# Empirical Rule in Statistics

### Empirical Rule:

The empirical rule is referred 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).

- Empirical rule can be applied for a symmetrical bell shaped frequency distribution
- Empirical rule is known as the three sigma rule
- The range within which approximate percentage of values of the distribution are likely to fall within a given number of standard deviation from the mean is determined below:

- Approximately 68.26% of the data is within one standard deviation of the mean.
- Approximately 95.44% of the data is within two standard deviations of the mean.
- More than 99.72% of the data is within three standard deviations of the mean

#### Examples of the Empirical Rule:

**Example 1:**

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:

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

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. Half lies above 14.6 and half lies below 11.6. So, the probability of the animal living for more than 14.6 is 16% (calculated as 32% divided by two).

**Example 2:**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:

- Standard deviation (µ ± σ): 8.6 to 11.4 years
- Standard deviations (µ ± 2σ): 7.2 to 12.8 years
- 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%

#### Limitations of Empirical Rule

- Empirical rule applies only to normally distributed data
- It has a wide range of applications, but in cases where distribution is not normal or the shape of distribution is not known, its application is restricted

- Chebyshev’s inequality is the best alternative to empirical rule.

Author: Mohit T (Algae Study)

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