Probability that one observation comes from the same population as a given sample I’m working with sports performance data and I have a sample of 10 observations ($x_1,\dots,x_{10}$) regarding the total number of ... per match. The according values are 15, 18, 24, 12, 13, 16, 28, 20, 14, 25.
How do I compute the probability that the value of an additional observation (e.g. $x_{11}$ = 7) comes from the same population as the given sample?
Is this even possible without fitting a distribution and estimating the parameters (which is difficult in this case due to the low sample size)?
Edit: The 10 observations are from one player only. There are more observations from other players available and I modeled them using a Negative Binomial distribution which describes the data very well (overdispersed count data, therefore not Poisson).
 A: As already stated in the comments, you may need more information then your sample to estimate the probability. The only thing that you can estimate from the sample is probability of $x_{11}$ coming from the same distribution as your sample given the empirical distribution of your sample. Basically you would be assuming that distribution of the population is similar to the distribution of your sample (and if you have doubts if the small sample you have is representative, then it seems that you are not willing to make such assumptions).
The value $7$ out of range for your data so obviously you cannot use cumulative empirical distribution function to calculate the probabilities, the same with using bootstrap in here.
Small sample size does not make estimating parameters of distribution harder, it only makes it less reliable, so there is no reason why you couldn't use parametric distribution.
However if you are not interested in fitting parametric distribution, you can use nonparameteric approach such as using kernel density. In this case you also have to estimate (or pick) a parameter for bandwidth and make few other decisions such as picking kernels, or other parameters, but you do not have to make decisions about form of parametric distribution for your data. Below you can see kernel density estimate based on your data.  

