# Sig figs - How strict to be as a grader?

• Derrick Palmiter
In summary, when it comes to applying the rules of significant digits, it is sometimes unclear when it makes sense to do so and when to just round to the nearest number. Different calculations may call for different approaches, depending on the situation.
Derrick Palmiter
I'm in my first year teaching Physics. My students all understand the importance of significant digits and the rules for applying them, but in some more complicated calculations, like multi-step vector component problems that also require unit conversions, it's a bit unclear, even to me, when it makes sense to apply the rules of significant digits, and when to wait. Just looking for some input from more experienced teachers, as I am not sure how to grade my student's work. I can, in some cases, see legitimate reasons for two separate answers, depending on when rounding due to significant digits is invoked.

Many thanks in advance for any insights.

Derrick Palmiter said:
My students all understand the importance of significant digits and the rules for applying them, but in some more complicated calculations, like multi-step vector component problems that also require unit conversions, it's a bit unclear, even to me, when it makes sense to apply the rules of significant digits, and when to wait.
You always keep all the significant digits that your calculator/Mathematica/Matlab can handle in all middle steps (although you should not write them down if you report middle steps). Do not round your results until the very end or you will introduce errors that may grow in coming steps.

When you do unit conversions, you should still keep the rules of significant digits in mind. Nobody would measure their waistline to be 91.44 cm - although 36'' would be a reasonable measurement - or 91 cm.

I find the subject of significant digits a bit lacking anyway. In a sense, it is just a poor man's error analysis and learning/teaching arbitrary rules of thumb always seemed a bit annoying to me.

I think you need to encourage common sense and show some common sense. If the right answer is 91 cm, I would treat 91.4 better than 91.4400023181236.

CalcNerd and symbolipoint
Orodruin said:
You always keep all the significant digits that your calculator/Mathematica/Matlab can handle in all middle steps (although you should not write them down if you report middle steps). Do not round your results until the very end or you will introduce errors that may grow in coming steps.

I think Orodruin has this exactly right.

Derrick Palmiter said:
I'm in my first year teaching Physics. My students all understand the importance of significant digits and the rules for applying them, but in some more complicated calculations, like multi-step vector component problems that also require unit conversions, it's a bit unclear, even to me, when it makes sense to apply the rules of significant digits, and when to wait. Just looking for some input from more experienced teachers, as I am not sure how to grade my student's work. I can, in some cases, see legitimate reasons for two separate answers, depending on when rounding due to significant digits is invoked.
I tend to do calculations the way Orodruin described, but I find most of my students don't. What they'll do is calculate some value, write this intermediate result down, and then use the number they wrote down in subsequent calculations. They won't store intermediate results in the memory on their calculator. Given this behavior, I suggest that they first figure out how many sig figs they need at the end and keep one extra digit during calculations. For example, suppose the values in a projectile problem are given to two significant figures. If their calculator says the time of flight for a projectile is 0.451753951 s, they should report 0.45 s for the time of flight, but if they need to use that quantity to find, say, the range of the projectile, they should use 0.452 s in that calculation.

If you want to make it even simpler, since most textbooks tend to provide values with two or three sig figs, just tell students to keep four digits while calculating and round to the correct number of sig figs at the end.

I do make the distinction that V50 noted. If a student has one extra sig fig, I'll let it slide, but if they write down 10 digits in their final result, I'll take a point off, which can get quite costly for them.

I find the traditional approach to significant digits both lacking motivation and lacking rigor. I tended to tell my students to carry two more significant digits than required in intermediate results (and write them down), and to only round the final answer according to the traditional approach. I don't recall ever deducting points unless there were too few sig digs to determine correctness or waaaaaaay too many. I'd just circle errors in red and make a note.

I try and convey a more realistic yet practical sense of handling uncertainty. In the lab, one approach is to make a measurement multiple times and compute a standard error of the mean for the multiple trials. In the classroom, this includes both ideas relating to significant digits and simplifying assumptions. Sure, using g to three sig digs might reduce uncertainty to < 1%, but probably not when neglecting air resistance in a projectile motion problem.

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Derrick Palmiter said:
I'm in my first year teaching Physics. My students all understand the importance of significant digits and the rules for applying them, but in some more complicated calculations, like multi-step vector component problems that also require unit conversions, it's a bit unclear, even to me, when it makes sense to apply the rules of significant digits, and when to wait. Just looking for some input from more experienced teachers, as I am not sure how to grade my student's work. I can, in some cases, see legitimate reasons for two separate answers, depending on when rounding due to significant digits is invoked.

Many thanks in advance for any insights.

There are two questions here- 1) what is the 'correct' way to perform a numerical calculation, and 2) what are the 'correct' expectations for your students?

The first question has been answered, more or less. The second question depends a lot on your student population- what grade are you teaching? what do your peers expect from their students? what are your goals for your students?

Andy Resnick said:
There are two questions here- 1) what is the 'correct' way to perform a numerical calculation, and 2) what are the 'correct' expectations for your students?

The first question has been answered, more or less. The second question depends a lot on your student population- what grade are you teaching? what do your peers expect from their students? what are your goals for your students?

Thank you for the response. I'm teaching junior and senior level high school physics. I'm not really aware quite what my peer expectations would be, hence, my posting here. I'm trying to educate myself on what may typically be expected by other high school science teachers. In general, I would expect my students to be proficient enough with working with significant digits and their application in multi-step physics problems, that upon entry to college level physics work (engineering physics) they do not have to struggle at all with keeping track of significant digits or to deal with rounding errors. I want them to be proficient enough that they can focus on the new concepts they will be introduced to at the undergraduate level and not have to waste any time relearning how to work with significant digits.

I hope this helps clarify what I'm looking for. I can give you an example problem if that would help.

Thank you to all for your insightful responses. I've read several of your recommendations in other places, but was not sure how prevalently held these opinions were. Saving the rounding for the end result makes good sense to me.

I'd like to thank Vela, especially, for pointing out, that for me it seems simple to follow Orodruin's advice, and as someone with experience working with college level higher math and science, this might seem obvious, but students at the high school level are not always formed to handle/treat formulas and calculations purely symbolically in that way. I try to encourage them to do this, and will usually work example problems according to that method, but it's difficult to overcome the inertia of all of their other math and science classes, or their at times "elementary school" level of abstract ability.

Should we round the intermediate results? This seems like it should be an easy question. Yet I have seen this question answered 'yes' by some teachers and 'no' by other teachers on their websites. So maybe it's not an easy question after all?

I have this problem in my notes somewhere. I did not create the problem. No doubt we can come up with similar problems. This is a simple problem which could be done by hand.

The problem:

(1.8 x 2.10) x (1.542) = ?

The solution:

First I do it without any intermediate rounding.

1.8 x 2.10 = 3.780

Actually if I do this on my calculator it reports the answer as 3.78, but I'm doing it by hand also.

3.780 x 1.542 = 5.828760

Here again my calculator drops the final zero and gives 5.82876.

Since 1.8 has only 2 sf I round the final answer to 5.8.

Now here is what happens if I round during the calculation.

1.8 x 2.10 = 3.780

Round to 2 sf gives 3.8

3.8 x 1.542 = 5.8596

Round to 2 sf gives 5.9

Which answer is correct? I say 5.8, which means rounding only the final answer is correct in this example. Of course there are counterexamples where this type of intermediate rounding does not change the final answer.

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Dr. Courtney
P.S. to answer your question more directly, I think the only fair thing for the students is to establish clear rules for sfs so they know exactly what they are expected to do. If there are exceptions to the rule in a particular problem, then mention that in the problem. Then grade accordingly.

If neither the textbook nor the teacher makes the rules clear about this, then it's not fair to penalize the students if they don't know what to do, particularly since even teachers argue about this issue.

David Reeves said:
Should we round the intermediate results? This seems like it should be an easy question. Yet I have seen this question answered 'yes' by some teachers and 'no' by other teachers on their websites. So maybe it's not an easy question after all?

I have this problem in my notes somewhere. I did not create the problem. No doubt we can come up with similar problems. This is a simple problem which could be done by hand.

The problem:

(1.8 x 2.10) x (1.542) = ?

The solution:

First I do it without any intermediate rounding.

1.8 x 2.10 = 3.780

Actually if I do this on my calculator it reports the answer as 3.78, but I'm doing it by hand also.

3.780 x 1.542 = 5.828760

Here again my calculator drops the final zero and gives 5.82876.

Since 1.8 has only 2 sf I round the final answer to 5.8.

Now here is what happens if I round during the calculation.

1.8 x 2.10 = 3.780

Round to 2 sf gives 3.8

3.8 x 1.542 = 5.8596

Round to 2 sf gives 5.9

Which answer is correct? I say 5.8, which means rounding only the final answer is correct in this example. Of course there are counterexamples where this type of intermediate rounding does not change the final answer.

Yes, that's just it. Exactly the type of issue I'm dealing with, except in a multi-step physics problem, the intermediate answer can be off by quite a bit more than just a tenth. Today in class we looked at a Force vector problem or two involving friction, where, depending on where the significant figures were applied, we had 3 possible answers of 510 N, 530 N, and 560 N, with the last answer being that attained by following the recommendation of only symbolic manipulation until the final calculations. A difference of 50 N out of 550 is nearly 10%! That seems like a significant problem, depending on application. This was a problem that came directly from the textbook, which reported 530 N as the answer.

Derrick Palmiter said:
Yes, that's just it. Exactly the type of issue I'm dealing with, except in a multi-step physics problem, the intermediate answer can be off by quite a bit more than just a tenth. Today in class we looked at a Force vector problem or two involving friction, where, depending on where the significant figures were applied, we had 3 possible answers of 510 N, 530 N, and 560 N, with the last answer being that attained by following the recommendation of only symbolic manipulation until the final calculations. A difference of 50 N out of 550 is nearly 10%! That seems like a significant problem, depending on application. This was a problem that came directly from the textbook, which reported 530 N as the answer.

That's quite a difference! Can you post the details of this problem? Also, what does the textbook say about significant figures?

Derrick Palmiter said:
Yes, that's just it. Exactly the type of issue I'm dealing with, except in a multi-step physics problem, the intermediate answer can be off by quite a bit more than just a tenth. Today in class we looked at a Force vector problem or two involving friction, where, depending on where the significant figures were applied, we had 3 possible answers of 510 N, 530 N, and 560 N, with the last answer being that attained by following the recommendation of only symbolic manipulation until the final calculations. A difference of 50 N out of 550 is nearly 10%! That seems like a significant problem, depending on application. This was a problem that came directly from the textbook, which reported 530 N as the answer.
10% might not be that problematic. It all relates to the precision of your measurements and you might have a good lesson to teach here.

Let's take the example by @David Reeves . A good estimation for the reading error is ±1 on the last digit. So the numbers of the example are:
• 1.8 ±0.1
• 2.10 ±0.01
• 1.542 ±0.001
Therefore the extreme cases (rounded to 2 sf) are:

1.7 * 2.09 * 1.541 = 5.5
1.9 * 2.11 * 1.543 = 6.2

So nobody could say more precisely that the answer is somewhere between 5.5 and 6.2. When rounded to 3 sf, one might also say 5.83 ± 0.36, which is ±6% or a 12% range !

Well, I always like to remember Mt Everest in my comments about approximation. During the1920-1940s time frame, Everest was surveyed to 29,000 ft. Well, how accurate is That!? Well, with this oddity, it was to 1 ft. But how to convey that? Simple, the survey team added a bogus 1 foot, so that during this time the official height was 29,001 ft. It was wrong, but the right answer could imply an error of 10 or 100 foot, depending upon implied accuracy. Today, we KNOW the exact height is actually 29,029 ft, but that is changing by a few mm every year as well.
.
Same principle would apply to a library boosting a million books. What if said library really had exactly a million books. Best to buy one more, so that you can express your count as 1,000,001 or else have others ridicule your hyperbole.

CalcNerd said:
Everest was surveyed to 29,000 ft. Well, how accurate is That!? Well, with this oddity, it was to 1 ft. But how to convey that?
What is wrong with ##29000\pm 1##?

David Reeves said:
That's quite a difference! Can you post the details of this problem? Also, what does the textbook say about significant figures?
Sure, here it is:

A man is pushing a 150.0 kg desk across a floor. The frictional coefficients between the desk and the floor are μs = 0.45 and μk = 0.35. If the desk is already moving, how much force must the man exert to keep the desk accelerating at 0.30 m/sec2?

The textbook's rules for sig figs are thus:

A digit within a number representing a measurement is considered to be significant if:

I. It is non-zero OR
II. It is a zero that is between two significant figures OR
III. It is a zero at the end of a number and to the right of the decimal point.

When adding and subtracting with significant figures, round your answer so that it has the same precision as the least precise measurement in the equation.

When multiplying and dividing measurements, round the answer so that it has the same number of significant figures as the measurement with the fewest significant figures.

That's really all the textbook says about it, aside from discussing how to determine how precise a measurement is intended to be when speaking of precision and accuracy with measurements.

No rules are really given as to when to apply the rules of significant digits, but from the example problems, the author seems to apply them every time he makes an intermediate calculation; however, I've noticed some inconsistency there. He tends to not work things out entirely symbolically. It's frustrating for my students.

So, in the problem given above, if I round to account for significant digits at all intermediate steps, I come up with, as the author does, 580 N as the final result. But if I work symbolically and only apply significant digits at the end, I get 560 N. I guess the numbers weren't as far apart as I remembered, looking at it a second time, but I think the way I posed the question to the students originally also may have required a unit conversion...giving them the mass of the desk in slugs or something. This added another intermediate step with more rounding and gave a result like 530 or 510. It's also possible that I'm mentally combining two separate problems that we worked.

In any case, the trouble remains the same. I've had a few problems like this now where it really makes a significant (in my mind) difference in the solution. I understand, of course, that that may or may not be an actual problem, depending on what application is being made of the figures in reality. Calculating particle physics is not the same thing as pushing a desk around the floor, or measuring the elevation of Mt. Everest.

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Derrick Palmiter said:
In any case, the trouble remains the same. I've had a few problems like this now where it really makes a significant (in my mind) difference in the solution.

If the student had set up the force balance equation correctly, turned the crank correctly but because of inconsistent rounding obtained a slightly different answer, how much partial credit would you award the student? Personally, I would deduct somewhere between 0% and 10%, depending on the actual answer.

I personally teach my students not too many, not too few (three is a good number). I don't teach error analysis and i hate that sig figs are seen as so important (chemistry classes in my old school would spend weeks on sig figs alone).

Like others have said, the significant digits rule is a rule of thumb and may fail in some cases. That should be the important lesson to teach. Here is a good example:
$$1.7 \times 11 = 19$$
$$1.7 \times 9 = 20$$
The important lesson is that you cannot just blindly put numbers into an equation, follow the rules to the letter, and take the result for granted. You must understand the process and its limitations.

Andy Resnick said:
If the student had set up the force balance equation correctly, turned the crank correctly but because of inconsistent rounding obtained a slightly different answer, how much partial credit would you award the student? Personally, I would deduct somewhere between 0% and 10%, depending on the actual answer.

My practice has been to deduct approximately 5% of the point value for the problem. So if the problem is 5 pts, I usually award 3 for correct set-up, and 1.5 for accurate calculations and half a point for significant digits/units/etc.

Andy Resnick
Derrick Palmiter said:
My practice has been to deduct approximately 5% of the point value for the problem. So if the problem is 5 pts, I usually award 3 for correct set-up, and 1.5 for accurate calculations and half a point for significant digits/units/etc.
Half a point out of 5 total points is 10%, double the 5% you said you usually deduct.

Mark44 said:
Half a point out of 5 total points is 10%, double the 5% you said you usually deduct.

Yeah, but to one significant figure... <ducking and running>

jack action and Derrick Palmiter
Mark44 said:
Half a point out of 5 total points is 10%, double the 5% you said you usually deduct.

It's a half point for sig figs /units, or any other info that might be needed. So, it's a quarter of a point for each taken individually. Sorry if that was unclear.

Yeah, but to one significant figure... <ducking and running>

Haha...nice...

Just some perspective. In my first year of graduate school and as a TA, my course instructor required us to take account some students were solving the numerical part of the problem with a slide rule, and most were using a pocket calculator. Intermediate steps using slide rule can lose a lot of precision. Even now, (I suppose, because I have not been teaching for a while), you may have to account for different constants.

For example, my course instructor told us, if your students want to use g = 10 m/s2, that is OK. We are not interested in the numerical answer, but if they know the method. I dislike computer scoring for similar reasons. The computer can sometimes require the student to be "exact", and be unforgiving.

I prefer to take a relaxed attitude towards significant figures. I think you should bring to the student's attention that the number of places the calculator supplies is (usually) to many places. I would bring up this issue with the course instructor and your fellow grad TA's considering you want uniformity in course grading amont all the TA instructors, anyway.

mpresic said:
I prefer to take a relaxed attitude towards significant figures. I think you should bring to the student's attention that the number of places the calculator supplies is (usually) to many places.

I have the same attitude- I only get cranky if the student simply copies out the 12 digits their calculator spits out.

## What are significant figures and why are they important?

Significant figures are digits that are used to represent the precision or accuracy of a measurement. They indicate how certain we are about a particular value and are important in scientific calculations and communication.

## How do you determine the number of significant figures in a measurement?

The rules for determining the number of significant figures can vary, but generally, all non-zero digits are considered significant, as well as any zeros between non-zero digits. Trailing zeros after a decimal point are also significant. Leading zeros are not significant unless they are followed by a decimal point.

## How strict should I be as a grader when it comes to significant figures?

It is important to be consistent and follow the rules for determining significant figures. However, it is also important to consider the context and purpose of the measurement. If the measurement is being used for a rough estimate, then being less strict with significant figures may be appropriate. If the measurement is being used in a precise calculation, then being more strict may be necessary.

## Can I round off numbers to the nearest whole number to simplify the use of significant figures?

Rounding off numbers can be useful in certain situations, but it is important to remember that this may affect the precision and accuracy of the measurement. It is generally recommended to keep all digits until the final calculation, and then round off the final answer to the appropriate number of significant figures.

## Are there any exceptions or special cases when it comes to significant figures?

Yes, there are some exceptions and special cases when it comes to significant figures. For example, when adding or subtracting numbers, the final answer should have the same number of decimal places as the number with the least number of decimal places. Additionally, when using logarithms, the number of significant figures in the mantissa (the part before the decimal point) should match the number of significant figures in the original value.

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