# What probability distribution is this?

When plotted on a graph the part above about 14 looks symmetrical, the part below about 14 is not symmetrical. The curve looks to me as if it bulges outwards on both sides about half-way down more than a Gaussian would. On closer inspection even in the symmetrical part, the left hand side is higher than the right hand side.

Data: x,y
1, 12.55
2, 14.3
3, 25.05
4, 29.34
5, 35.27
6, 29.17
7, 24.53
8, 13.8
9, 7,97
10, 4.96


Might it be, for example, an exponentially modified Gaussian? If there is not enough data to decide what it is, what could it be?

• There are not enough data in the world to say the sample is drawn from some distribution. You can perhaps get a good sense of many distributions that are inconsistent with the data. The sample probably isn't drawn from any neat, simple-form distribution; the real world is usually not so neat. It could be drawn from any of an infinite number of distributions. If you really need to identify a distribution that would make for a reasonable approximation, you may be better to contemplate theoretical considerations (if any) and previous studies. Jun 20 '18 at 17:35

This is to illustrate graphically the difficulty of identifying the distribution of a population based on a sample of only ten observations from that distribution.

The sample mean of your ten observations is 19.7 and their sample standard deviation is 10.3. A Shapiro-Wilk test does not reject the null hypothesis that the data may have come from a normal population, but with only ten observations the test has very poor power.

Suppose we guess that the population is normal and the population mean and variance match the sample values. If we make histograms of six samples of size ten from the distribution $\mathsf{Norm}(\mu=19.7,\, \sigma=10.3),$ the figure below shows the results:

In each panel, the red curve is the density function of $\mathsf{Norm}(19.7, 10.3).$ Notice the variety of results, none of which "looks" especially normal. One of the samples has a negative value, even though your original sample does not have negative values.

I made a python program a while back that bruteforces all distributions implemented in scipy.stats and uses AICc to determine what distribution fits the data the best. The code can be found here.

The smallest (best) AICc values from all the distributions were (normal distribution included as reference)

ncx2                 41.06
chi                  49.47
gamma                53.46
...
norm                 98.05


Links to the distributions: ncx2, chi and gamma

Of course this is just the distributions that fits the data the best, with a penalty on model complexity. The underlying distribution might be something very different.