The Standard Deviation

The root-mean-square deviation of x from its average is called the standard deviation. For a set of discrete measurements, the standard deviation takes the form

for discrete measurements of x

and

for continuous x
where < > implies average.

Determining the average or mean in the above expression involves the distribution function for the variable.


Example of free particle in a box
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Standard Deviation for Particle Position

A free particle which is constrained to be between x=0 and x=L has a distribution function which is just a constant. The relationship for the standard deviation of the position is the square root of the integral


Normalizing the distribution gives the value for C.


The mean value of x is


Using this, the standard deviation becomes


The average in the square root is


The resulting standard deviation for the free particle is


Use in 3-D box calculation
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Distribution functions

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