Gaussian distribution in Astronomy 
*

*What does it mean for a data set to have gaussian distributed noise?

*What does an error bar on a data point really imply? 

*What does a 5-sigma result mean? How often is a 5-sigma result wrong?

*What does a 3-sigma upper limit mean?

 A: *

*One believes that the measurement differs from the true value by a random variable chosen from a Gaussian distribution.  Many things can contribute to the errors of measurement.  The Gaussian distribution is convenient, but often wrong out on the tails-the tails are too small.

*The author should tell you.  It will usually be some number of standard deviations in the Gaussian approximation.

*A 5 sigma result is a measurement that is that far from some background level.  There may be a broad source of light that you can put a smooth curve through and a bright spot in the middle of it.  The null hypothesis is that there is nothing there and the bright spot is a fluctuation in the noise.  A 5 sigma result says the null hypotheses is ruled out to that level.  If you believe the Gaussian model, it would only be wrong about one time in $10^{12}$.  Experience says otherwise, however.

*A 3 sigma upper limit is given when your measurement is consistent with zero.  It is the value that would have to fluctuate down by 3 sigma to match your data.  You are trying to convince the world that the true value is less than this.
