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.

Note: Generally, to say "within A of B" means "between B-A and B+A." For instance, "within 2 of 5" means "between 5 - 2 = 3 and 5 + 2 = 7," in short "between 3 and 7."


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.


Another way of looking at it!

This is merely a consequence of the 68-95-99.7 rule.


Want to see an example? Check out the distribution of weights of adult males.