# Propagation of uncertainty in an experiment

1. Apr 1, 2013

### MeissnerEffect

I performed an experiment using different wavelength lasers to calculate the distance between the pits of a DVD by measuring the angles formed by the resulting diffraction pattern,but now I'm unsure on how to calculate the uncertainty of the final result.

I took 9 measurements each with their own uncertainty, found an average and calculated the standard deviation, but I'm unsure of how I'm supposed to combine the uncertainties of each individual result with the standard deviation.

I suspect that I should do something like get the average of the uncertainties and add it to the standard deviation, although it's pretty much just a guess.

Thank you

2. Apr 1, 2013

### ehild

See http://ned.ipac.caltech.edu/level5/Leo/Stats4_5.html, for example.

If xi is the mean and σi is the standard deviation/uncertainty of the result of the i-th experiment, the weighted mean of n data is

$$\bar x = \frac{\sum_1^n {x_i/\sigma_i^2}}{\sum_1^n {1/\sigma_i^2}}$$

and the standard deviation/ uncertainity of the mean is

$$\bar\sigma =\sqrt{\frac{1}{\sum_1^n {1/\sigma_i^2}}}$$

ehild