# Implicit error margins based on significant figures

• B
• johann1301h
In summary, the conversation discusses the topic of significant figures, error margins, and measurement in the context of physics. The main question raised is why textbooks often provide numbers without error margins but require answers to be written with a specific number of significant figures. The convention of expressing accuracy and precision implicitly through significant figures is mentioned, and it is suggested that this convention is used more in textbooks (with hypothetical situations) than in real-world situations (practice). The importance of considering the precision of the variables in a computation is also highlighted. The conversation ends with a recommendation for a textbook on error analysis and a humorous exchange about the use of pedantry as a unit of measurement.
johann1301h
TL;DR Summary
A discussion on the relationship between error margins and significant figures.
WARNING: Topic is very pedantic.

I have used a set of different physics books over the years, and they have all had a focus on the topic of significant figures, error margins and measurement. I have never quite understood these concepts fully and the relationships between them.

One aspect I want to address in this discussion is measurements written without error margins. To be clear, a measurement with error margins could be 201 ± 7, and without would just be 201. The main reason motivating this, is that 99% of the physics problems I have encountered provide their numbers without any error margins. The books providing these problems all seem to be in the opinion of writing the answer with a certain amount of significant figures.

If the books don’t provide any error margins, why do they want us to answer with a specific number of significant figures?

My current answer to this, is that the numbers are assumed to be the result of some (hypothetical) experiment and that the margins must be implicit (sort of like how 4 implicitly means +4).

I did find a statement on Wikipedia regarding implicit error margins, which reads as follows;

A common convention in science and engineering is to express accuracy and/or precision implicitly by means of significant figures. Where not explicitly stated, the margin of error is understood to be one-half the value of the last significant place. For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 843 m would imply a margin of error of 0.5 m (the last significant digits are the units).

Im wondering how much this convention is used, where I can read more about it, and if there are anyone here who can confirm that this is a valid and used principle. For instance, where does it come in use?

I started reading An introduction to error analysis, but so far, the book seems to skips the detail regarding implicit error margins.

Anyway, is anyone here pedantic enough to address how physics problems often require/motivates a constrained answer with significant figures, while at the same time, the problems only provide numbers without any error margins at all.

I can understand that it would be very cumbersome if all the problems would always include error margins since it would require more effort of calculating the answer, and in terms of additional print space for each problem. But still, when doing real life physics, in a lab, can these non-margin problems really compare at all?

It seems to me that these problems we find in physics books only address the topic of error analysis half-way.

In my - current - opinion, if they don’t address the topic in a realistic way, why address the topic at all. That is, why should we care to write the answers in a constrained way with “a correct” amount of significant numbers.

Merry Christmas, and hope 2021 will be a better year for you all!

It seems to depend if the book is a textbook (with hypothetical situations) vs real world situations (practice) where the precision of the variables is very much important. An explicit section on error margins should bring up the points that you do.

So for instance, a textbook might reference sin(3 degrees) which is given with 1 digit of precision*, but the computation is not expecting you to use 5e-2. The text is probably expecting 0.052336 to be used, and I only stopped there because it would take two more digits to be meaningfully more precise.

In practice, the precision to which the 3 degrees is known makes all the difference in the world as to the precision with which the resulting computation is to be considered meaningful. Yes, the convention you quote is the one typically used. Precision is often known to x many digits where x can vary by halves, so a measurement of 0.1276 meters is considered to be 3.5 digits of precision.

Also, the precision of one measurement might result in a computation that is known to more or fewer significant digits. For instance, cos(8.2 deg) ±0.05deg) is has 2 digits of input precision but 3.5 digits of output precision. Thus it is sometimes a mistake to carry all computations to some minimum precision (weakest link so to speak).

*Anything in degrees is actually automatically 3 digits of precision left of the decimal. 3 degrees is no more or less precise than 73 or 245891 degrees.

Last edited:
atyy
johann1301h said:
. . . is anyone here pedantic enough . . .
How do we measure that?

atyy
Halc said:
It seems to depend if the book is a textbook (with hypothetical situations) vs real world situations (practice) where the precision of the variables is very much important. An explicit section on error margins should bring up the points that you do.

The book I am currently reading is Serway Physics for scientists and engineers. It does not adress the topic of implicit error margins, but does adress how to calculate error from a non error margin number set.

Yes, the convention you quote is the one typically used. Precision is often known to x many digits where x can vary by halves, so a measurement of 0.1276 meters is considered to be 3.5 digits of precision.

Are there any good textbooks covering this?

sysprog said:
How do we measure that?

Whatever the SI unit of pedantry is, I'm sure a different unit is in common use in the USA.

sysprog
Whatever the SI unit of pedantry is, I'm sure a different unit is in common use in the USA.
We use an inverse scale calibrated in [insert political figure's name here]s.

hutchphd and sysprog
johann1301h said:
Are there any good textbooks covering this?

Taylor, An Introduction To Error Analysis. Is quite good. You can judge a book by its cover:

However, significant figures are more a rule of thumb than a rigorous calculational tool, so nobody would write an entire book about them. Think of them as a way to keep you from writing anything completely idiotic. "The dinosaurs died 65,000,002 years ago."

hutchphd and atyy
Taylor, An Introduction To Error Analysis. Is quite good.

Thanks for the response, but as I wrote in the OP, this book skips the topic of implicit error margins. (based on my initial reading)

And, for the reasons I mentioned above, I don't think you will find a book that does this.

... so nobody would write an entire book about them.

johann1301h said:
[challenge accepted]
If you're contemplating writing a book on that, (shameless unsolicited plugging of PF here) maybe you could try writing an Insights article on it here first?

However, significant figures are more a rule of thumb than a rigorous calculational tool,
But a very useful rule of thumb when using a slide rule.

hutchphd
Nugatory said:
using a slide rule

What is this "slide rule" of which you speak?

atyy
johann1301h said:
Im wondering how much this convention is used, where I can read more about it, and if there are anyone here who can confirm that this is a valid and used principle. For instance, where does it come in use?

What is stated on Wikipedia is a conventional rule of thumb. You can find it taught in many high school science textbooks.
https://www.bbc.co.uk/bitesize/guides/z3dt3k7/revision/4
https://www.bbc.co.uk/bitesize/guides/zv3rd2p/revision/5
https://www.dartmouth.edu/~genchem/sigfigs.html

Nugatory said:
But a very useful rule of thumb when using a slide rule.
When I was a kid (early '70s) the Math teacher told us that: for a slide rule, two more digits of accuracy at the same visual scale would require a hundred times the size, whereas a calculator with two more digits of accuracy could still fit in a coat pocket.

sysprog said:
When I was a kid (early '70s) the Math teacher told us that: for a slide rule, two more digits of accuracy at the same visual scale would require a hundred times the size, whereas a calculator with two more digits of accuracy could still fit in a coat pocket.
Two significant digits is plenty for many interesting and important real world problems.
I've seen an engineering professor from the early calculator days quoted as saying that if you're looking at the second significant digit of the tensile strength... you've already made a mistake.

sysprog
Nugatory said:
Two significant digits is plenty for many interesting and important real world problems.
I've seen an engineering professor from the early calculator days quoted as saying that if you're looking at the second significant digit of the tensile strength... you've already made a mistake.
From https://www.discovermagazine.com/technology/the-first-nerd-tool:

The five-inch Keuffel & Esser Deci-Lon, the quintessential 1960s pocket slide rule, cost $12.50. Recently one sold on eBay for$180. Chemist Linus Pauling used to astound freshmen at Caltech by multiplying numbers to six decimal places on his pocket slide rule. "He calculated the last two digits in his head," says Caltech math professor emeritus Tom Apostol.

And a slide rule was present at the creation of the nuclear age—Enrico Fermi used his to calculate exactly how far a cadmium control rod should be pulled from a graphite and uranium pile as he demonstrated the first controlled nuclear chain reaction at the University of Chicago in 1942. Moments after making the final calculation, Fermi put away his slide rule, smiled broadly, and announced, "The reaction is self-sustaining."
Even with my Flying Fish log log vector slide rule, I'm still not accomplishing that much ##-## what's up with that?

I believe the GUM actually discusses this, at least briefly

https://www.bipm.org/en/publications/guides/gum.html

Note that since the GUM is the official guide from BIPM is it -quite literally- always the primary source as long as you are dealing with measurement results expressed using SI units
Also, the GUM and its various supplements is surprisingly readable.

## 1. What are significant figures and why are they important in determining error margins?

Significant figures are digits that represent the precision of a measurement. They are important in determining error margins because they indicate the level of uncertainty in a measurement and help to ensure that calculations are not overestimated.

## 2. How do significant figures affect the calculation of error margins?

Significant figures determine the number of decimal places in a measurement and therefore affect the precision of the measurement. This precision is taken into account when calculating error margins, as a more precise measurement will have a smaller error margin.

## 3. Can significant figures be used to estimate error margins for all types of measurements?

Yes, significant figures can be used to estimate error margins for all types of measurements, as long as the measurements are quantitative and have a numerical value. This includes measurements of length, mass, time, and volume.

## 4. How are implicit error margins based on significant figures different from explicit error margins?

Implicit error margins are based on the number of significant figures in a measurement, while explicit error margins are provided explicitly as a range of values. Implicit error margins are more commonly used in scientific calculations, as they are a more accurate representation of the precision of a measurement.

## 5. What are some common sources of error that can affect the accuracy of implicit error margins based on significant figures?

Some common sources of error that can affect the accuracy of implicit error margins based on significant figures include rounding errors, measurement errors, and limitations of the measuring instrument. It is important to minimize these errors as much as possible to ensure accurate calculations.

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