Empirical Rule Calculator

Instructions: This Empirical Rule calculator will show you how to use the Empirical Rule to compute some normal probabilities. Please type the population mean and population standard deviation, and provide details about the event you want to compute the probability for. Observe that not all events can have their probability computed with these technique. For a general normal probability calculator, please check here .

Population Mean (\(\mu\))
Population St. Dev. (\(\sigma\))
≤ X ≤
X ≤
X ≥

More About the Empirical Rule

The Empirical Rule states that the area under the normal distribution that is within one standard deviation of the mean is approximately 0.68, the area within two standard deviations of the mean is approximately 0.95, and the area within three standard deviations of the mean is approximately 0.997.

It needs to be observed that these are approximations only. Using the exact normal distribution tables, for example, the area within two standard deviations of the mean is more like 0.954500, instead of 0.95, although 0.95 is a simpler to remember number.

Using this Empirical Rule only a handful number probabilities can be computed. For the general case, use this normal probability calculator .

Relationship between Chebyshev's theorem and the Empirical Rule

Notice that the empirical rule is applicable only to normal distributions. For the case of general, non-normal distributions, you should use instead our Chebyshev's inequality calculator , or even Markov's inequality for non-negative random variables.

Observe that sometimes the empirical rule is referred as the 68-95-99.7 Rule Calculator, because of the probabilities associated with the rule.

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