After my post the other day about rounding errors, I got a ton of

requests to explain the idea of *significant figures*. That’s

actually a very interesting topic.

The idea of significant figures is that when you’re doing

experimental work, you’re taking measurements – and measurements

always have a limited precision. The fact that your measurements – the

inputs to any calculation or analysis that you do – have limited

precision, means that the results of your calculations likewise have

limited precision. Significant figures (or significant digits, or just “sigfigs” for short) are a method of tracking measurement

precision, in a way that allows you to propagate your precision limits

throughout your calculation.

Before getting to the rules for sigfigs, it’s helpful to show why

they matter. Suppose that you’re measuring the radius of a circle, in

order to compute its area. You take a ruler, and eyeball it, and end

up with the circle’s radius as about 6.2 centimeters. Now you go to

compute the area: π=3.141592653589793… So what’s the area of the

circle? If you do it the straightforward way, you’ll end up with a

result of 120.76282160399165 cm^{2}.

The problem is, your original measurement of the radius was

far too crude to produce a result of that precision. The real

area of the circle could easily be as high as 128, or as low as

113, assuming typical measurement errors. So claiming that your

measurements produced an area calculated to 17 digits of precision is

just ridiculous.

As I said, sigfigs are a way of describing the precision of a

measurement. In that example, the measurement of the radius as 6.2

centimeters has two digits of precision – two *significant
digits*. So nothing computed using that measurement can

meaningfully have more than two significant digits – anything beyond

that is in the range of roundoff errors – further digits are artifacts

of the calculation, which shouldn’t be treated as meaningful.

The rules for significant figures are pretty straightforward:

- Leading zeros are
*never*significant digits. So in “0.0000024″, only the “2″ and the “4″ could be significant; the leading

zeros aren’t. - Trailing zeros are only significant if they’re measured. So,

for example, if we used the radius measurement above, but expressed

it in micrometers, it would be 62,000 micrometers. I couldn’t

claim that as 5 significant figures, because I really only measured

two. On the other hand, if I actually measured it as 6.20 centimeters, then I could could three significant digits. - Digits other than zero in a measurement are always significant

digits. - In multiplication and division, the number of the significant

figures in the result is the*smallest*of the number

of significant figures in the inputs. So, for example,

if you multiple 5 by 3.14, the result will have on significant

digit; if you multiply 1.41421 by 1.732, the result will have

four significant digits. - In addition and subtraction, you keep the number of

significant digits in the input with the smallest number of

*decimal places*.

That last rule is tricky. The basic idea is, write the numbers

with the decimal point lined up. The point where the last significant

digit occurs first is the last digit that can be significant in

the result. For example, let’s look at 31.4159 plus 0.000254. There

are 6 significant digits in 31.3159; and there are 3 significant digits in 0.000254. Let’s line them up to add:

31.4159 + 0.000254 ------------- 31.4162

The “9″ in 31.4159 is the significant digit occuring in the

earliest decimal place – so it’s the cutoff line. Nothing

smaller that 0.0001 can be significant. So we round off

0.000254 to 0.0003; the result still has 5 significant

figures.

Significant figures are a rather crude way of tracking

precision. They’re largely ad-hoc. There is mathematical reasoning

behind these rules – so they do work pretty well most of the time. The

“right” way of tracking precision is error bars: every measurement has

an error range, and those error ranges propagate through your

calculations, so that you have a precise error range for every

calculated value. That’s a much better way of measuring potential

errors than significant digits. But most of the time, unless we’re in

a very careful, clean, laboratory environment, we don’t really

*know* the error bars for our measurements. Significant digits

are basically a way of estimating error bars. (And in fact, the

mathematical reasoning underlying these rules is based on how

you handle error bars.)

The beauty of significant figures is that they’re so incredibly

easy to understand and to use. Just look at *any* computation

or analysis result described anywhere, and you can easily see if

the people describing it are full of shit or not. For example, you

can see people claiming to earn 2.034523% on some bond; they’re

not, unless they’ve invested a million dollars, and then those last

digits are pennies – and it’s almost certain that the calculation

that produced that figure of 2.034523% was done based on

inputs which had a lot less that 7 significant digits.

The way that this affects the discussion of rounding is

simple. The standard rules I stated for rounding are for

rounding *one* significant digit. If you’re doing a computation

with three significant digits, and you get a result of

2.43532123311112, anything after the 5 is *noise*. It doesn’t

count. It’s not really there. So you don’t get to say “But

it’s *more* than 2.435, so you should round up to

2.44.”. It’s *not* more: the stuff that’s making you think it’s

more is just computational noise. In fact, the “true” value is

probably somewhere +/-0.005 of that – so it could be slightly more

than 2.435, but it could *also* be slightly less. The computed

digits past the last significant digit are *insignificant* -

they’re beyond the point at which you can say anything accurate. So

2.43532123311112 is *the same* as 2.4350000000000 if you’re

working with three significant digits – in both cases, you round off

to 2.44 (assuming even preference). If you count the trailing digits

past the one digit after the last significant one, you’re just using

noise in a way that’s going to create a subtle upward bias in

your computations.

On the other hand, if you’ve got a measured value of 2.42532, with

six significant figures, and you need to round it to 3 significant

figures, *then* you can use the trailing digits in your

rounding. Those digits are *real* and *significant*.

They’re a meaningful, measured quantity – and so the correct rounding

will take them into account. So even if you’re working with

even preference rounding, that number should be rounded to three sigfigs as 2.43.