68% of Values Lie Between 8 and 12

95% of Values Lie Between 6 and 14

97.7% of Values Lie Between 4 and 16

95% of Values Lie Between 6 and 14

97.7% of Values Lie Between 4 and 16

This calculator identifies the expected range of a normally distributed variable, expressed as multiples of the observed standard deviation of the population. The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. This is also referred to as the three-sigma rule or the 68-95-99.7 rule.

This rule can also be used as a quick and intuitive test of the normality of a sample distribution. A normally distributed variable should have 68% of data points within one sigma of the center, 95% of data points within two sigmas of the center, and almost everything within three standard deviations. If the sample you are looking at is notably different that this pattern, you may be dealing with something other than a normal distribution. Alternatively, you may be looking at a sample that would eventually converge on the normal distribution but is constrained by low sample sizes from doing this in practice. This latter pattern is quite common in certain supply chain problems (average order size at the customer level is normally distributed, but you don't have enough homogeneous customers buying the same item for the joint distribution to converge on a normal pattern).

For a deeper academic discussion of these principles, check out this article about the empirical rule.

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