Probability and Statistics > Statistics Definitions > Empirical Rule

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## Definition of the Empirical Rule

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts:

- 68% of data falls within the first standard deviation from the mean.
- 95% fall within two standard deviations.
- 99.7% fall within three standard deviations.

The rule is also called the 68-95-99 7 Rule or the **Three Sigma Rule**.

## When do we use the Empirical Rule?

The Empirical Rule is often used in statistics for **forecasting**, especially when obtaining the right data is difficult or impossible to get. The rule can give you a rough estimate of what your data collection might look like if you were able to survey the entire population.

This rule applies generally to a random variable, X, following the shape of a normal distribution, or bell-curve, with a mean “mu” (the Greek letter &mu) and a standard deviation “sigma” (the Greek letter σ). The rule doesn’t apply to distributions that are not normal, but you can apply it to other distributions using Chebyshevâ€™s Theorem.

## Empirical Rule: Notation

When applying the Empirical Rule to a data set the following conditions are true:

- Approximately 68% of the data falls within one standard deviation of the mean (or between the mean – one times the standard deviation, and the mean + 1 times the standard deviation). In mathematical notation, this is represented as: μ±1σ
- Approximately 95% of the data falls within two standard deviations of the mean (or between the mean – 2 times the standard deviation, and the mean + 2 times the standard deviation). The mathematical notation for this is: μ±2σ
- Approximately 99.7% of the data falls within three standard deviations of the mean (or between the mean – three times the standard deviation and the mean + three times the standard deviation). The following notation is used to represent this fact: μ±3σ