The 68-95-99.7 Rule
For Normal Distributions

This rule applies generally to a variable X
having normal (bell-shaped or mound-shaped) distribution
with mean "mu" (the greek letter) and standard
deviation "sigma" (the greek letter). However,
this rule does not apply to distributions that are not
normal.

Approximately 68% of the observations fall within 1
standard deviation of the mean

Note that the range "within one standard
deviation of the mean" is highlighted in green. The area
under the curve over this range is the relative frequency
of observations in the range. That is, 0.68 = 68% of the
observations fall within one standard deviation of the
mean, or, 68% of the observations are between (mu -
sigma) and (mu + sigma).

Below the axis, in red,
is another set of numbers. These numbers are simply
measures of standard deviations from the mean. In working
with the variable X we will often find it
necessary to convert into units of standard deviations
from the mean. When the variable is measured this way,
the letter Z is commonly used.

Approximately 95% of the observations fall within 2
standard deviations of the mean

Approximately 99.7% of the observations fall within 3
standard deviations of the mean

Only a small fraction of observations (0.3% = 1 in
333) lie outside this range.