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The Width of the Normal Distribution
The parameter describing the width of the Normal Distribution,
has the greek letter ``sigma''.
The formula of this function is:
The first two parameters are the mean value and the width.
The equations to get the right parameters for our coin example are:
In our case, N is equal 10 and p is 1/2,
which gives in numbers 5 and 1.58.
You can relate the width with the
probability to get a result that deviates from the expected mean
more than a given value.
If you chose this difference value to be sigma itself,
the Normal Function will have 31.7 % of the total area
(which is 1, since it is normalized) outside.
Interpretations of the Width
We were asking ourselves, how probable it is
to get a certain number of heads when you throw a coin ten times.
Using the Normal Distribution,
we find that the mean expected value is 5
and that 68.3 % of the area is inbetween the range (3.42,6.58).
We can interpret that number as an expected fluctuation intervall,
so that we are not concerned,
if a single ``measurement'' gives 4 instead of 5 heads.
Here we have to make a choice,
what level of deviation would gives us concern.
In statistical analyses, you have to state that in some way,
normally it is called confidence level.
If you want to be very cautious, not to suspect something is wrong too often,
you would call all outcomes ``normal'' that are not further away from
the mean value than 3 sigma.
In the coin example that would be the intervall (0.26,9.74),
so only when you get no or 10 heads
you would suspect something is wrong
with your coin.
On the other hand,
if you do your measurement often enough,
you will get a result outside that intervall,
even with perfect coins!
In the plots you get using the
forms,
each counting measurement has an ``error bar'' attached.
The size of this bar is one sigma of the expected width.
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CLEO WEB PAGES
Updated: 26. December 1995
Author: Andreas H. Wolf (ahw@mps.ohio-state.edu)