Math Wizards: Sig Figs

Factual Questions
21

JS Princeton: When computing a final answer, the biggest mistake you can make is not “reporting too many significant digits.” It is failing to calculate & report an uncertainty value. As long as my technician properly calculates measurement uncertainty, I don’t pay much attention to the no. of “significant digits,” as long as they’re reasonable (typically 4 to 5 for most measurements).

Let’s say I have a temperature readout that has a readability of 0.001 °C and an uncertainty of ± 0.01 °C. Now let’s say a fancy correction factor is applied which extends the digits. Does it really matter if my technician reports 5.123 ± 0.01 °C or 5.1232 ± 0.01 °C? Not really. Of course, she would normally just use the same no. of significant digits (5.123 ± 0.01 °C), but I won’t lose sleep if more were used.

Let me repeat: readability (a.k.a. resolution, significant digits) should not be used to establish or communicate measurement uncertainty. You should not waste your time worrying about significant digits. Everyone who sells data - including NIST - simply reports a “reasonable” number of significant digits when performing computations, and often reports more than needed just to be safe. (You can always round later.) No one pays much attention to it, unless of course the number of sig digits is unreasonably small. By contrast, lots of energy is spent determining measurement uncertainty…

22

Sailor, I did not “misspeak,” and I am not wrong. The standard inch is completely and utterly irrelevant to my point. The SURVEY inch is defined so that 1 meter (the same meter that everyone uses) is 39.37 inches. Therefore, if you’re going to say “an inch is 2.54 cm,” you need to specify WHICH INCH. This is why I said that YOUR post was “true, but incomplete.”

Learn to read.

Yes, all this means is that a standard inch is different from a survey inch, by two parts in one million. Thanks to the dopers who pointed this out to the illiterati.

23

This talk reminds me of a professor I had in engineering school. He’d devise elaborate problems, requiring many steps in calculations (unit conversions, matrix inversions, etc.). After the homework was handed in he’d always post the answers out to, say, 10 or 15 figures.

Unrealistic, yes, but you knew you had the problem nailed when you got the same answer as his out to 10 digits.

This was in the late 70’s, when pocket calculators were just beginning to become common. ( I never learnt on a slide rule).

Crafter-man, your first few lines really put me off, but after reading your posts, I think you got this stuff down.

24

Crafter, I think to some extent we’re talking past each other. I agree completely that what matters is uncertainty and error and sig figs are basically an aesthetic gesture.

But they’re an important aesthetic gesture if we’re going to get people to understand the concept of precision without appealing to measurements of uncertainty. Most people couldn’t care less about uncertainty until they enter a technical field, so to go through and try to teach paradigmatic calculations or theory one usually uses sig figs as a general guide.

The best example of this is in chemistry where, in the lower levels, the error is almost entirely due to imprecise instrumentation (and gross incompetancy, but be that as it may…) To ask a student to figure out how to measure the error in, say, a graduated cylinder is a waste of time. Better to let them carry proper significant figures through and not worry about whether or not the uncertainty is 5.0 mL or 7.0 mL. However, if the student reports they’re expected yield to twelve significant figures, then there seems to be a communication problem. The alternative is to assign them errors on every instrument they use (since measuring them is not practical) and then force them to go through the error propagation to come up with uncertainty estimates. These calcuations, as you know, can often get well out of hand, and most students aren’t equipped with the mathematical prowess to deal with these problems in Chemistry 101. Instead of having them make errors up, we get snippy over significant figures to get them thinking about the problems. When they start working for you, then they can actually do the proper uncertainty analysis and prevent themselves from being consigned to the dreaded mailroom position.

25

>> Therefore, if you’re going to say “an inch is 2.54 cm,” you need to specify WHICH INCH.

An “inch” defined as 2.54 cm. exactly
A "survey inch"is defined as 1/39.37 m exactly
A foot = 12 inches exactly
A survey foot = 12 survey inches exactly
A mile = 5280 feet exactly
A survey mile = 5280 survey feet exactly
A Nautical mile = 1852 M exactly

You said “depend on the definition of 1 meter equaling exactly 39.37 inches” and this is mistaken. “1 meter equaling exactly 39.37 survey inches” would be correct. A survey inch is not an inch just like a nautical mile is not a mile. They are different things with different names. When dealing exclisively with surveying it may be undertstood that you are talking about survey units but to say in a general forum inch instead of survey inch is wrong just like a mile is not a nautical mile.

26

JS Princeton: I wouldn’t have any heartburn if someone wanted to fiddle with significant digits just for fun. And I understand your point as it being a purely “academic” or “appreciative” exercise. But I can’t emphasize enough that it will never fly in the real world.

Here’s another example: Let’s say you’re trying to figure out how fast a car is traveling. Using a stop watch and measuring tape you record the following data:

distance = 101 feet
time = 4 seconds

So how fast is the car traveling?

Hmm, that easy. 101 ÷ 4 = 25.25 feet / second, right?

Oh wait… there’s only 1 significant digit in the time! So is the “correct” answer 30 feet / second? But the technician said the time was “precisely” 4 seconds. So should we state the time as 4.0 or 4.00 seconds? Argh!!!

Do you see the problem? Too much ambiguity. And if you ask me, too much nonsense. Pure mailroom talk.

They key thing missing in this problem is uncertainty. Let’s redo the problem, but this time we’ll do some basic uncertainty analysis:

distance = 101 feet ± 1/16th inch = 101 feet ± 0.00521 feet
time = 4 seconds ± 0.01 seconds

The speed is 25.25 feet / second. But what’s the uncertainty in speed? It is not 0.00521 ÷ 0.01 = 0.5208 feet / second. It is (101/4)*[(0.00521/101)+(0.01/4)] = 0.0644 feet / second. Therefore, the answer is:

speed = 25.25 ± 0.064 feet / second

Note that the speed (25.25) is displayed with a resolution of 0.01. I came up with this based on the uncertainty (±0.064), not the number of significant digits in the original numbers. If the uncertainty came out to be ± 0.64, I might be tempted to display the answer with a resolution of 0.1. But then again I might not, simply because I hate throwing away digits. But it wouldn’t really matter…

Anyway, I guess my point (again) is that, in the real world, you should never determine the number of significant digits in the final answer based on the number of significant digits in the original values. Everything, including the no. of significant digits to display, is based on uncertainty.

27

Crafter_Man, I treat significant figures as a rule of thumb rather than a hard-and-fast rule. However, it irks me when I see far too many significant figures used. I see this not so much in professional life, but rather in newspapers and such.

For example, I often see statements like “British officials say that these tariffs would result in a total loss in revenue of 15.237 billion dollars.” (I just made that up.) Well, really, they estimated a loss of 10 billion pounds, with no uncertainty reported. The US reporter just ran that number through a currency conversion and gave the results. In this case, the number reported in US newspapers ($15.237 billion) implies that the British government has a much more exact idea of actual revenue loss than they really do. Since there’s really no way for the reporter to report the uncertainty of the loss (since the British government didn’t give it) he ought to do the next best thing and round the number given to $15 billion.

And, yeah, 10 billion = 1 sig fig and 15 billion = 2 sig figs, but rule of thumb and reasonable estimation and tendency to round to the nearest multiple of five and all that. Fifteen billion is still a much better representation of the actual value than $15.237 billion is.

28

Crafter_Man: I’m assuming you just made one of those hurried-post mistakes, but your example uncertainly calculation:

is incorrect. The resulting uncertainty in the speed is:

(101 m /4 s)*sqrt[(0.00521/101)[sup]2[/sup]+(0.01/4)[sup]2[/sup]] = 0.0631 m/s

The only reason your answer is close is that the speed uncertainty is dominated by the time uncertainty, so my expression reduces (in approximation) to yours.

Now, the meat of the problem:

1. When you write a number down (say in a newspaper or a lab report or a scientific paper), you are attempting to communicate some information. Depending on what information you hope to convey, you might quote a number one way or another. If I were to write in the newspaper about a football game, and I wanted to discuss the number of passing yards Joe Quarterback had, it would make no sense for me to say, “Joe passed for 115 +/- 2 yards, where the uncertaintly reported is for a 1-sigma confidence level…” I must know both who the audience is and what they will likely get from my number. Zut gave the nice example of reporting $15.237 billion – the reporter should have recognized that his audience would interpret that precision to mean that the Brits knew exactly how much revenue was lost.

English is full of ambiguities that are handled by the context. Language isn’t science.

2. Having said that, science is science. A number reported in a scientific setting should have a proper uncertainty reported along with it. Statistical errors are ofter reported as x +/- s[sub]x[/sub], where this is to be read as: “We report a value of x, and 68.27% of experiments just like this one would report a value between x+s[sub]x[/sub] and x-s[sub]x[/sub].” (The 68.27% is the fraction of the total area under a Gaussian curve that is within one standard deviation of the mean: 0.68268949 or so.)

3. Education is an iterative process. A 9th-grade chemistry student can’t be expected to calculate the partial derivatives necessary to propagate uncertainties. Further, if a) the equations used only involve multiplication, division, addition, and subtraction, and b), a single uncertainty dominates, then the sig. figs. method presents a reasonable approximation to a more correct error propagation method. Although it is only an approximation, introducing students early to the concept of uncertainty in invaluable. If they grasp the concept that writing “23.445234892494 cm” is silly, they have learned something.

(Of course, if either of the above two conditions are not met, using sig. figs. can result in grossly underestimated errors…)

29

If I sit at the only door to the “Fat Guy” convention, and touch my electronic counting device to open the gate once for each fat guy that enters, I find out that 6000 fat guys entered the convention center. Now, fat guys weight about 250 pounds each, so how much does the aggregate convention of “Fat Guys” weigh?

The problem is that we have one number, which has four significant figures, and is exact. There is no uncertainty. Our other number is an estimate. How do you decide how much uncertainty there is in that type of figure? Well, there are very specific ways of addressing that, adopted by the NIST, and explained at their web site, http://physics.nist.gov/cuu/Uncertainty/index.html]here. Lots of luck understanding it, by the way.

But, however you do your assessment, an estimate is a wild assed guess, made by someone you happen to think guesses very well. But the count is no such quantity. It is exact. And even though the last three digits are zeros, they are significant figures. Saying that 6.0 x 10[sup]3[sup] people entered the gate is not the same. It is imprecise. It would be correct to say that 6.000 x 10[sup]3[sup] people entered.

Suppose we find out that the Fat Guy club requires that its members weigh at least 241 pounds as a condition of membership, and that members over 260 pounds become eligible members of the Very Fat Guy club, and do not attend this convention. We can now say that we have some better chance of assessing the uncertainty of our weight figure. We know that 1,554,000 lbs of fat guys is the most we can have, and 1,446,000 lbs is the least. Unless we know the distribution of exact weights, we can only guess about where in that range the actual value lies. The values even round off differently, no matter where we decide to put the significance. The figure 250 lbs. gives us 1500000 lbs. The difference between the median figure, and the extremes is 54,000, so, unless we have further information, our answer is 1.50 (+/- 0.054) x 10[sup]3[sup].

Tris

“In my opinion, there’s nothing in this world,
Beats a '52 Vincent, and a red headed girl.” ~ Richard Thompson ~

30

or here.

Preview is my friend.

31

Exactly. :wink:

So the number 6000, in this example, being exact, does not have four significant figures. It has an infinite number of significant figures.

If you knew the average weight of the 6000 fat guys to 9 significant figures, you could properly express their aggregate weight to 9 significant figures.

(Just noticed this thread again, after it was cited.)

32

I take issue with the precision assumed on this measurement. Every measurement will have an associated standard error; if your fat-guy-count is 6000 +/- 0.001, then this statement is not self-evidently absurd given the quantization of fat-guys; it simply means (very approximately) that once in a thousand times you will have miscounted by one (actually, you have to do some statistical work, with a t-test, to determine the confidence you have that you have really counted 6,000 FGs and not miscounted 6,001 or 5,999. In order to do this, you will have to make an assumption on the distribution of your error - errors at my level of expertise are always normally distributed, but would be different if, for instance, the most common error was two fat-guys entering at once and being counted as one extremely fat guy.

A very, very good example of the uncertainties in measurement was the Bush-Gore dispute over the Florida count. One can (and perhaps should) say that the measurement error in Florida’s count was unacceptably high and should be reduced; but there is no possibility of a polling system always giving the right answer: some elections will be determined with less than 90% confidence.

As an aside, I’ve always rather enjoyed considering the question of significant digits of logarithms. In your example, we have 6000 fat guys at 250 pounds each; to do the calculation with logarithms we would use log(6000) = 3.7782, which has 4 significant figures. Only the mantissa (0.7782) counts for this calculation, because the characteristic (3.0) serves only to locate the decimal point.

Snotty undergrads can have fun with this trivia. I know!