Empirical Rule: Definition, Formula, Example, How It's Used

What Is the Empirical Rule?

The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all observed data will fall within three standard deviations (denoted by σ) of the mean or average (denoted by µ).

In particular, the empirical rule predicts that 68% of observations falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

Understanding the Empirical Rule

The empirical rule is used often in statistics for forecasting final outcomes. After calculating the standard deviation and before collecting exact data, this rule can be used as a rough estimate of the outcome of the impending data to be collected and analyzed.

This probability distribution can thus be used as an interim heuristic since gathering the appropriate data may be time-consuming or even impossible in some cases. Such considerations come into play when a firm is reviewing its quality control measures or evaluating its risk exposure. For instance, the frequently used risk tool known as value-at-risk (VaR) assumes that the probability of risk events follows a normal distribution.

The empirical rule is also used as a rough way to test a distribution's "normality". If too many data points fall outside the three standard deviation boundaries, this suggests that the distribution is not normal and may be skewed or follow some other distribution.

The empirical rule is also known as the three-sigma rule, as "three-sigma" refers to a statistical distribution of data within three standard deviations from the mean on a normal distribution (bell curve), as indicated by the figure below.

Examples of the Empirical Rule

Let's assume a population of animals in a zoo is known to be normally distributed. Each animal lives to be 13.1 years old on average (mean), and the standard deviation of the lifespan is 1.5 years. If someone wants to know the probability that an animal will live longer than 14.6 years, they could use the empirical rule. Knowing the distribution's mean is 13.1 years old, the following age ranges occur for each standard deviation:

• One standard deviation (µ ± σ): (13.1 - 1.5) to (13.1 + 1.5), or 11.6 to 14.6
• Two standard deviations (µ ± 2σ): 13.1 - (2 x 1.5) to 13.1 + (2 x 1.5), or 10.1 to 16.1
• Three standard deviations (µ ± 3σ): 13.1 - (3 x 1.5) to 13.1 + (3 x 1.5), or, 8.6 to 17.6

The person solving this problem needs to calculate the total probability of the animal living 14.6 years or longer. The empirical rule shows that 68% of the distribution lies within one standard deviation, in this case, from 11.6 to 14.6 years. Thus, the remaining 32% of the distribution lies outside this range. One half lies above 14.6 and the other below 11.6. So, the probability of the animal living for more than 14.6 is 16% (calculated as 32% divided by two).

As another example, assume instead that an animal in the zoo lives to an average of 10 years of age, with a standard deviation of 1.4 years. Assume the zookeeper attempts to figure out the probability of an animal living for more than 7.2 years. This distribution looks as follows:

• One standard deviation (µ ± σ): 8.6 to 11.4 years
• Two standard deviations (µ ± 2σ): 7.2 to 12.8 years
• Three standard deviations ((µ ± 3σ): 5.8 to 14.2 years

The empirical rule states that 95% of the distribution lies within two standard deviations. Thus, 5% lies outside of two standard deviations; half above 12.8 years and half below 7.2 years. Thus, the probability of living for more than 7.2 years is:

95% + (5% / 2) = 97.5%