# Re: about the standard Deviations

From: Alberto Fasso' (Alberto.Fasso@cern.ch)
Date: Mon Apr 14 2003 - 16:09:51 CEST

In principle it is true that a standard deviation could be obtained
with a single Monte Carlo run, analyzing the distribution of scores due to
different particle histories. Some programs offer this possibility.
However, implementing such a feature would imply doubling all storage
requirements, since for each value scored (each bin, each energy interval, etc.)
one should score also its square. In addition, the distributions of
contributions to a given quantity from each history are not always
easy to handle, because they are not necessarily Gaussian. On the contrary, they
are often strongly asymmetric, with a few large and many small
contributions. In many cases even, most histories are likely to produce zero
contribution.

The use of a number of independent runs largely avoids all the above problems,
since the distribution of results from independent runs tends to be normally
distributed (Central Limit Theorem), and there is no need to add up the squares
of each score. The runs, however, should be made of sufficient size that
the average value of the desired quantity be non-zero for each of them.
(A zero value would be an artefact due to insufficient sampling which would
spoil the Gaussian distribution).

Having said that, I do not understand why you say that "The averages or standard
deviations got from a number of independent runs seem not the results what I
want". In the limit of a large number of cases, the averages and standard
deviations obtained with two methods tend to coincide.

Alberto Fassò

On Tue, 11 Mar 2003, wuzhen wrote:

> Dear fluka users,

> I think the standard deviations of each physical quantity should be got from
> an independent run using a random number seed. For an example, if I have
> simulated 10000 particles in a run using a random seed, I think the standard
> deviations can be got from these 10000 particles. The averages or standard
> deviations got from a number of independent runs seem not the results what I
> want. Can you explain it for me?
>
> Best regards,
> Zhen wu
>
>
>

```--
Alberto Fassò
CERN-EP/AIP, CH-1211 Geneve 23 (Switzerland)
Phone: (41 22) 767 2398    Fax: (41 22) 767 9480   Alberto.Fasso@cern.ch
```

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