Please explain significant figures to me

[Turek]

@Stranger On A Train

Four or five places was the limit, even with the top quality slapstick…

Heh, that typo cracked me up.

Thanks for your earlier clarification, Stranger. It’s been a few years since science class.

[Thrasymachus]

@Crafter_Man

Learning how to “carry significant digits” is worthless. Actually, it’s worse than worthless, since it teaches you to do the wrong thing. (I’ve been a metrologist going on 10 years now, and I’ve never seen anyone do this. In fact, if anyone in my field did do this they’d be laughed at.)

Two or three significant digits for the final uncertainty value is typical.

I couldn’t agree more. Sig fig counting is a bunch of bunkum, and one of my pet peeves.

Consider this example, which I love tossing at sig fig folks:

A cube measures 23" on a side. What is it’s volume?

Calculator: 12,167 cubic inches
Sig figs of 2 gives: 12,000 cubic inches

Ok so far. But wait. The 3 is uncertain, so it could really be 22.5 or 23.5. This gives bounds of:

Lower bound: 11,390.625 cubic inches
Upper bound: 12,977.875 cubic inches

which means an actual volume of 13,000 is possible. So it’s really only one sig fig accurate now. Why?

Because the type of function being calculated is the determinant of the final accuracy, not the number of digits that you’re sure about.

Calculating volume means three multiplications, so small errors in measuring mean a big change in the result. Conversely, operations like fractional exponentiation or trig (in certain ranges) change only a little with a big change in the input. Take cosines near 0 for an example. Cosine of 0 is 1, cosine of 0.1 is .9998…

Any formal work about predicting intervals of certainty involves deriviatives - I've never heard it done any other way except in high school.  To use the cube example again:

uncertainty = Ds (d/ds) = .5" * d/ds s^3 = .5" 3 (23)^2 = =+/- 793 cubic inches

Still approximate (since cube is, well cubic, where its slope at a point is linear) but it’s does a hell of a lot better than counting on fingers. “Real world” functions are usually much more complicated in terms of how the output changes vs. the input(s). Like vibration analysis for example.

In many fields (such as mechanical engineering; my field), 1% accuracy is extremely fine stuff considering loads and material properties data is mostly guesses or 10% at best, so we commonly stop at 4 figs for results. Besides, we’re just going to double it at least before building the thing!

Don’t get me wrong, the lesson of sig figs is a good one in general, that the calculator can give you more digits than are worthwhile based on the inputs. But that counting stuff? Not useful for anything more than the next hour exam and best forgotten after that.

[Nava]

Significant figures aren’t “behind the decimal”, they’re total.

1000000 could have 7 figures… or only one. Depends on whether you’re trying to get that one in a million chance that always works, an accountant, or an engineer

One example that I like: budgets for public works can be something like $203,546,127.23 - yep, cents and all. But then they go 50% overbudget, so all that apparent exactitude is bogus (you can’t even believe the first figure!)

[Little_Nemo]

But learning signifigant figures is useful because it provides an example of how you can’t get more information out of a solution than you put into it. Manipulating data does not create data.

[Stranger_On_A_Train]

@Little Nemo

But learning signifigant figures is useful because it provides an example of how you can’t get more information out of a solution than you put into it. Manipulating data does not create data.

Clearly you aren’t an accountant for a major corporation.

Stranger

[easy_e]

@Crafter_Man

But I suspect the OP is referring to this question: How many significant digits should be displayed for the final answer? To understand how to do this you need to know that, in the real world, every measurement value has an uncertainty value associated with it. If you have to make a bunch of calculations, each and every variable in the beginning, middle, and end has an uncertainty value associated with it. This is because uncertainties propagate through the calculations according to well-established rules. The number of significant digits to display for the final answer depends on the uncertainty value of the final answer. If, for example, the final answer is 763.4227543 ± 0.1, then you would display it as 763.4 ± 0.1 or perhaps 763.42 ± 0.1. If the final answer is 763.4227543 ± 0.001, then you would display it as 763.423 ± 0.001 or perhaps 763.4228 ± 0.001. If the final answer is 763.4227543 ± 0.0132, then you would display it as 763.4228 ± 0.0132 or perhaps 763.42275 ± 0.0132.

Whew, you’re taking me back to p-chem lab. In every report we’d have to do error propagation to show how we came up with the tolerance on our final result.

Thank OG We only had 6 or 7 lab reports that semester.

[Chronos]

Of course, to be truly pedantic, one might point out that a value with a +/- error estimate attached to it is still inadequate. That’s implicitly assuming some distribution on the errors (probably Gaussian). Strictly speaking, the only complete way to express an output set of values is to give a distribution on the space of all possible vectors of outputs. If you want something a little more manageable than that, then you can give distributions on each output variable and a correlation matrix between them. Or diagonalize your set of output variables, and give a distribution on each new variable. Or assume a Gaussian distribution and just give +/- errors for either of those cases. Or give a distribution on each output variable without the correlations. Et cetera. Anything less than the full distribution over the entire vector space is a shortcut. Using the number of significant figures to indicate precision is just one more shortcut.