This depends on the parameter you are investigating, your model and the population you are studying and a number of other things (See wikipedia, search: sampling distribution for more details). E.g. if your parameter is a square, you might expect e.g. the chi-square distribution.

If you estimate the simple mean from a repeated simulated samples and resulting sampling distribtion does not appear normal, then you might consider some bias to arise from the conditions listed in the wikipedia page (e.g. sampling procedure, non-normal population distribution)

a large number of Monte Carlo simulations should have given indeed a normal distribution. If not, then there are several interpretations to your results (mainly two):

1) either the random number generator is skewed, so it will never give you a normal distribution.

2) or the phenomena that you are studying IS skewed, on which case, the average will not be at the middle, and usually the median is more useful or more telling. Skewness will tell you a sort of bias, which you will have to bear in mind to properly draw conclusions about the distribution of your population.

I am using a random number generator of James 1990, which has

been proved to be a good random number generator since among other things it provides a repetition period of > 10**(18).

Because of Central Limit Theorem, for any skewed sample distribution of size N>30, their derived distribution of all possible means should be approximately normal.

I have an observational sample of unknown distribution (size sample around 50), I want to estimate some parameters values and uncertainties under a set of different conditions. Each set of conditions constitutes a model. When I perform a KS-test, which is a goodness-of-fit to compare the output simulated data against the output observed real sample under non-parametric assumptions, the mean sample of all the KS-test applied to all possibles outcomes does not follow a normal distribution, in some models I obtained really skewed KS-test distributions. Of course I determine the best model by the best fit according to the median/mean of the KS-test distribution. However, I cannot understand why normal distribution is not satisfied.

Hi;

I think your main population members haven't the same distribution. Because the random variables must be identically distributed.

Lester LIpsky - University of Connecticut

Have you considered the possibility that your underlying distribution is "power-tailed?

That is,

Pr(X > x) ==> c / x^a,

and that a < 2. In this case, the measured variance is unboundedly large,

and the distribution of means is inherently skewed! These converge to

"alpha-stable," or Levy-Pareto distributions. The Normal distribution is a

special case for a > 2. See books by Taqqu for details.

Hi,

you can perform a clustering analysis and then try to analyze each cluster and further you can try defining any other relationship to redefine your initial population distribution

My main population is binomial, constituted by two kind of elements x1 and x2, which have different probability of being observed, but satisfies normal distribution when it comes to distribution of means of each element. This is because my size N population is large enough so that Np>6 and N(1-p)>6 (p=probability of success x1). One important point is that the output samples simulated do not have exactly the same size but similar.

The thing is that, I want to validate a theory, so I perform a K-S test to compare the observational population with the outcomes population of My Monte-Carlo simulations. I compare the data simulated y with the observational data yo. The y-values corresponds to a certain variable associated to my x-values which I have found applying certain equations whose physical parameters I want to constraint. I do different models changing the input parameters.

I don't see why the distribution of average of the KS-tests does not follow a normal distribution. Might it be due to the fact that the output samples have not the same size?. It appears that some certain output sample are more likely to be yield.

My critical point is whether I can validate a theory by a KS-test derived from the distribution of KS-test means which does not fit to a normal distribution.

Even more, i am wondering if theories can be validated from non-normal populations.

I am not a statistician, but I have often to do with both normal and skewed distributions of motoneuron interspike intervals and this differences can be explained by motoneuron physiology. You do not give any details about the variable you measure, but its skewed distribution may also bear some important information on the nature of underlying process.

Central Limit Theorem (CLT) does not work for arbitrarily distributed random numbers. They have to be identically distributed (iid) AND their variances should be finite to make CLT work. For example, the Cauchy distribution, with pdf ~ 1/(x^2 + a^2), or the Pareto-Levy distribution mentioned earlier, are good examples not satisfying those conditions and invalidating CLT.

One small comment on some of the above comments. The CLT DOES work for

a mixture of different distributions IF:

a) All the variances are finite, and

b) none of the variances are much larger than the others (this has to be

quantified better).

Lester, you are right, of course. However, if you allow different distributions to enter the game then at the same time you have to restrict them to the class with well behaving third central moment. Its finiteness does not suffice, if I'm not mistaken.

Judit, don't stick to normal distribution too firmly. The world around us is full of Zipf's law examples: population of cities, rain falls, earthquakes, popularity of names or movies, stock market exchange rates, and even egg-shell fragmentation. Maybe your particular case is just an another example?

Well, if we must, let us list a few more real-world distributions with unbounded

variance:

1) CPU times in research environments;

2) File sizes in research environments;

3) solar flares;

4) internet traffic (probably because of 2));

5) Jobs in unstable environments that must RESTART;

6) volcano eruptions;

7) I've lost a few others.

As Marek says, the list can go on forever. as we look further.

Zipf's law and Pareto (power-talied) distributions are complements of

each other. Zipf's law refers to the "Order statistics" of the largest members

of a finite set of numbers drawn randomly from a Pareto distribution.

( I hope that means something to the reader).

As pointed out previously, CLT does work for variables with distributions that are not identical. There are, in fact, several versions of CLT, some of them requiring iid variables, some of them only independen variables but with certain conditions over moments and some other special conditions, and even some for dependant variables . Check for Lyapunov or Lindeberg or lindeberg-feller versions.

Less skewed sampling distribution, closer to normality.