be) based on measurements. Even the practice problems with made-up numbers found in textbooks should be thought about in that light. Unfortunately, these questions never give uncertainties with their values. So in a certain sense, there is no value in worrying about the significant digits in these situations. I used to teach rules about how to carry an appropriate numbers of “significant digits” through a calculation. Unfortunately, these rules are internally inconsistent and do not contain the seeds of the correct understanding. A new and improved rule shouldn’t start from the flawed premise that a single written number tells us anything about the significance of its digits. From a practical point of view, we do need some sort of guidelines so students will avoid two problems: the over-rounding of results and the mindless copying of every digit the calculator gives. The first problem is a real and serious one – the usefulness of their results will be lost with too much rounding. The second problem is a minor one and more a matter of convenience.

The first new rule for the results of calculations without uncertainties is: express a final result with three significant digits. Three digits give a result roughly reliable to one part in one thousand. This is more than sufficient for most purposes at the high school level. Without the guidance of uncertainties, don’t even attempt a more complicated rule – what would you gain from it? The second new rule is: record the results of the middle steps in calculations using one or two extra digits, or guard digits. These digits help to protect the final result from a loss of accuracy due to rounding. Voila. Together, these are simple, clear and reliable rules that don’t cloud students’ minds with faulty reasoning.

- Estimate an uncertainty for the final result and use that to determine how many digits to write down

- Record final results with three significant digits, as determined by writing the number in scientific notation

- Record intermediate results with one or two extra guard digits

I encourage you to try out this approach to numbers in science. I hope you will agree that it encourages better scientific thinking and reduces the number of annoying questions from students. Remember: all those “annoying” questions are really students’ way of pointing out the weaknesses of what we are teaching teach them.

Many of the ideas I have presented here were adapted or inspired from the work of John Denker. I encourage you to explore his article: Uncertainty as Applied to Measurements and Calculations

- P.S. Don’t call them significant figures. Why? Ask a random student what a figure is. Then ask what a digit is. No further justification should be needed.

- P.S. Don’t use the term error anywhere! Ask a random student ... you know the drill. Use the term uncertainty.

Here is the complete lesson and corresponding homework that trains our grade 12 students in these techniques. Click here to download it in pdf format.

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