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Statistical Basics
All data taken with an ``counting method'' at HEP is subject to random fluctuations! This means, if you would have a second set and measure the same thing again, you will get a slightly different answer.
You can ``modell'' this with throwing a coin or rolling a dice. If you have a coin or anything with two sides available now, try this:

Throw your coin and count, how often you got a certain side, let's say ``Heads''. If your coin is ``working properly'' you expect half of the throws to be ``Heads'', right? Try it ten times, and you should get five.
Well, as you know it is also possible to only get four, or get six. But how probable is this? When do you doubt a coin? When you get 3 heads?

The Probablity Function
A function is an algorithm, that computes a value for a certain set of ``parameters''. The Probability Function gives you the probability of a certain outcome of your ``measurement'', like throwing a coin.
Let's construct the probability function for the coin step by step. We should give this function a name, why not P(N;k)... N is the number of throws, and k is the number of heads.

• For one throw each side has the same probability.
  So P(1;0) = 1/2 and P(1;1) = 1/2, right?
  One important rule for all probability function is, that the sum of all probabilities is one (this is called ``normalized'').
• The next step is two throws. Here we have two ways to get exactly one head: At the first throw or at the second. There is only one way each, to get no head or two heads. Since each way has the same probability, we get:
  P(2;0) = 1/4, P(2;1) = 1/2, and P(2;2) = 1/4
  
• Another way to find that result is, to use the independent outcome of each throw. Then you say:
  P(2;0) = P(1;0)*P(1;0)
  which means to get no head, you have to have no head in each independent throw. It also follows:
  P(2;1) = P(1;0)*P(1;1) + P(1;1)*P(1;0)
  P(2;2) = P(1;1)*P(1;1)
  As you can see you get the same result! ``Independence'' is a very important fact, and is widely and falsely ignored by the popular approach to gambling.
• Using the last approach, we can easily construct the proability function for three throws:
  P(3;0) = P(2;0)*P(1;0)
  P(3;1) = P(2;1)*P(1;0) + P(2;0)*P(1;1)
  P(3;2) = P(2;2)*P(1;0) + P(2;1)*P(1;1)
  P(3;3) = P(2;2)*P(1;1)
  or
  P(3;0) = 1/8
  P(3;1) = 3/8
  P(3;2) = 3/8
  P(3;3) = 1/8
  
• Try to work your way up to ten throws.
  This is the result as a picture:
  

The Normal Distribution
In the case of a counting measurement, like the mass distributions in the first interactive form, we need a more general description of the probability function.
In HEP you will find the Normal Distribution (also called ``Gaussian Function'') as the modell function of a measurement. This function has two parameters:

• The mean value, which gives you the mean expected outcome of the measurement. For our example above, it would be 5.
• The width of the distribution, which gives you the expected ``spread'' around the mean value.

There is a fixed algorithm, that relates our example case with this function. It gives you the mean and the width of this function, that describes the outcome of the ten coin tosses in the best way.

shows the distribution using the Normal Function drawn together with the correct result. As you can see, it reproduces the correct results. The main difference is the ``smooth shape'' instead of the steps. The Normal Function does not ``know'' that there are only natural numbers possible.

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Updated: 26. December 1995
Author: Andreas H. Wolf (ahw@mps.ohio-state.edu)