At this point, it is useful to try some examples. First, we may have
a uniform probability distribution localized in the range 
,
This distribution is sketched in figure 4. (The units in this figure
are not units of 
but are units of the standard deviation,
which we will show how to compute presently.)
Note that we have taken proper care that the distribution is normalized. X must have some value, so
which is true in this case. Clearly also, 
. We thus have,
is now non-zero and clearly increases with the spread
in the distribution 
.
We could also have a random variable Y distributed according to an exponential distribution
as in Figure 5.
This distribution is also properly normalized (<1>=1) with zero average (<Y>=0). Now,
which again is non-zero and grows with the width of the distribution.
Finally, we may have a random variable Z distributed according to the normal distribution,
as in Figure 6.
Here,
In all three of these cases we found that the variance measures the
square of the parameter controlling the width 
of the
distribution. This leads us to our final definition, that of the
standard deviation, which is meant to give a direct measure of
the width of a distribution. We define the standard deviation as the square
root of the variance,
Because of its form, the standard deviation is sometimes referred to as the root mean square (rms) deviation. This is the unit that was used along the horizontal axis of the plots for the distributions of X, Y, and Z in Figures 4-6. From those figures you can see that the standard deviation does correspond to what we would qualitatively associate with the approximate width of these distributions.
The table below summarizes our results for these three distributions.
TABLE II: WIDTHS FOR THREE BASIC DISTRIBUTIONS
