Solve Significant Figures Homework

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SUMMARY

This discussion focuses on the proper representation of experimental results with significant figures and uncertainties. Participants emphasize that significant digits should reflect the precision of the measurements, with specific examples illustrating correct formatting. For instance, 4.12734 ± 1.357 should be rewritten as 4.1 ± 1.4, while 0.4321273 ± 0.00169 is simplified to 0.4321 ± 0.0017. The importance of consistency in scientific notation and avoiding mixed formats is also highlighted, particularly when dealing with large numbers and varying orders of magnitude.

PREREQUISITES
  • Understanding of significant figures in experimental data
  • Familiarity with scientific notation
  • Basic knowledge of error propagation
  • Ability to round numbers correctly based on precision
NEXT STEPS
  • Research the rules for significant figures in scientific measurements
  • Learn about error propagation techniques in experimental physics
  • Explore best practices for presenting data in scientific notation
  • Study the implications of rounding errors in data reporting
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Students in physics or engineering courses, researchers handling experimental data, and anyone involved in reporting scientific measurements accurately.

Silviu
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Homework Statement


I am given some results of certain experiments with errors and I need to rewrite them correctly in term of significant digits.

Homework 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).

The Attempt at a Solution



4.12734 ± 1.357 --- 4.1 ± 1.44.12734 ± 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[/B]
 
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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:
Silviu said:
2.7e-6 ± 3e-7
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.
BvU said:
(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 ##.
Should be 0.002, not 0.02. I would not shift the decimal dot around like that just to get a multiple of 3.