Advice with Significant Figures
Posted by: Scott 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?
15 Responses to "Advice with Significant Figures"
January 19, 2012 at 10:18 am
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.
January 19, 2012 at 12:59 pm
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
January 19, 2012 at 9:19 pm
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.
January 19, 2012 at 9:25 pm
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.
January 20, 2012 at 9:00 pm
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?
January 20, 2012 at 9:40 pm
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.
January 22, 2012 at 10:20 am
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.
January 22, 2012 at 11:15 am
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.
January 23, 2012 at 8:05 am
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.
February 22, 2012 at 9:19 pm
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.
February 22, 2012 at 10:13 pm
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.
February 23, 2012 at 10:44 am
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!”)
What do you think? Your opinion matters, leave a reply.
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January 18, 2012 at 11:35 pm
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.
January 20, 2012 at 10:36 pm
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
January 23, 2012 at 8:02 am
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.