The Standard DeviationThe rootmeansquare deviation of x from its average is called the standard deviation. For a set of discrete measurements, the standard deviation takes the form
and
Determining the average or mean in the above expression involves the distribution function for the variable.

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Standard Deviation for Particle PositionA 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

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