# How to define fluctuations for a given probability distribution?

I have the following problem:

Suppose I have a Gaussian probability distribution with mean $\mu$ and variance $\sigma^2$. I want to compare the flucuations of the corresponding random variable with the fluctuations of another probability distribution that has, e.g., two separate peaks but the same mean and same variance as the Gaussian one.

How can I define a meaningful measure for the fluctuations? Variance obviously does not work here as it is the same in both cases. I guess I need to take into account the higher moments of the distribution(s). Is there an analytic way of doing this? Or should I just find the quantiles of this distribution that contain 68% around the median of the distribution, in analogy to the standard deviation of the Gaussian distribution? Or is the coefficient of variation (standard deviation divided by mean) already taking this into account?

In fact, I am looking for this measure of fluctuations for a Poissonian distribution in the simulation of a physics experiment with photons, so ideally, I would like to have a closed expression that I can just plug into my simulation.

I did not find the answer so far; sorry if it is a duplicate question after all. Thanks for your help!

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Edit: I thought about it a bit more and will now try to clarify my question: What I thought is the following: In the following picture, I show a Gaussian distribution with a given width, which I here call $\delta n$.

I want to know how to calculate the $\delta n$ for a given distribution, so that say 68% of the total population lie within $\bar{n} \pm \delta n$, where $\bar{n}$ is the mean of the distribution. For the Gaussian distribution it is obvious.

However what happens if I have an asymmetric distribution like this one:

First, now there are two different values $\delta n$ and $\delta n'$, because the distribution is asymmetric. My question: given the distribution, is there an analytic way formula for these values so that $P(\bar{n}-\delta n < X \leq \bar{n}+\delta n')=0.68$? Here, $P(X)$ is the probability distribution for the random variable $X$.

I would then like to use these values $\delta n [P(X)]$ and $\delta n'[P(X)]$ in some further calculation.

I have the feeling that probably the solution is very simple and I just don't see it...

• What are "fluctuations" in your context? Can you provide a simple example / dataset & point out what you want to detect? – gung Feb 1 '17 at 13:40
• Thanks for asking: I edited my question, trying to clarify what I mean. – kof Feb 1 '17 at 14:51
• The technique described for Gaussian distributions at stats.stackexchange.com/a/127541/919 applies to all continuous distributions without modification. It also applies to all distributions when translated in terms of the distribution function (rather than the density function). – whuber Feb 1 '17 at 15:41