Significant figures

Factual Questions
1

In another thread I asked about graphing Excel data. Reply graciously helped me by researching how to do it, and then doing it for me. I noticed that the calculated mpg numbers on the tables next to his graphs are out to six decimal places. My data includes gallons to three decimal places (which is printed on my receipts) and calculated mpg to one decimal place.

That reminded me that I’ve always had trouble with significant figures, so I thought I’d ask about them.

For example, let’s say I have a value of 0.12345. IIRC, the 5 at the end is an estimation. It might be exactly 5 (i.e., the value 0.000050), or it might be rounded. That we don’t know it it’s exact or rounded is a limitation of the measuring device. Is the 5 a significant figure? Or, since it’s uncertain, is the value to the correct number of significant figures 0.1235?

2

I used to work in a metrology lab, so I can say with some authority that you should not worry too much about the “significant digit” issue. Folks make way too big a deal out of it. Instead, this is what you should do:

  1. Record every digit displayed by the measurement device. If (for example) the device is a digital voltmeter and reports 0.12345 VDC, record 0.12345 VDC.

  2. After writing down the reading, write down the *uncertainty *of the reading. This can be located in the instrument’s latest calibration record. As an example, the calibration record may say the instrument’s uncertainty is ± 0.00003 V. (The actual uncertainty will probably have an absolute and relative component, but I’m just trying to keep things simple here.) In which case you would write 0.12345 ±0.00003 VDC. If the calibration record is not available, use the uncertainty value located in the instrument’s operating manual in the Specifications section.
    The above should be done at a minimum. When we record data in the lab, we also take into account the confidence interval of the uncertainty (usually 95% or 99%), systematic uncertainty (bias), and perform propagation of uncertainty calculations when required.

3

even if a last digit is rounded it is still significant. in analog devices you would do rounding in taking the reading.

4

OK, how about this: 1.987 x 6.05

That comes out to 12.02135.

The smallest number of significant decimal places is in the latter number. So it looks like 12.02135 should be rounded to 12.02. But 6.05 has three significant figures. In the product, three sig. figs would be 12.0. If the product was 12.07, then the number expressed to the correct number of sig. figs would be 12.1, since there are three sig. figs in 6.05, right?

5

you have numbers with 4 and 3 significant digits. the product should have 3 significant digits.

6

It’s starting to come back.

Just wait. Something will dislodge a particle in the folds of my brain, and I’ll be asking about something else! :stuck_out_tongue:

7

Yes… no… maybe. As I explained above, to do it properly you must know the uncertainty of each value. You would then multiply each value to get the answer, and then add the uncertainties in quadrature (RSS of relative uncertainties). The number of significant digits to display – for both the answer and the uncertainty value – will depend on the final uncertainty value.

8

If i was multiplying 1.987 by 6.05, I’d keep at least one more digit as a guard digit in case I wanted to do any more processing, giving 12.02, and probably another one, since the leading digit is a 1, or 12.021 (e.g. 9.99 isn’t really any less accurate than 10.01, even though the later has one more digit. Crafter_Man’s accurate method doesn’t have that issue.)

Actually, I’d probably have it stored as a computer file, and keep all the digits. I don’t think there’s ever any harm in keeping extra digits, and the computer can handle 12.02135 as easily as 12.0.

9

This is all absolutely the right thing to do in a metrology lab, but it’s overkill here. Johnny L.A. is just trying to calculate his gas mileage, and realistically speaking, the difference between this method and the approximation based on sig figs just isn’t going to affect whatever decision he ends up making.

10

Actually, that was just the jumping-off point that reminded me of sig figs. This thread is more along the lines of A) How to determine how many sig figs to keep (or rather, how many digits are sig figs); and B) How sig figs are determined in actual use, such as in a laboratory or other scientific endeavour.

11

The short answer is yes, as your last example did, round to the least number of significant digits.

The more complex, more accurate answer, is to determine the actual error and use that in the calculations.
Measurement A gives error Delta(A) or dA
Ditto for B, dB

For addition or subtraction, it’s absolute - A+B has an error of dA+dB
Obviously, in the case of subtraction, the error may be greater than the number.

For multiplication or division, it’s relative - AxB has an error of (AB)(dA/A+dB/B)
Which works out to BdA+AdB
Generally, the largest error tends to dominate. When finished with calculations, round everything to the same number of significant digits as the

If you’re more graphically oriented, addition or subtraction on the number line:
A number includes a bar “plus or minus”. Glue the two numbers together on the number line, then put the glued-together error over the result.

FOr multiplication graphical, draw a square on the plane, from 0,0 to A,B.
But A and B have a an error range. Think of that as A±dA B±dB
That error results in a wide border around the top and right of the box. The area of the box is the product, the area of the error border is the total error.

Generally, the more math with uncertainties, the worse you result gets.

If I say I’ll give $90 to $110 to about 20 or 30 people, what amount of money am I promising?
$100±10, 25±5 people. Best case $3,300 worst case $1800.
100x25=$2500.
(10/100+5/25)x(2500)= (0.1+0.2)x(2500)=750, round,
so the answer is: (5+, round higher)
$2500±$800

Notice this gives the range from $1700 to $3300, not from $1800 - another side effect of calculations is that the calculated result may not match the obvious one - but because the result say “plus or minus” the correct answer is within the range.

Notice also we’ve gone from an error of about 10% and 20% (plus or minus) to an error of almost 30%. Errors tend to compound if the math is not good.

I remember a place I worked at once, they measured ore coming in on a conveyor belt. They had created a mathematical model that told us to 7 digits how much metal was in process. The head engineer laughed and said “the converor belt weightometer is 2% accurate, but is rarely more than 10% full - so the number it reports is accurate plus or minus 20%. The lab assays are good to 2 digits, maybe - so we have an error of about 2% to 5% there. That 7-digit number contains maybe 2 significant digits if we are lucky. But that’s what management wanted to see…”

12

In any serious scientific endeavor, people are not using sig figs, because they’re only an approximation for the error propagation calculations that Crafter_Man laid out. That’s what people actually do.

13

To me 0.12345 implies that the real value is between 0.123445 and 0.123455. If your original data isn’t precise enough for that. you should round back to where you do know the value. An alternative is to state a plus or minus value.

I was also taught to look at parts per thousand. In both 6.05 and 12.02 the last digit isn’t far off from one part per thousand. In the real world I would accept 12.02 as a result of a calculation starting with 6.05. You were correct to question 12.02135. In ye olde days, I would have set the index of my Pickett on the 605 line and the hairline just short of the 199 line and read 1202 or maybe 1201 or 1203 and reported it as such. Everybody knew the last digit could be off by one or 2. That was the nice thing about slide rules, they always truncated figures to about 1 part per thousand.

I have to wonder if modern gas pumps are really good to the 5 significant figures they report. I think odometers out of the factory are better than they used to be, but as soon as the car hits the road, tire wear becomes a factor, plus inflation pressure and speed, mess with the tire circumference.

I love the way they report racing speeds to 6 significant figures. We have very accurate time keeping devices. We can even measure the dimensions of the track quite accurately. Now, the exact line any one car follows in any lap? Even using a highly precise odometer, I don’t see 6 significant figures. On a 2 1/2 mile track, they would need to be good to an inch.

14

I only wish that were true. In reality, even most PhD’s are taught that in class, then rarely practice it in publications. I’ve seen error analysis done mind bogglingly wrong, then published in respected journals. Of course, this is largely dependent on the field. I’m sure that physicists do a very thorough analysis, since that math is trivial. It’s not so true for organic chemistry. I wouldn’t trust any error bars in an organic paper anywhere.

It’s even worse in industry, where accurately representing your results is not encouraged. In industry, I’ve seen people get one positive result, then send it down the line for product development. Then it’s up to the engineering team to make a product from something that never worked in the first place.

15

To be fair, I did say “any serious scientific endeavor”, not “science as it is practiced”. :wink:

16

I agree that people overstress this. On the other hand, it is often amusing to see things reported with far too many significant digits. For example, I cracked up when watching a science fiction movie in which the protagonist stated ominously that he was only one mile from the asteroid belt.

Unless you are doing the calculations by hand, there is no reason to truncate any digits until it is time to report the final result.

The number of digits you should keep in the final result depends on the context. If you are quoting a new value for a fundamental constant, it is normal to show several digits beyond the “significant digits” and then show the estimated error. If you are quoting the percentage results of a poll of 100 people, it would be silly to quote three significant figures, since the casual reader will think all those digits are significant and not understand that the error bars are about 10%.

17

Oh I love that, 66.667% of 100 people.

18

Quoth Crafter_Man:

You’re still making some possibly unjustified assumptions here: You’re assuming both that the errors on the two measurements are independent, and that they’re Gaussian (with the error bars presumably indicating some number of standard deviations). Really, to be completely thorough about it, one ought not to quote merely a single error value, but a full distribution.

19

Speaking from industry, as part of an R&D group that includes several PhDs - yeah, actual uncertainty reporting is terrible. Sig figs are routinely ignored, usually to put a good result on a data sheet.

I’m an engineer. I actually got in a yelling fight with one our scientists a few years back because he sent a report out to customers stating something like “our material has a tensile modulus of 434,432.1243 +/- 4233.234322 psi!”

I was like, don’t you realize how stupid that makes you look? You know damn well the variability from sample to sample is huge. And I know your equipment, and there’s no way you’re getting that precision. Just report like a few sig figs. 430k +/- 4k, that’s realistic.

He looked at me like I was from the damn moon.

Fucking PhDs.

:smack:

20

He needs an error estimate on that error estimate:
434,432.1243 +/- (4233.234322 +/- 41.9279483932)

There, that’s better! :slight_smile: