The empirical rule (also known as the three-sigma rule or the 68-95-99.7 rule) is a rule frequently used in many situations to calculate intervals within which some percentage of the data values lie. The empirical rule is widely in the theory of statistical quality control.

**Statement of the Empirical Rule:**

The empirical rule states that given a normal distribution with some mean and standard deviation then:

- 68% of the data lies within one standard deviation of the mean
- 95% of the data lies within two standard deviations of the mean
- 99.7% of the data lies within three standard deviations of the mean

The empirical rule can be used in situations where the underlying data is distributed normally.

This can be checked by seeing whether the distribution is symmetric and has a single “peak”. The rule can also be applied if the data is not terribly skewed.

**Example:**

Suppose that we know that the average height of a particular population is 1.72m with a standard deviation of 0.5m. Generally, the heights of people tend to be distributed normally so we can apply the empirical rule. Hence we can say that:

- 68% of the people have heights lying within one standard deviation of the mean that is, between 1.72-0.5=
**1.67m**and 1.72+0.5=**1.77m**. - 95% of the people have heights lying within two standard deviations of the mean that is, between 1.72-2*0.5=
**1.62m**and 1.72+2*0.5=**1.82m**. - 99.7% of the people have heights lying within three standard deviations of the mean that is, between 1.72-3*0.5=
**1.57m**and 1.72+3*0.5=**1.87m.**

**Applications of the empirical rule:**

The empirical rule is also used in quality assurance. Suppose you are manufacturing a device according to some specifications. After production, you check a sample of the devices to see if the specifications are being followed.

But how do you determine the sample size? The empirical rule helps you to determine the sample size required for 99.7% quality assurance. That is you can obtain a nearly 100% guarantee for quality by only checking a sample and not the entire population of the manufactured devices.

Sometimes we use the six sigma rule instead of 3 Sigma in quality assurance which gives us a 99.99% guarantee of quality.

The empirical rule was first stated historically by De Moivre who observed the rule empirically by repeatedly tossing a coin a large number of times