I did a quick ~~experiment~~ measurement tonight: I measured the length and width of a sheet of printer paper in centimeters. I came up with \( l = 27.95\space \text{cm} \) and \( w = 21.60\space \text{cm} \). Each of the measurements I believed to be within \( \pm 0.05 \space \text{cm} \).

If I want to find the area, what value should I report? \( l \times w = 603.72 \space \text{cm}^2 \) without regard to the number of figures being reported.

Now, if I use the "crank three" method to get the range of values for the area, so that I can report the uncertainty in my area calculation, I would have:

$$l_{max} \times w_{max} = a_{max} = 605.1175\space \text{cm}^2 $$

$$l_{min} \times w_{min} = a_{min} = 601.245\space \text{cm}^2 $$

Then the uncertainty in the area calculation should be:

$$\Delta a = {a_{max} - a_{min} \over 2 } = 1.93625 \space \text{cm}^2 $$

Which means, that if I am a student just learning how to do this, I might report my answer as:

$$ 603.72 \space \text{cm}^2 \pm 1.93625 \space \text{cm}^2 $$

If I'm a really good student, I might remember that my instructor mentioned something about the uncertainty indicating how many digits should be reported in the answer. Maybe I even have in my notes a simple example from class. Hmmm, now I'm just confused. It seems like there should be a way to round my answer (both the value and the uncertainty) appropriately. But, how? (Remember, I'm still a beginning physics student.)

Here's my (the physics instructor, not the student, now) point: if you hate the rules or guidelines surrounding the traditional way of doing sig figs, that's fine with me. I'll even hop on that bus with you most of the way. But at some point, there has to be an actual discussion about the significance of the digits in the answers and the uncertainty. From there on out, we can choose whatever (appropriate) method we want for finding uncertainty, right?

If I want to find the area, what value should I report? \( l \times w = 603.72 \space \text{cm}^2 \) without regard to the number of figures being reported.

Now, if I use the "crank three" method to get the range of values for the area, so that I can report the uncertainty in my area calculation, I would have:

$$l_{max} \times w_{max} = a_{max} = 605.1175\space \text{cm}^2 $$

$$l_{min} \times w_{min} = a_{min} = 601.245\space \text{cm}^2 $$

Then the uncertainty in the area calculation should be:

$$\Delta a = {a_{max} - a_{min} \over 2 } = 1.93625 \space \text{cm}^2 $$

Which means, that if I am a student just learning how to do this, I might report my answer as:

$$ 603.72 \space \text{cm}^2 \pm 1.93625 \space \text{cm}^2 $$

If I'm a really good student, I might remember that my instructor mentioned something about the uncertainty indicating how many digits should be reported in the answer. Maybe I even have in my notes a simple example from class. Hmmm, now I'm just confused. It seems like there should be a way to round my answer (both the value and the uncertainty) appropriately. But, how? (Remember, I'm still a beginning physics student.)

Here's my (the physics instructor, not the student, now) point: if you hate the rules or guidelines surrounding the traditional way of doing sig figs, that's fine with me. I'll even hop on that bus with you most of the way. But at some point, there has to be an actual discussion about the significance of the digits in the answers and the uncertainty. From there on out, we can choose whatever (appropriate) method we want for finding uncertainty, right?

## 5 comments:

As an instructor, I don't mind the plethora of digits on both the value and the uncertainty listed. However, the uncertainty helps one see which digits are not really needed. I would actually prefer students to plot the result as a point with error bars so they can see the physical meaning of the result. If they do it by hand, they'll see how careful (or not) they need to be. If they do it with some software package, the computer doesn't care about the extra digits.

Okay, I can live with that. At some point (for most physicists, at least) a calculation is going to be compared to a real measurement, and the two should be compatible with each other.

I just prefer that this thought process happens earlier in the student's physics journey, rather than later.

I'm not really a sig-fig nut job, I swear!! It just causes me to wince a bit when I get 10 decimal places on an exam question.

Andy,

I like the idea of using the hand-plotting of error bars to show that a rule of thumb of roughly 2 significant digits on calculated errors makes good sense.

One place where I am mildly strict about sig-figs is in formal reporting of results (papers, posters, talks) in my 2nd year and higher lab courses. I see sig-figs, units, proper citations, spelling and grammar all to be part of the same package. I'm reasonably loose with my rules for what is acceptable for most of these things, but there is a definitely a point at which one's use of 5 sig-figs on a calculated error or a grossly incomplete citation will lose a student some marks. In these courses I am trying to help the students learn scientific communication skills and adhering to conventions in their field is part of that.

Joss, where were you last night when everyone else was ganging up against sig figs? :-)

Yeah, that was a GPD meeting I wished I didn't have to miss. I was busy putting family time ahead of physics time.

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