Help me understand sig figs. According to this old thread, the first reply claims you simply do not count zeros at the beginning or end of a number - carte blanche. But, isn’t this a gross oversimplification? While I can see 19300 is 3 sig figs, what about 19300.0? Is that 6 sig figs? Isn’t that decimal place significant? To me, it means someone measured something to one decimal place of accuracy. And, what about 0.01? Later in the same thread, one claims it has 3 sig figs, as well. If correct, why is the leading zero significant?! For goodness sake, it is there whether you write it or not! How can it possibly be significant?
More importantly, what is the solution to the following examples?
(no, this is not homework, it is my trying to understand)
a) 52330.0 x 85.2
b) 52764.0 x 34.1
c) 56086.0 x 23.8
d) 60060.0 x 39.2
Or, if you hesitate to reply, would you agree a and d have the same number of sig figs? And, a and d have less sig figs than b and c? In short, if the products of all these were shown in a table, then the table may have some results showing one decimal place while others could be rounded off to the nearest 10 or 100? …all in the holy name of sig figs? (To me, that would look quite odd.) What sense does that make? Why didn’t science simply say don’t show more decimal places in the output than shown in the output?
Significant figures make a lot more sense when you actually consider what they are intended to convey - the precision of a particular measurement.
In your two posts you’ve kind of mixed up the sig figs.
For example:
52764 has 5 sig figs
52764.0 has 6. The “.0” implies that the number has a precision to that last decimal place.
52330 is a trickier example. A few textbooks maintain that 52330 has four sig figs, because it can be translated into 5.233E+05. They claim that if it was five sig figs the number should be recorded as “52330.” (note the ending period).
It’s all in whatever you choose to use as a convention in that case. Personally, the last digit despite being a zero is recorded, so I would say that:
They are incorrect - this is only accurate to 1 significant figure.
The solutions to all of these questions are only accurate to 3 significant figures. For example, the lack of accuracy for the second number in question b) means the answer may have any value from 1796612.5 (for 52763.95 x 34.05) to 1801892.3 (for 52764.05 x 34.15). The highest accuracy you can use with any certainty is 3 significant figures (1.80*10^6).
a) Oh, so my output (result) is limited by the input with the least no of sig figs?
b) Scientific notation may be the best way to handle this. I like that!
c) Thanks for the overall clarification! (Maybe it is hard to understand because others are equally confused and it propagates?)
a) Yes. If, say, you’re trying to work out your car’s fuel economy, it’s all very well measuring the petrol consumption to the nearest ounce, but if you’re only taking your distance travelled to the miles digit on the odometer, that’s going to be the big restriction on calculating your accuracy… if your meter was reading 12345 when you started and 12346 when you finished, you might have travelled anything from 0.1 mile to 1.9 miles. (Better if you travel 100 miles to the nearest mile, obviously; that’s a smaller proportion of error.)
However, “significant figures,” in and of themselves, are meaningless. What’s important is the context, that context being the *precision *of the measurement represented by the number. The concept of “significant figures” is a quick shorthand used for keeping track of approximate accuracy.
Malacandra’s example is an excellent illustration–if you’re really interested in understanding the concept, then go through the exercise of calculating your fuel economy. Fill up your vehicle, then note the odomenter mileage AND reset your tripometer (if your car’s like mine, the tripometer has a “tenths” place digit while the odometer does not). Drive around for a while, then fill up again, and note the gallons from the gas pump as well as the tripometer/odometer readings.
Now play with the numbers. The actual mileage on your odometer is essentially +/-0.5 miles. That’s pretty precise when you have 100,000 miles, on your car. However, subtract two readings and you’re +/-1 mile. Still pretty precise compared to 100,000 miles, but not so much when you’ve only driven 30 miles (for example) between fill-ups.
So now calculate your fuel economy. How precise do you think that number is? Does it depend on whether you use the odometer or tripometer? It might–precision of the answer depends on the precision of each measurement (mileage and gallons), and *that’s *what significant figures are used to keep track of.
Another challenge: Figure out the volume of a telephone pole. You can measure its diameter with the aid of a ruler (or its circumference with a tape measure) and take a fair stab at its height by any of several ingenious methods. The formula for the volume of a cylinder is p r[sup]2[/sup] h - and there is no sense whatever in quoting your answer to 15 decimal places even though your calculator will quote p to such precision. At best you know the height to maybe six inches, and so the volume to the nearest 5 - 10 cubic inches: no closer. So no answer with any decimal places in it is remotely supportable.
This is key. If I have a long ruler that is marked off marked only to inches, then I will measure an object that is 100.4 inches long and get 100 inches. This measurement has only 3 significant digits.
If I have a really crappy ruler that is only marked every 10 inches, then I will still get 100 inches, but this time it has only 2 significant digits. That’s because someone looking at my measurement will say that the actual object could be between 95-105 inches and still be measured as 100.
So you can’t just look at the naked number, you have to know where it came from.
This is why scientific notation is helpful in keeping track of significant figures — you could represent your measurements as 1.00 x 10[sup]2[/sup] and 1.0 x 10[sup]2[/sup] cm, respectively.
When I was a kid we were also taught “engineering notation,” which is like scientific notation but the decimal point was placed so that the exponent was always a multiple of 3. Does anybody else remember that, and even more important, has it ever been actually used?
As an example, the coefficient of thermal expansion is often expressed as a number X 10[sup]-6[/sup]/[sup]o[/sup]C, regardless of the actual size of the number. See, for example, the chart in the linked article, which give coefficients ranging from 0.56 X 10[sup]-6[/sup]/[sup]o[/sup]C to 77 X 10[sup]-6[/sup]/[sup]o[/sup]C.
I suspect the reason for sticking with a single exponent is to prevent errors (by accidentally misreading a suprescript) when comparing numbers. Note, though, that while the practice of using a single exponent when discussing coefficient of thermal expansion (and other properties such as viscosity, for example) is common, it’s not universal.
But if you’re being that careful, you should really be recording your errors explicitly.
So don’t write 1.0 x10[sup]2[/sup], write 1.0 +/- 0.5 x 10e2. Or really figure out your error and make it 1.15 +/- 0.12 x10e 2 or whatever.
Really, significant figures is just a quick and dirty rule of thumb for roughly tracking errors. And it’s a pretty good quick and dirty rule of thumb, but like most quick and dirty approximations it runs into trouble in some cases.
I wonder whether teaching it doesn’t do more harm than good, instead of starting with real error notation.
There is a fully developed treatment of accuracy in measurements that includes following them through various calculations, dealing with non-Gaussian distributions, distinguishing between possible and likely ranges, and so forth. The simple custom of only showing digits as nonzero if they are believed correct, or variations on that theme, isn’t the core of all this. It’s just a custom. There are good reasons to use other customs.
For example, I often keep all the available digits with a number as it goes through various treatments and calculations, although teachers say not to. This custom has rescued me several times, because all the extra (and practically random) digits serve as a kind of fingerprint, like a comment field that helps trace the number backwards when something has gone wrong and we need to find why. When you’re using computers, doing this is often essentially free.
I think that the real advantage of significant figures is not that they give a quick-and-dirty way to express relative error, but that they give a quick-and-dirty way to propagate errors from measured quantities to derived quantities. It’s important that science students learn, as quickly as possible, the GIGO nature of derived measurements like Malacandra’s fuel-economy example. You can, of course, propagate errors using more correct error estimates like (1.15 +/- 0.21) x 10[sup]2[/sup], of course — but error propagation of this type involves some rather nasty formulae and really requires calculus to get a handle on, which most high-school science students don’t have.
I do agree, though, that once students reach college there’s not really an excuse for using significant figures any more. Maybe it’s acceptable for first-years, but by the time you’ve been doing college-level math & science for more than a year you should be able to use grown-up error propagation.
On this topic, I really wonder about the accuracy of fuel pumps. Displaying down to a thousandth of a gallon really seems questionable to me. Granted, it’s suggesting it is accurate to 3 milliliters, which would be a lot to be off, but pumps display over five orders of magnitude and I just find that a bit questionable, that it has dispensed 25.021 g accurate to five significant figures.
Wasn’t there a recent dispute about how the pumps were calibrated? Were motorists being charged for mass or volume, I think it was volume, which is significantly affected by the thermal expansion of gasoline.