How to understand reproducibility in statistics?

I'm having problems to find info about reproducibility in statistics. The books I have with an starting level in statistics doesn't even mention it.

At college, one of my previous professors explained it like something you just have to remember. She gave me a list of distributions with this property and the parameter that is involved with reproducibility. I don't like that because I have a really bad memory, in fact, I don't remember that list anymore. I think if I can understand properly why this happens and maybe reading a demonstration I will remember it much better and I will be able to use it faster.

I wonder if I'm using the wrong name for this property. I'm talking about: $$X_i\sim N(\mu,\sigma^2) \quad\forall i=1,\ldots,n \implies \sum X_i \sim N(n\mu,n\sigma^2)$$ Some distributions only fulfill this property for one of its two parameters and some others doesn't fulfill it. I would like to know where is this property coming from.

Thanks

• I've never heard of this being called reproducibility, but it reminds me about the concept of a stable distribution. Commented May 30, 2015 at 3:04

3 Answers

I haven't heard it called reproduceability, but the property you are referring to appears to be that of stable distributions:

independent sums of random variables from a stable distribution family will remain in that family, within a location-scale transformation.

The key idea of reproducibility is that the variance is declining as a percentage of the mean as you scale up the experiment (n is the scale). This corresponds to the idea that adding together more experiments ought to mean better understanding, i.e. lower overall variance if we combine them all. For a normal distribution with mean $n\mu$ and variance $n\sigma^2$ ($n$ = number of experiments), the coefficient of variation is the standard deviation over the mean, which is $\frac{\sigma} {\sqrt{n}}$, which gets smaller as $n$ gets bigger. Thus a normal distribution does model reproducibility.

I would call that "linearity" rather than "reproducibility". A function f(x) of a single variable is linear if and only if f(ax+ by)= af(x)+ bf(y).