# Is there any way to measure of Gaussian-ness?

I have some sampled data that has $1/f$ noise in it, with departures from the mean. These are long term departures. I could use something like a median filter but the window length would be longer than I would like.

• Is there any way to measure its departure from a Gaussian distribution?
• Or measure how 'Gaussian-like' a time series statstical sample is?
• The Gaussian distribution has some properties that are useful. You can start here. – Envidia Apr 3 '17 at 21:44
• I have troubles understanding was you really are looking for: a measure of gaussianity? Characterization of $1/f$ noise? Why is your original data Gaussian? Why does a median filter come into play? – Laurent Duval Apr 3 '17 at 21:54
• I have taken a few statistical courses and I'm aware of the math. I want to know what other people use – Voltage Spike Apr 4 '17 at 4:57

## 3 Answers

One way to measure the similarity between two distributions is the Kullbeck-Leibler divergence. Granted, it is not a norm because it is not symmetric, but it is a way of quantifying the distance between two pdfs.

EDIT: There is a topic called "Normality testing" or "Gaussianity testing" that is devoted to that very task. Perhaps some tools from that field will help.

A classical measure of "gaussianity" is the kurtosis of your random variable (RV). Kurtosis is the forth order cumulant of a RV. Say $y$ is your RV with zero mean, the kurtosis can be defined as: $$kurt(y)=E[y^4] - 3(E[y^2])^2$$ If $y$ is gaussian, $E[y^4]=3(E[y^2])^2$ and therefore $$kurt(y)=0$$

Have a look at https://doi.org/10.1063/1.3504369

In this reference the authors check for gaussian behavior of the 1/f noise of a resistor using a fourth order frequency spectrum.