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.

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