The most common measure of the fluctuations of a set of numbers
away from the average is the standard deviation, defined as follows.
(a) For each atom in the five-atom toy model of Figure 6.5, compute the deviation of the energy from the average energy, that is, .
Call these deviations .
(b) Compute the average of the squares of the five deviations, that is, .
Then compute the square root of this quantity, which is the root-mean-
square (rms) deviation, or standard deviation. Call this number . Does
give a reasonable measure of how far the individual values tend to stray
from the average?
(c) Prove in general that
that is, the standard deviation squared is the average of the squares minus
the square of the average. This formula usually gives the easier way of
computing a standard deviation.
(d) Check the preceding formula for the five-atom toy model of Figure 6.5.