# Uncertainty of an average

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1. Sep 30, 2014

### maves

1. The problem statement, all variables and given/known data
I am analysing intensities of some pictures. There are specific regions of higher intensities and so far i have managed to locate those regions automatically in Mathematica and the program returns the average intensity and standard deviation on this regions ("circles"), so I can also determine the number of circles.

2. Relevant equations
I would like to calculate the average intensity of the whole picture and its uncertainty. How can I make this, using mean values of intensity and their deviations (uncertainties)?

3. The attempt at a solution
To get the average value of the whole picture, I of course use the formula for average on my measured mean intensities:
$$x_{avg}=\frac{1}{N}\sum_i x_i\,.$$
But what about the error? One way would be to calculate standard deviations of these mean intensities regarding the calculated "mean of the mean", using a standard formula for std.dev.:
$$\sigma=\sqrt{\frac{1}{N}\sum_i(x_i-x_{avg })^2}$$
If this is correct, should it be N-1 in the denominator? And for error i use $$\Delta x=\sigma/\sqrt{N}$$?
But that does not anyhow include the uncertainties of my measurements that I already have (and I also don't think it's ok as my circles are not of equal size). I believe that the uncertainties should be used somehow-? But the "average" of those uncertainties doesn't seem to be the right way.

Last edited: Sep 30, 2014
2. Oct 5, 2014