Gaussian distribution in Astronomy

Question

1. What does it mean for a data set to have gaussian distributed noise?
2. What does an error bar on a data point really imply?
3. What does a 5-sigma result mean? How often is a 5-sigma result wrong?
4. What does a 3-sigma upper limit mean?

Answer 1

1. 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.

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

3. 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.

4. 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.