statistical test to compare a single value with a simulated set of random values I would like to know how I can test a real value against a population of random values to know the likelihood that the real value could have occurred by chance.  
I have a geographical dataset of 57 archaeological site locations. 19 of these sites coincide (overlap) with mapped mines and quarries. I believe that the presence of the archaeological sites is related to the distribution of mines and quarries (I.e the sites were discovered because the quarrying revealed them) but I cannot be certain. So I wrote a script to generate 57 randomly located sites in the same study area, and analyse and report the number of times the random sites coincide with the same mines and quarries. I have run the script 100 times (i.e. for 100 different sets of 57 randomly generated points). Not once in all 100 simulations do as many as 19 of the 57 random points coincide (the highest number of coincidences I obtained in the simulations is 6). Although this seems to suggest that the number of coincidences in the real dataset is unlikely to have occurred as a result of chance, I would like to show this with a simple statistical test. I was looking particularly at the standard score or z test. 
Heres my data (I'm using R, but my question is of course not R specific) – the number of coincidences (variable “coins”) ordered from lowest to highest for each of the 100 simulated sets of 57 points. 

coins
[1] 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2
[38] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3
[75] 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6

So, to use the standard score, I calculated the mean (pop_mean) and sd (pop_sd) for the 100 simulations: 
mean = 2.21
SD = 1.35
and used 19 as my single value sample, like so:

pop_sd <- sd(coins)*sqrt((length(coins)-1)/(length(coins)))
pop_sd
[1] 1.351259
sd(coins)
[1] 1.358066
pop_mean <- mean(coins)
z <- (19 - pop_mean) / pop_sd
z
[1] 12.42545

Which is off the upper end of the z-value comparison table and hence suggests that it is very unlikely that the real value of 19 could  have been drawn from the random dataset “coins”. 
But at this point I become confused. Since dataset “coins” is the entire population, and I know that the value 19 does not appear in it, I am not clear if my test has any validity. Perhaps I do not need a test at all -   I just show my simulation results and say that at 0.01 confidence level (100 random simulations), the value 19 is not likely to have occurred by chance. I would really appreciate some advice here from someone with a stronger background in statistics.  
 A: In the Journal of Quality Technology article "The Usefulness of Monte Carlo Tests", a similar question was asked.  The solution given was based upon a method by Clarke and Evans (1954) "Distance to Nearest Neighbor as a Measure of Spatial Patterns in Biological Populations" Ecology 35, pp. 445–453.
For each of the N points, the nearest neighbor was found.  The statistic evaluated is the sum of those N distances.  Small values are used to indicate clustering as a one-sided test.
For your test, random locations would be compared to the mining sites.  Distances to nearest neighbors is found, and the distances are summed.  Replications of random tests are conducted with the observed statistic.  The significance level is obtained from the position of the observed statistic in the ordered list.  (With 999 trials and the observed statistic, measures as low as .01 can be obtained.  Trials would need to increase by factors of ten to increase the levels of significance.)

You may want to consider a $\chi^2$ goodness of fit test or Binary Logistic Regression to solve this problem or to give greater meaning to your results.

If you want to continue with your coins example, you would need to use a Binomial distribution.  To calculate a Z score relative to this distribution based upon proportions, then $$Z=\frac{\left(p-p_0\right)}{\sqrt{\frac{p_0 \left(1-p_0\right)}{2}}}$$ where the target proportion is $p_0$ and the process proportion is $p$ when $nP>5$.
