Posted by 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 cm2.
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.
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