# What distribution best describes multiple, sequential normal distributions: What is the sum of more than two normal distributions?

I am curious as to what describes the following distribution. If we were to record some data which are all from a normal distribution, but the standard-deviation changes for blocks of points recorded. For example, in the following plots we can see normally distributed data where each colour represents a different values of $$\sigma$$.

In the histogram all of the data have been plotted, which forms something that vaguely resembles a Student-T distribution -- although I don't think this is representative. I have also plotted the PDF's of each parent distribution on top of the histogram.

It clearly isn't a convolution of Normal distributions as the convolution of normal distributions, should itself be a normal distribution.

If I however sum the four normal distributions, the result perfectly envelopes the entire histogram:

That is to say $$\frac{1}{N}\sum_{i = 1}^{N}G(\mu, \sigma_{i})$$ where each $$\sigma_{i}$$ represents different standard deviation. In the example I have shown here $$\mu = 0$$ for all cases.

Is there a name for this distribution?

• There is a reason such mixtures of Normal distribution may resemble $t$ distributions: see stats.stackexchange.com/questions/52906.
– whuber
Commented Oct 11, 2020 at 15:52
• Thanks whuber, that's quite useful. Is there such a distribution as I wrote with my expression? Commented Oct 11, 2020 at 15:57
• Yes, as I wrote earlier: It's called a "Gaussian mixture" or "Normal mixture" distribution. At stats.stackexchange.com/a/451862/919 and stats.stackexchange.com/a/429877/919 I posted proofs that such mixtures are never Normal unless they have just one component. It's even easier to show that any finite mixture of Normals can never be exactly a $t$ distribution (because the mixture has finite moments of all orders but no $t$ distribution has all finite moments).
– whuber
Commented Oct 11, 2020 at 16:09
• This is extremely useful. Thanks. Do you know of any approximations that can describe a Gaussian mixture, if one is not sure of how many Gaussian are mixed. I essentially would like to demonstrate that my data is Gaussian mixed. Commented Oct 11, 2020 at 17:34
• That is going to be very difficult because the uncertainties are huge. I don't see how anyone could make such a demonstration: the best you can hope for is to estimate the components and show that the mixture is a usefully accurate approximation.
– whuber
Commented Oct 11, 2020 at 18:51