Physics & Physical Science Demos, Labs, & Projects for High School Teachers

Advice with Significant Figures

Posted on: January 18, 2012


Teachers – how are you handling significant figures?  I’m a bit at odds with my textbook and I’m wondering what the rest of the world is doing.  I’ve discussed my issues with our chemistry teacher, he tends to agree with me, but it’s just the two of us.  Let me explain.

I teach from Holt Physics.  The book treats sig figs mostly okay.  When they provide numbers for problems, they are always precise.  Usually the numbers are in scientific notation, so you know where you stand with your given information.  The book does state that 1500 could be 2 or 3 or 4 significant figures because we don’t know about the two trailing zeroes.  I tell the students to err towards caution in those cases and treat that number as though there are 4 sig figs.  The book correctly states that the answers are rounded to the least significant number of figures.  You all know what I mean.

Here is where we part ways.  I teach my students to carry an extra place while doing calculations.  For example, if I’m dividing 35 by 62, my working answer is 0.565.  If this is my answer to the problem, I would round this to 0.57.  If I’m using this number in another calculation, I would use all three digits.  The textbook rounds this here along the way even when it is used later.  I’ve even seen problems where they have rounded more than once in the same problem.  (There is no way I can remember the actual problem right now.)   The results are often an error of about 10% difference between my answer key and my calculations.

While we are at this, I have a question I’m stuck on.  Suppose you read a meter stick and you get a reading of 8.65 cm.  That is three significant figures.  Now you move a little ways up the ruler and read 22.40 cm.  The accuracy of the ruler hasn’t changed, but I’m now working with 4 significant figure versus 3 before.  I wouldn’t round the second number, it is as accurate as the device, but the first number isn’t 8.600.  How do you account for this when you are dealing with the significant figures of a problem?

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27 Responses to "Advice with Significant Figures"

I have been teaching significant figures as part of an IB physics class I am teaching (the first year I’m teaching the course) and have run into many of the same issues. It’s nice to know I’m not the only one.

As was explained to me, and what I tell my students, is never round until the end of the problem. My textbook does this as well and I’m surprised to hear the Holt book rounds in each step.

As for measuring on a meter stick, you are correct that you could have 3 sig figs in one measurement and 4 sig figs in another. The accuracy of the ruler is still the same however because each measurement is to the same decimal place, in this case the hundredth’s place. Also, when you are adding and subtracting the total number of sig figs don’t matter only the least significant digit. For example, say you want to find the difference between you two measurements of 8.65 cm and 22.40 cm. That would leave you with 13.75 cm. This is fine because the least significant digit is still the hundredth’s place and your answer goes to that decimal. Any other math operation requires your answer to use the least number of significant figures.

I’m curious to hear how other people handled some of these issues as well, or corrections to my thoughts.

Matt,
I agree and that makes sense for addition and subtraction, but if you are multiplying, your answer is no longer the 4 digits, it’s down to three. What is really happening with the meter stick is the accuracy is read to the tenths place and estimated to the hundredths. The number of digits preceding the decimal isn’t a factor, yet it is depending upon where we are reading the ruler or meter stick.

I have a 100 meter long tape. The accuracy of the markings are the same throughout. I can accurately read and estimate 7.65 cm, 17.65 cm, 107.65 cm, and 1007.65 cm. Is my tape accurate to 3 sig figs or 6? The answer here is that it is accurate to two decimal places, but we don’t have any rules to deal with that using multiplication, do we?

Scott

I’m with you on everything. Don’t round off to the proper sig figs until you are giving your final answer. And 1500 is two sig figs unless otherwise noted. Placeholder zeroes don’t count because they would not be there in scientific notation unless a decimal is added after them.
About the single digit meter stick dilemma, I have no thoughts, it happens on triple beam balances, and pretty much all other instruments as well. Thats just the bummer of how it goes I suppose.

I have always been taught, and now I teach that students should not round in any intermediate steps. Along that same line, I also teach my students that when all is said and done they should use the least amount of significant figures given in the original problem as the number of sig figs for their answer.

Many of my students do struggle with sig figs still however. With some rounding too early and other not rounding at all. It is definitely a constant struggle to try and get them all on the same page.

depending on which math operations you are doing:
addition/subtraction -> keep least sig (precise) number of decimal places
any other operation (mult/div/square…) count number of sig figs.

I tell students keep at least one more than number of sig figs for calculation, in lots of science classes you can use technology to keep ‘extra’ digits for calulation ie speadsheets, graphing calcs…

the 1500 example has rules to follow
– if counting things (natural number) its ‘infinitly’ signficant
– if measured it’s only 2 unless on of zeros has a bar over it or the use of the decimal 1500 -> 2 sig figs 1500. -> 4 sig figs

as far as advice on how to handle your textbooks misuse of sig fig, point your students to an authorative source and say ‘we are going do it right’ even if the text doesn’t (ASTM E29-06b)

or a lot less authoritative http://en.wikipedia.org/wiki/Significant_figures

I think significant figures are more a matter of level of actual accuracy than some convention or style about how many numerals you show. So for rulers, the rule so to speak, is to round to the accuracy (nearest tick mark) not “how many digits” are in the number’s representation. Looks like others are thinking in similar ways.

I tend to agree with the comments above. When I teach sig figs, I always tell them to continue the operation in their calculator (by using the [2nd], then [Ans]) so that they do not round until the very end.

I also think its important to note that when performing calculations based on specific equations, I instruct my students to ignore sig figs of constants within the equations. For example in Chemistry, students always use 6.022E23, which is technically 4 sig figs. But if they are working with more or less sig figs (to where the answer needs, say only 3 sig figs), just ignore the Avogadro’s # sig fig and only go with what’s given in the problem.

The rules for sig fig in our chemistry textbooks are for addition and subtraction keep the least amount of places past the decimal ex:
24.6+4.23 would be 28.9 When rounding the answer you also need to follow the rounding rules for sig fig. multiplication and division you would keep the least amount of sig fig. I’m not sure how you can get two different amounts of significant figures using the same meter stick. When taking measurements the rules are all known digits and one estimated. I’m assuming the last zero in your example was maybe an estimate?

Yelena, I think the issue is the existence and use of non-contextual standards for SDs that specify how many to use, like “three digits” which would indeed be different from consistency with what actual power of ten is relevant in the digits. Hence 348 and 3.48 both have three digits but the latter is being specified to hundredths. If we applied the same absolute standard to the first, it would be 348.00, etc.

Well, I’m going to go against the grain here. I spend very little time on significant figures in class. I mention them. I explain how they are found. I tell the students that when they get to college, they may have a profession that cares a whole lot about them, but that for our purposes we’re just going to round to 2 decimal places.

I do teach that when you are using a measuring device that does not have electronic output you should report out one more place than is directly read and that it’s an approximation.

Kathy, I don’t think you are necessarily wrong and I wouldn’t mind doing the same. But how do you deal with numbers where your answer is 0.0065 kg? Does that become 6.50 grams or 0.01 kg?

I just had a similar discussion with my honors physics class this week where I explained that when I taught Conceptual Physics, we used g=10 m/s^2. They were outraged. I explained to them that you could do a lot of the math in your head and it is only a 2% error from 9.8. We often get larger errors than that by rounding to soon.

The real issue is getting the kids to understand the accuracy of the measuring tool. Just because a stop watch reads 22.71 seconds doesn’t mean my reflexes make the time accurate to two decimal places.

I’m still struggling with this whole subject. Maybe telling them how many digits to round each problem is the way to go.

No one is going to tell them to what precision to round if they ever do their own scientific investigation. I think it’s better to teach them proper sig figs and be honest about the ambiguity when you get to specific instances. My wife is working on a graduate thesis right now working with lead concentrations in urban soils. If no one ever taught her sig fig rules when reporting data I’m sure she would be quite upset. Its not perfect but at least its something to go with.

The problem here is that (I believe) sig figs are a cheap way to get around doing actual uncertainty analysis. The point of sig figs is that you can’t know any particular value precisely enough for it to be ‘perfect’ when measuring. However, when doing word problems this is an artificial problem as the numbers are all made up anyway.

To solve this problem I ignore sig figs in problems (just round to 2 decimal places, as stated above) but I hit them hard in terms of lab (did you really just measure that velocity to a tenth of a mm/s?). This is the natural habitat for significant figures.

To answer the question as to why 22.40 would have more sig figs than 8.65 cm; the relative uncertainty (uncertainty/value) is lower (more precise) and sig figs approximate that idea. The last sig fig denotes the uncertain figure, so lets assume an uncertainty of .05 cm. Therefore the relative uncertainties are 0.05/22.40=.22% and 0.05/8.65=0.5%. Taking a more extreme example, if you are measuring something that is 2 mm long with this type of ruler, your relative uncertainty is .05/.2=25%; your value is much more uncertain with the same uncertainty value if the value itself is smaller, hence the need to switch instruments as you have smaller and smaller measurement needs.

Casey, That’s a really good explanation of the problem. I agree we are working around the uncertainty analysis, but it’s better than stating on every problem “round to xx places” when the numbers are all made up. I know I explained sig figs and their purpose in the beginning of the year. We even did a lab to help them understand uncertainty, but now I think they do sig figs because I require it. I would bet they have forgotten the reasoning behind it.

Sig figs is probably my best choice for now. It is, after all, high school physics; we have to pick and choose our battles. I’ll continue with what I’m doing and, as always, look to improve upon my methods in the future.

I totally agree about picking and choosing battles; this is one I move away from in my general physics class. However, I still emphasize the purpose of sig figs as a lab technique, and it pops up later in the year (S: “I got 192 and she got 194, why is that?” Me: “Well, which figure is uncertain?” S: “Oh, it’s rounding errors!”)

I had to read all the comments to see if anyone mentioned uncertainty analysis. I can honestly say I did not understand significant figures or uncertainty until I took physics in college. It was definitely a stretch for me to be able to do uncertainty, and the TA’s were never very helpful.

I completely agree that significant figures are there to help keep track of +/- values (i.e. the +/-0.05 cm). I wonder if it would be worth exploring the idea of having students report uncertainty values (not necessarily percent analysis) with every instrument they use, but primarily with meter sticks, balances, and spring scales. At least then they would realize that they cannot report any lab data that has more digits than the +/- value. (Does this +/- value have a name by the way? just “uncertainty”?)

I think sig figs also depends on the year and level of the class. I don’t even go near sig figs in 9th grade physical science. I used to teach it at the beginning of the year for chemistry and I taught all the math operations with it, but then I never used it. I remember hating that part of chemistry in high school. Always having to be aware of the digits drove me nuts. As a teacher I feel like kids worry so much about the answer and being right that they often ignore the process, eventually I did away with sig figs unless it was Honors level chemistry. Recently, however, I’ve been trying to find better ways to make students of aware of how to recognize errors, including the explanation of experimental error and suddenly I’m feeling like I need to revisit my stance on sig figs, at least for the 11th grade chemistry kids.

I ignored sig figs the first couple years teaching physics. As I got better at teaching the course, I found a way to teach it and maintain it throughout the year without it becoming a burden. For one thing, every sample problem is rounded, so they see it several times a day from me. When their answer differs from the key, they know that sig figs and rounding is the first place to look.

On my standards-based quizzes, I don’t worry about sig figs on the C-level problems, but they have to be correct on the B and A-level problems. I know they don’t get it completely, but at least it is on their mind when they are writing down their answer.

On a related but slightly different note, I have an issue with your rounding of 8.65 to 8.7. This is a common “error” that I see all the time including in textbooks. 8.65 is no closer to 8.6 than it is to 8.7 so why always round up??? Most people do this. As a statistics and science teacher, I remind students that this “rule” will tend to skew a large data set to the high side artificially. The “rule” to avoid this is to always round to the even number when the only digit being eliminated is a 5. For example, 8.65 rounds to 8.6 and 8.75 rounds to 8.8. By doing this, you will round up about half of the time and down the other half…..keeping the mean unchanged in a large set of data. I know this is picayune but it is one of those things that brings out my stats training.

I don’t think I’ve ever heard that before, but I like it. Thanks, I’ve learned something today.

Interesting take on it. I was not a statistics student ever, so this gives me something to think about and consider.

I also teach my student this rounding rule and many comment that they have never seen this before. Once I explain the reason to them most easily accept it.
As far as sig figs go; I make my students use them only in lab work which is where they are relavent. What confuses many student is using sig figs when they are not relavent. At the start of the year, I ask student why they think they do not learn about sig figs in math class. I then point out that they are not important in Algebra class because the numbers are just numbers they are not measurements and therefore there is no need for accuracy. Lets face it in math class they are told an answer like Sqr Root of 7 meters is acceptable, because that is the most exact answer.
On word problems I always tell them to give me a reasonable answer which is usually 2 or 3 places after the decimal point.

Great discussion. Before entering the teaching field I was a research chemist where significant figures DID matter. We had some very complicated rules for rounding and other mathematic operations to avoid propagation of errors. Now, enter the world of teenagers. Much different story. Quite often they use the rounding/sig figs issue as a small battle to be “more right” with their answer. The best approach I have found is to show the students that the expression of a measurement is more of a communication tool. In other words, if I record a measurement of 8.65 cm or 22.60 cm, I am communicating the precision of my instrument to be ± 0.01 cm. Or if you decide to convert to mm, the precision would need to be expressed as ± 0.1 mm. Also, they learn something about consistency in measurement. If their data says 5.40 cm for one measurement and then 4.3 cm for another, I challenge them by asking them if they used a different ruler than the first measurement. This gives me an opening for talking about specifications and how to look them up in a manual or online. The idea of using the right instrument for the right measurement is also a concept I believe to be helpful. Using a triple beam balance doesn’t make sense for a watermelon because the mass is on the order of kg, using an instrument with precision of 0.01 g would not be wise scientifically and could cause damage to the more sensitive equipment. This allows me to talk about the possibility of stress on an instrument. Where using it beyond its limits may compromise its internal integrity and overall precision for future use.

Howard, Great comments.

I tried responding to your email but there is something wrong with your address. Email me with a working address.

Scott

It’s really cool! Your blog is really informative! I am also Physical Science teacher at an International School in the Philippines. Since I am new here, I find their practice hard. It is progressive and aims for experiential learning. You understand that some of the topics in Physics cannot be associated with an activity, mostly if it is in line with Math! I hope I can always reach you out whenever I have a complex topic! I am hoping for your reply!

You can always use the “contact me” page to reach me. I’m always around and usually quick to reply.

Your example with the ruler of having a measurement of 8.65 cm and 24.00 cm shows the “precision” of the instrument. In other words, it lets all who read your data know that the instrument you used had centimeter divisions and millimeter divisions but you had to estimate the next value. The actual number of significant digits is irrelevant unless you need to multiply or divide these numbers. If you are adding or subtracting them then it is the number of decimal places you have that is important. For example if you needed to find the area of rectangle with length and width using those values, your answer on a calculator would be: 207.6 cm^2…. but since you are multiplying the least number of significant digits is important, so the answer would be rounded to 208 cm^2. If these values are position measurements and wish to know the displacement starting at 8.65 cm and ending at 24.00 cm you would subtract and get this on your calculator: 15.35 cm. What’s important here is the number of decimal places… you have two in both of your data measurements, therefore your answer remains with two decimal places. What sometimes is confusing is if you had to use measuring tools with different precision and you had 24.0 cm and 8.65 cm. Your calculator answer would still be 15.35 cm, but since the 24.0 cm measurement only has one decimal place, your answer with significant digits would be 15.4 cm.

And to Howard, I like how you use precision as a way to talk about the integrity of the instrument the students are using. This is core of why significant digits are important.

And to Scott about rounding with “5”. One doesn’t round with a 0, 1, 2, 3, or 4 and does round up with a 5, 6, 7, 8, or 9. In other words exactly half the time one rounds up… why do you make a specific rule for “5”? If you are rounding, you must remember this value that influences whether you round or not is of no significance.. and that “0” may not truly be a zero, so we have a rule in place that the lower 5 numerals (0-4) don’t change the value of your answer and the upper 5 numerals (5-9) do round up. Exactly half the time one rounds up and exactly half the time one doesn’t, and no you don’t heavily influence rounding up by doing this.

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