Calculating error on averages with uncertainties in meas.

[Adoniram]

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 textbook 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
or
δMavg = √(14)/3

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

Any help is appreciated!

 

[haruspex]

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.