They say that the absolute uncertainty in my measurement should always be reduced to 1 significant figure but I don't understand why.

  • $\begingroup$ There is no such rule. It is wrong, whoever is insisting on that. In real world, all that matters is standard deviation and the actual measuring instrument. If you balance can weigh up to four decimal places, and you weigh 100 times, the standard deviation can be +/- 0.0010 g [I am assuming a very bad balance]. In reality it could be +/- 0.0003 g. Perhaps this is the origin of this rule of thumb to keep 1 significant figure. $\endgroup$ – M. Farooq Jul 5 '19 at 14:06
  • $\begingroup$ Notice the convention of providing absolute uncertainty of physical constants in Wikipedia. E.g 1.2345(67) = 1.2345 +/- 0.0067 $\endgroup$ – Poutnik Jul 6 '19 at 6:10

The key point is estimation of standard deviation of estimation of measurement standard deviation on Stats SE.

The exact formula is complicated with extensive Gamma function involvement. The essential info is the relative uncertainty of $s$ decreases very slowly with number of measurements.

It can be approximated by involving Stirling approximation for factorial of large numbers.

$$SD(s)=s\cdot \sqrt{\mathrm{e}\cdot \left(1-\frac1n\right)^{(n-1)}-1 }$$

SD is Standard deviation
s is estimation of standard deviation $\sigma$
e is the natural logarithm basis
n is the sample number

The estimation of relative error of $s$ is

  • about 33% for 5 samples
  • about 20% for 13 samples
  • about 10% for 50 samples.

So 1 significant digit for $s$ is a reasonable rule, unless there is a large enough measurement sample set.


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