Executive summary:
a) Sig figs are an abomination.
b) There are some constructive suggestions at the bottom of this
message.
Longer version: Let's start by redoing one of the worked exercises,
namely item (e) on page 20, in section 1.7, which is entitled
"Significant Digits".
When I do it, I get that h = 0.138 ± 0.065, where the error
bar is the HWHM of a rectangular (not Gaussian) distribution.
This stands in contrast to the value given in the book, namely
0.1, which I take to mean 0.1 ± 0.05. Note that my central
value differs from the claim in the book by more than 3/4ths
of the claimed error bar. More than 40% of the actual data
falls outside the claimed error bars. The situation is shown
in the attached diagram.
This sort of data destruction is all-too-common when using "sig
figs". Actually, worse destruction than this is quite common.
For this reason among many others, people who care about their
data do not use sig figs.
On page 18 it says "In such cases, you need to round your
answer to the correct number of significant digits."
I insist no, in such a situation you do not "need" to round
off the number. You can round for convenience if you want,
but you don't "need" to. Roundoff error is an error.
A few lines later it says "it is best to wait until you have
obtained the final result in a multistep calculation before
rounding."
That's mostly a good idea. I suppose your suggestion involves
marking each intermediate number according to where you /would have/
rounded it if sig-figs rules were used ... but not actually rounding
it at all.
It seems odd that none of the worked examples in either book do
this. Why do you not follow your own advice?
Alas, there's an even worse problem: Nobody knows what you mean by
"final". Your bottom-line result is not "final", unless you think
it will never be used by anybody. In contrast, if your results are
to be used by a teammate, your "end" is the other guy's "beginning"
... and from the overall team point of view, your number is an
intermediate result. In this case, evidently the best policy is
"do not round at all".
Seriously: If you have a situation where not rounding "at all until
the end" is a viable policy, then not rounding /even at the end/
would be an even better policy, incomparably better than rounding
the "final" result according to sig-figs dogma.
This is the sort of thing that gives ivory-tower busywork a bad name.
The only scenario I can think of where there is an "end" is where
homework calculations are graded but never used for any practical
purpose. Sometimes I think the origin and entire purpose of sig figs
is to round things off so brutally that everybody in the class gets
the same numerical answer, with no statistical fluctuations. Every
gets a bogus answer, inconsistent with reality, but everybody gets
the /same/ bogus answer, so "mission accomplished". Some people
seem to think that conformity is far more important than getting
the physics right.
This also leaves us with the question, significant to whom? Often
you don't know the purpose(s) to which the results will be put, so
you cannot really know the significance. The point here is that
insignificance is not the same thing as uncertainty.
Furthermore, there is nothing wrong with writing down things of
the form 2.305617 when the uncertainty is 0.1 ... assuming the
uncertainty is clearly expressed, as it always should be. The
"extra" digits to not overstate the uncertainty unless you believe
in the sig-figs dogma, which I absolutely do not.
How bad does the sig-figs nonsense have to get before people stop
pretending it makes sense?
By the way, I guarantee that your students have figured this out.
They know sig figs are garbage. They learn it by rote anyway.
For them, this is just another reminder that critical thinking
is not tolerated on school grounds.
Page 18 clearly implies that you can look at a number such as
9.75609756 and determine the associated uncertainty ... which is
provably not true. Consider for example the number 2.54. Can
you estimate the uncertainty? You "might" be able to, if you
knew the number resulted from roundoff, but what if it didn't?
For all you know, 2.54 could be exact, with no uncertainty
whatsoever. It could be the number of cm per inch.
We should also consider null experiments. Suppose the answer
s 0.1 ± 17. How are you going to represent that using sig figs?
Last but not least, in physics and in other disciplines, it is
common to have signals that are very much smaller than the noise.
If you write down only the digits that you are "pretty sure of",
you throw away the signal.
Let's be clear: Roundoff error is an error. The sig-figs dogma
requires you to keep rounding off until the roundoff error becomes
the dominant contribution to the overall error, which is horribly
bad practice. In a well designed experiment, roundoff is virtually
never the dominant contribution to the uncertainty.
The last sentence on page 18 is something I mostly agree with:
"The rigorous way to deal with uncertainties in measurements is
to recalculate any computed value by using the high and low
uncertainties for each value in the calculatoin, which is a
time-consuming task."
My remarks:
a) I call that the Crank Three Times™ method.
b) It's not particularly laborious, especially if you
are using a spreadsheet. In many cases it is less
laborious than trying to propagate the sig figs from
step to step during the calculation.
c) It is not fully rigorous, but it is incomparably
more reliable than the sig figs method could ever
hope to be.
d) It is a valuable pedagogical stepping stone toward
methods that are even more powerful (and not much
harder), including out-and-out Monte Carlo.
Constructive suggestion: Get rid of nearly all of section 1.7.
Replace it with a few simple, practical rules:
1a. Use many enough digits to avoid unintended loss of significance.
1b. Use few enough digits to be reasonably convenient.
Keep in mind that there are plenty of cases where
there is no uncertainty, and/or the uncertainty is too
small to be worth worrying about.
If you need to state the uncertainty, state it
separately and explicitly. Don't try to use one
numeral to represent two numbers.
You can do a lot with the Crank Three Times™ method.
ericmazur
commented
9 years ago 
Executive summary: a) Sig figs are an abomination. b) There are some constructive suggestions at the bottom of this message.
Longer version: Let's start by redoing one of the worked exercises, namely item (e) on page 20, in section 1.7, which is entitled "Significant Digits".
When I do it, I get that h = 0.138 ± 0.065, where the error bar is the HWHM of a rectangular (not Gaussian) distribution. This stands in contrast to the value given in the book, namely 0.1, which I take to mean 0.1 ± 0.05. Note that my central value differs from the claim in the book by more than 3/4ths of the claimed error bar. More than 40% of the actual data falls outside the claimed error bars. The situation is shown in the attached diagram.
This sort of data destruction is all-too-common when using "sig figs". Actually, worse destruction than this is quite common.
For this reason among many others, people who care about their data do not use sig figs.
On page 18 it says "In such cases, you need to round your answer to the correct number of significant digits." I insist no, in such a situation you do not "need" to round off the number. You can round for convenience if you want, but you don't "need" to. Roundoff error is an error.
A few lines later it says "it is best to wait until you have obtained the final result in a multistep calculation before rounding."
That's mostly a good idea. I suppose your suggestion involves marking each intermediate number according to where you /would have/ rounded it if sig-figs rules were used ... but not actually rounding it at all.
It seems odd that none of the worked examples in either book do this. Why do you not follow your own advice?
Alas, there's an even worse problem: Nobody knows what you mean by "final". Your bottom-line result is not "final", unless you think it will never be used by anybody. In contrast, if your results are to be used by a teammate, your "end" is the other guy's "beginning" ... and from the overall team point of view, your number is an intermediate result. In this case, evidently the best policy is "do not round at all".
Seriously: If you have a situation where not rounding "at all until the end" is a viable policy, then not rounding /even at the end/ would be an even better policy, incomparably better than rounding the "final" result according to sig-figs dogma.
This is the sort of thing that gives ivory-tower busywork a bad name. The only scenario I can think of where there is an "end" is where homework calculations are graded but never used for any practical purpose. Sometimes I think the origin and entire purpose of sig figs is to round things off so brutally that everybody in the class gets the same numerical answer, with no statistical fluctuations. Every gets a bogus answer, inconsistent with reality, but everybody gets the /same/ bogus answer, so "mission accomplished". Some people seem to think that conformity is far more important than getting the physics right.
This also leaves us with the question, significant to whom? Often you don't know the purpose(s) to which the results will be put, so you cannot really know the significance. The point here is that insignificance is not the same thing as uncertainty.
Furthermore, there is nothing wrong with writing down things of the form 2.305617 when the uncertainty is 0.1 ... assuming the uncertainty is clearly expressed, as it always should be. The "extra" digits to not overstate the uncertainty unless you believe in the sig-figs dogma, which I absolutely do not.
How bad does the sig-figs nonsense have to get before people stop pretending it makes sense?
By the way, I guarantee that your students have figured this out. They know sig figs are garbage. They learn it by rote anyway. For them, this is just another reminder that critical thinking is not tolerated on school grounds.
Page 18 clearly implies that you can look at a number such as 9.75609756 and determine the associated uncertainty ... which is provably not true. Consider for example the number 2.54. Can you estimate the uncertainty? You "might" be able to, if you knew the number resulted from roundoff, but what if it didn't? For all you know, 2.54 could be exact, with no uncertainty whatsoever. It could be the number of cm per inch.
We should also consider null experiments. Suppose the answer s 0.1 ± 17. How are you going to represent that using sig figs?
Last but not least, in physics and in other disciplines, it is common to have signals that are very much smaller than the noise. If you write down only the digits that you are "pretty sure of", you throw away the signal.
Let's be clear: Roundoff error is an error. The sig-figs dogma requires you to keep rounding off until the roundoff error becomes the dominant contribution to the overall error, which is horribly bad practice. In a well designed experiment, roundoff is virtually never the dominant contribution to the uncertainty.
The last sentence on page 18 is something I mostly agree with: "The rigorous way to deal with uncertainties in measurements is to recalculate any computed value by using the high and low uncertainties for each value in the calculatoin, which is a time-consuming task."
My remarks: a) I call that the Crank Three Times™ method. b) It's not particularly laborious, especially if you are using a spreadsheet. In many cases it is less laborious than trying to propagate the sig figs from step to step during the calculation. c) It is not fully rigorous, but it is incomparably more reliable than the sig figs method could ever hope to be. d) It is a valuable pedagogical stepping stone toward methods that are even more powerful (and not much harder), including out-and-out Monte Carlo.
Constructive suggestion: Get rid of nearly all of section 1.7. Replace it with a few simple, practical rules:
1a. Use many enough digits to avoid unintended loss of significance. 1b. Use few enough digits to be reasonably convenient.
For the next level of detail, see https://www.av8n.com/physics/uncertainty.htm or equivalently http://www.av8n.com/physics/uncertainty.htm