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I Calculating error on averages with uncertainties in meas.

  1. May 10, 2017 #1
    Let's say I take 3 measurements, and each measurement has its own uncertainty:

    M1 = 10 ± 1
    M2 = 9 ± 2
    M3 = 11 ± 3

    I want to quote the average, and the net uncertainty. I understand that the uncertainty of the mean is:
    (Range)/(2*√N) where there are N measurements. So:
    (11 - 9)/(2*√3) = 1/√3
    which is taken from a text book I have that explains the use of the extra "2" for small measurement sets.

    However, this does not propagate the uncertainty of each measurement... Since the average is a sum of each measurement (over 3), I would think the propagated uncertainty would be:
    δMavg = √(δM12+δM22+δM22)/3
    δMavg = √(14)/3

    So... is my total error:
    δMerr = (1/√3) + √(14)/3 ≈ 1.82

    Any help is appreciated!
  2. jcsd
  3. May 10, 2017 #2


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    This is an example of heteroscedasticity (q.v.).
    The measurements should be weighted according to reliability in order to find the mean.
    A crude way is to replicate them in inverse proportion to the error range, so you could average 6 copies of M1, 3 of M2 and one of M3.
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