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A nice relationship between the uncertainty and average absolute deviation
for two averaged measurements with Gaussian distributed errors.
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It is a well known fact that for _large_ samples of two independent and
identically distributed random variables (fa, fb), the average rms in their
difference (i.e., the average over many sample pairs) is just the average
of their absolute difference divided by sqrt(2):

rms(delta) = <|delta|>/sqrt(2),

where delta = fa - fb.

For example, if fa, fb are signals (pixels or source photometry) measured
from separate images with approximately the same "sigma" (uncertainty), and
assuming fa, fb have errors distributed as *Gaussian*, then we also have the
well known relationship:

<|delta|>/sqrt(2) = sqrt(2/pi)*sigma.

Now, if a quantitity is some average combination of two measurements (e.g.,
from two images) where the combined signal is: 

favg = 0.5*(fa + fb),

we have:
sigma(favg) = sigma/sqrt(2). 

Combining this with the above relationship for Gaussian distributed errors
yields the final result:

sigma(favg) = sqrt(pi/8)*<|delta|>.

Therefore, an estimate for the uncertainty in the average of two independent
measurements with errors identically distributed as Gaussian is simply
sqrt(pi/8) multiplied by their average absolute difference.

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F. Masci, 12/3/2013