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Significant figures

  1. Feb 22, 2017 #1
    1. The problem statement, all variables and given/known data
    I am given some results of certain experiments with errors and I need to rewrite them correctly in term of significant digits.

    2. Relevant equations
    The professor explained to us that, for example, 1.12345 ± 0.5231 is not correct (or at least not the right way) because you already have an error on the first digit after . so adding more digits make no sense, so I should rewrite this as 1.1 ± 0.5 (or in some cases 1.12 ± 0.52).

    3. The attempt at a solution

    4.12734 ± 1.357 --- 4.1 ± 1.4


    4.12734 ± 0.0487 --- 4.13 ± 0.05

    0.4321273 ± 0.00169 --- 0.4321 ± 0.0017

    0.002163 ± 0.00032 --- 0.0022 ± 0.0003

    304479 ± 791 --- 3.045e5 ± 800

    728 ± 0.422 --- 728.0 ± 0.4

    511.24 ± 2.721 --- 511 ± 3

    383 ± 61.32 --- 383 ± 61

    987.12 ± 62.57 --- 987 ± 63

    6974.12734 ± 487 --- 6970 ± 490

    123456789 ± 2344 --- 1.23456e8 ± 2000

    0.000002723 ± 0.000000317 --- 2.7e-6 ± 3e-7

    12.4 ± 7.2 --- 12.4 ± 7.2

    These are the ones we have to rewrite and on the right of --- is my solution. Is it correct? I am a bit confused especially about the ones with big numbers and errors of order of 10^2. Thank you
     
  2. jcsd
  3. Feb 22, 2017 #2

    BvU

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    Gold Member

    Hi,

    In general, errors in experimental results are difficult to determine: they are estimates. Even with large numbers of observations and then averaging, statistics show that the relative accuracy of a standard deviation is about ##1/\sqrt n##. So with 10 measurements the error is only 30% accurate.
    Except in special cases, more than one significant digit is not really achievable.

    I learned that an exception is when the first digit of the error is a 1: then you give one more (in order not to have such a big step to the next value)

    You follow the guidelines nicely, but sense some discomfort when errors are > 1.

    I would present ##383\pm 61## as ##380\pm 60## without hesitation, idem ##987\pm 62 \rightarrow 990 \pm 60##.
    With bigger numbers, one way around is to report e.g. ##6974 \pm 487 ## as ## (6.9\pm 0.5)\times 10^3##

    Here another guideline comes in: powers of ten preferably in steps of 3 . But it's not a strict guideline at all.

    (on the next line you show some fatigue: last digit rounds off to 7). A mix of scientific and normal number format is uneasy on the eye; I would prefer ##(123.457 \pm 0.02) \times 10^6 ##.
    [edit] oh, 0.002 of course. Thanks mfb - and we see it's a matter of tastes differing (post below) -- not a strict guideline at all. My motivation: there's often a name for such a power, like ##M\Omega## etc.

    Then the next line is also uneasy because of the different exponents. I like ##(2.7\pm0.3)\times 10^{-7}## better. Taste ? [edit] oh, sorry ##10^{-6}## - even better. ; thanks mfb)
     
    Last edited: Feb 23, 2017
  4. Feb 23, 2017 #3

    mfb

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    2016 Award

    Staff: Mentor

    Don't use different exponents for central value and uncertainty. I would write it as (2.7±0.3) e-6, writing it as 2.7e-6 ± 0.3e-6 is possible as well.
    Should be 0.002, not 0.02. I would not shift the decimal dot around like that just to get a multiple of 3.
     
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