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In a calculation I have received a number of values in the range $1e4-1e6$. Now I would like to display the values with 3 significant figures according to the rules of using significant figures. But this causes some issues when the values vary in magnitude.

For instance, I have the values $34745$ and $211360$. It seems inconsistent to me to render these with 3 significant figures independently of each other, as $34700$ and $211000$, respectively.

On the other hand, choosing either as a baseline, rendering $35000$ and $211000$ or $34700$ and $211400$ seems to break the rules as well.

What is the standard course of action in this situation?

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    $\begingroup$ 34745 to 3 significant digits is 34700 not 34800. ;-) $\endgroup$
    – Andrew
    Jan 13 '17 at 11:51
  • $\begingroup$ Yikes, should I even be an engineer? $\endgroup$
    – Toivo Säwén
    Jan 13 '17 at 13:12
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    $\begingroup$ Depending on your time zone you could claim it was Friday afternoon. Silly mistakes are allowed on Friday afternoons and Monday mornings. $\endgroup$
    – Andrew
    Jan 13 '17 at 14:03
  • $\begingroup$ why do you think 34700 , and 211000 is inconsistent? Would you think 3.47E4 and 2.11E5 ok ? $\endgroup$
    – agentp
    Jan 13 '17 at 20:01
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The fundamental problem here is that you've got a bunch of numeric values which are not in engineering/scientific notation, so you have to make up some arbitrary guesses as to how many sig figs really are there.
Let's look at one of your values: 34745 . Where did it come from? If it's, say the reading on a digital display for a widget whose operating manual specifies an accuracy of +/-1 in the last digit, then you can use it as $(3.4745 \pm 0.00005)E5 $. But if it's the result of a calculator giving you a ton of digits after dividing two values which were accurate to, say, 1%, then you can't assign an accuracy to the result better than that of the inputs. Skipping the exact error propagation formula, you'd roughly be limited to $(3.47 \pm 0.03)E5$ .

Ultimately, the "rule" of sig figs depends on how you combined the source data to get the calculated result.

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It depends on what these different values are. If these different values are for different entities, then I would have each to 3 significant figures based on itself.

i.e. For the equation d = FL/EA, working in kN and m:

  • F = 5412, I would report as 5410
  • L = 6.911, I would report as 6.91
  • E = 210265, I would report as 210000
  • A = 0.043211, I would report as 0.0432

However, if I have a lot of deflections at different positions, then the relative deflection is important, and I would report all values based on which values are significant for the biggest deflection:

  • If Nodes 1-6 have deflections (in mm) of 0.000114, 0.256, 1.54, 17.9, 9.5211 and 0.0187, I would report them as: 0.0, 0.3, 1.5, 17.9, 9.5, 0.0

In this case, the accuracy of 0.000114mm for node 1 is unimportant: when node 4 moved 17.9mm, node 1 effectively didn't move at all.

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  • $\begingroup$ Effectively choosing the highest value as the baseline for significant figure rendering. This definitely makes sense to me. $\endgroup$
    – Toivo Säwén
    Jan 13 '17 at 13:24
  • $\begingroup$ note 210000 appears to have only two sig digits. Formally you might put a bar under the first zero or something to indicate it is significant. $\endgroup$
    – agentp
    Jan 13 '17 at 20:03
  • $\begingroup$ "Significant figures" are a very crude way to express error estimates. It doesn't usually make any physical sense to report values on a scale that distinguishes 9.97, 9.98, 9.99, 10.0, but then suddenly jumps to 10.1, 10.2, etc instead of 10.01, 10.02, etc. Using "best estimate $\pm$ error" is often a better way. $\endgroup$
    – alephzero
    Jan 14 '17 at 11:57

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