# How significant are sig figs?

1. Oct 26, 2007

### eli64

HI, please forgive me if I am introducing a topic that has been hashed about already, if so I would appreciate a link to the discussion (I tried searching the threads but didn't find one that really seem to talk abou this in a generalized way) or the thread moved to the appropriate forum.

I'm having a discussion with someone who is in science (engineering student who is an industry coop) but who adamantly believes that sig figs in the real world are useless. I don't doubt that in many instances that sig figs are tossed by the wayside when the impact of a few decimals here doesn't seem to make a difference but I was wondering how industry in general views this in their measurements.

His example was that in his job, his group measures the diameter of a groove in a disc, first with a ruler that is measures up to a certain distance and gives a value to 2 dec places, the rest of the measurement is done with a caliper that gives the remaining measurement to 3 dec places. The two measurements are added and they report the diameter to 3 decimal places. The issue I raised was that since the first instrument only measured to 2 decimal places, the whole measurement should have only 2 decimal places. He contends that it is more accurate to report to 3 decimal places since the caliper measured the critical remainder of the distance. They have a tolerance of 0.006 to adhere to.

What is your view of this practice? thanks for any responses

2. Oct 26, 2007

### FredGarvin

Sig figs are pretty important when it comes to data representation. Since I work in a test environment it is especially important to me.

Plain an simple, you can only measure to the accuracy of the instrument. The practice mentioned is not a practical one. The accuracy of the overall measurement should only be stated to level of the least accurate instrument. In your case the ruler.

Since a tolerance was mentioned, does your friend take into account the propagation of uncertainties into his calculations when he adds the two measurements together?

3. Oct 26, 2007

### eli64

Thanks for responding Fred, I'm not sure if my friend has ever done an error analysis since he has these views. I'm not sure if any of his physics classes required such an exercise. (Its been a long time but I do remember doing this when I took sciences) I don't think he understands that tolerances are uncertainties and that sig figs represent this.

(34.93 + 8.543)
(.725 - .676)

= 43.47 = 887 = 890
.049

890 was given as the correct answer and he contended that it should be 887 since this was mathematically more accurate and reflected the 3 sig figs of the numbers in the denominator.

Was the given answer indeed correct or is my friend right that even in math problems where add/sub and mult/div (different sig fig rules) the value with more sig figs is "more accurate".

4. Oct 26, 2007

### stewartcs

Well, I personally always carry as many decimal places as possible when calculating something. The more decimal places, the more accurate the end result. Significant figures can impose a fair amount of round-off error which could be problematic depending on what you are calculating.

Here is a link for a brief discussion on using them (toward the bottom of the page).

http://en.wikipedia.org/wiki/Significant_figures

5. Oct 26, 2007

### eli64

yes this is my friend's opinion and one that is dominant at his workplace. But I am wondering, how many decimal places have meaning if an instrument only gives accuracy to a certain number of decimal places? If that data is used for other purposes and the precision of the instrument is overstated by carrying many more decimal places that the measurement initially warranted, then I suppose the opinion is that it is safer to err on the side of overprecision than over-rounding as sig figs might impose.

Thank you for the link as well, it seems there is as much controversy what constituents significant figures as to what they are used for.

Then as Fred above implied, if measurements are only as accurate as the instrument used, all calculations that use measurements should be accompanied by an error analysis. (I remember doing this as well and hating every minute of it)

I think my next question is should error analysis, standard deviation and uncertainties be taught instead of sig figs?

6. Oct 27, 2007

### TVP45

In a for-profit industrial setting, you cannot use error analysis for every measurement anymore than you can afford to use GDT for every bolt hole. You must learn when to accurately calculate and when to use a "rule-of-thumb" like significant figures. I once worked with a very grumpy, very good, machinist who delighted in showing people why significant figures were important. When some newbie would insist on a measurement like, for example, 34.6781degrees on an angle of a simple part, the shop work might run two or three months with a charge of \$30,000. I never saw anyone repeat that mistake!

Many of us carry one extra sig fig in the middle of calculations and then discard it at the end. That's not quite right, but we get away with it.

Your young friend should be told that significant figures are standard practice and that he needs to do them or look for different work. His attitude will not serve him well in engineering.

7. Oct 27, 2007

### FredGarvin

The idea of carrying as many decimal places as you can to be more accurate is poor practice. Granted you don't want to get bogged down in doing sig figs all day when it's not necessary. However, simply taking something out to 10 decimal places is simply fooling yourself if you think that that is more accurate. You can only be as accurate as the data you are working with.

8. Oct 27, 2007

### eli64

well, I agree but I don't know how to convince my friend of this, especially when his place of work doesn't follow this practice and that I'm not an engineer (or in industry). I am, in fact, his teacher and wish to give some support to the material he is supposed to learn. As the responses show, this sig fig practice doesn't seem standard in some places and he happens to work in one of them. Whether or not, the company ever finds itself in some problem because of this practice, I don't know how else to convince him that sig figs ARE significant!

9. Oct 27, 2007

### stewartcs

I guess it would depend on the particular case. In some situations one may need to carry more decimal places than in others. If you are referring to a manufacturing enviornment, then I don't see the need to carry more than 3 decimal places (mainly for tolerances). But if you're perfoming some complex theoretical calculation you may very well need to carry more.

10. Oct 27, 2007

### eli64

stewartcs, are the instruments used for these tolerances always calibrated to 3 decimal places? for my friend, obviously it was not but the calculation was taken to 3 decimal places. Don't the tolerances imply that the instruments are matched to them?

11. Oct 27, 2007

### stewartcs

I'm not really sure about the calibration certificates. But to answer your second question, in order to machine a part to a specific tolerance, then the machine would have to be calibrated to that tolerance at a minimum.

12. Oct 27, 2007

### eli64

Does this not imply that the machine be as accurate as the other machines used? And tolerances are, in fact, sig figs in disguise?

I would think that theoretical calculations would be backed up by experimental data, measurements of constants like the speed of light can be calculated (this might be a bad example) but I wouldn't think it would really carry weight until it is experimentally measured, by instruments - very precise ones.

13. Oct 27, 2007

### stewartcs

I'm not exactly sure if this answers your question, but I will say that if a machinist is given a tolerance of +/- 0.003 inches (or whatever unit), then the machine he is using should be calibrated to that tolerance.

In my opinion, determining what is "significant" depends on the specific case. Essentially, significant figures equates to rounding off your answer. If you round off your final answer, then it's usually no big deal. For example, if you determine you needed a vent line to be 3.2454" in diameter then you would certainly round it off to 3.25". This implies that you need an instrument with 0.01" of accuracy in order to manufacture the pipe.

By the way, is this an academic question that you had given as an example?

Last edited: Oct 27, 2007
14. Oct 27, 2007

### eli64

yes, because we tell students that numbers in science most often represent measured data and so implies an uncertainty in the measurement - depending on the accuracy and precision of the instrument. I was wondering, by my friends attitude, why he was so against learning them. So I asked the question of all of you.

I am also confused, if you need a pipe with a diameter of 3.2454 inches, why would you round to 3.25 in? wouldn't you find a machine that could cut to .0001 inches? Or you figure that 3.25 inches would be wide enough with a little extra room and having that extra room is okay? I think we are looking at sig figs from opposite (but not exactly contradictory) views. Your sig figs would be the tolerance of the machine which will be used to make the part - if it wasn't practical to get a machine of .0001 tolerance. You reduce the sig figs because your machine to cut the pipe is not that accurate. I think of them as an uncertainty in the measured value.

15. Oct 28, 2007

### FredGarvin

Machining tolerances are a pretty straight forward thing. It does depend on the type of machine. For example, a good lathe operator can usually hold to .0005 to .001 depending on the type of cut. So, like Stewartcs mentioned, it depends on every situation.

In your example about the pipe, you can calculate the need for a pipe diameter of 3.2454 but the manufacturing tolerances of that pipe combined with the tolerances of the measuring instrument will more than obliterate the last two and possibly three decimal places. In that situation, from an engineer's perspective, going to a 3.25 diameter pipe would be the best thing to do. You could specify a pipe with that kind of diameter, but the cost would be trmendous. Also, it wouldn't be of any usable length since there is no manufacturing process that could hold that diameter for any appreciable length.

The sig figs and the uncertainty are compliments to each other. I don't think they are quite the same thing. I look at it like the measuring or machining tolerances help set up what sig figs you are going to use.

16. Oct 28, 2007

### stewartcs

Nice reply Fred. The costs would ridiculous, plus I don't know of a mill that would guarantee that kind of tolerance. The best I usually get from them is 8%.

17. Oct 28, 2007

### eli64

thanks Fred and Stewart, I understand this better now, but I still would like some way of helping my friend understand WHY sig figs are needed. Or he just suffers through this section and keeps losing pts because he feels he is making a statement. Maybe he just has to find himself in a project where they must be taken into account. (Fred do you have an opening for a bright but stubborn coop? ;) )

18. Oct 28, 2007

### vsingh165

Sig figs are important, especially when you're dealing with calculations involving long decimals or very large numbers (it makes things a little easier to deal with).