What is the Empirical Rule?

The empirical rule (also known as the three-sigma or 68-95-99.5 rule) is a statistical rule that states that almost all data from a normal distribution will fall within three standard deviations. (denoted with s) of either the mean (denoted using u).

The empirical rule predicts, in particular, that 68% of observations fall within the first standard deviation (u+- s), 95% within two standard deviations(u +-2s), and 99.7% within three standard deviations.


  • According to the Empirical Rule, 99.7% of data resulting from a normal distribution are within three standard deviations of its mean.
  • This rule states that 68% of data falls within one standard deviation, 95% falls within two standard deviations, and 99.7% is within three standard deviations.
  • The empirical rule uses three-sigma limits to establish the upper and lower limits of control in statistical quality control charts and risk analysis, such as VaR.
Empirical Rule

Understanding the Empirical Rule

Statistics often use the empirical rule to forecast outcomes. This rule is used to estimate the outcome of the data collected and analyzed after calculating the standard deviation.

As gathering the right data can be difficult or time-consuming, this probability distribution can be an interim tool. These considerations are important when a company is reviewing its quality control procedures or evaluating its risk exposure. Value-at-risk (VaR), a commonly used risk tool, assumes that risk events are predictable.

This empirical rule can also determine if a distribution is “normal.” If there are too many data points that fall below the three standard deviation limits, it is possible that the distribution may not be normal. The empirical rule, also known as the “three-sigma rule,” refers to statistical distributions of data within three standard deviations of the mean on normal distributions (bell curves), as shown in the figure below.

Examples of the Empirical Rule

Let’s say that a normal distribution of animals in a zoo has been established. On average, each animal lives 13.1 years. The standard deviation of this lifespan is 1.5. The empirical rule can determine if an animal will live more than 14.6 years. The distribution’s average age is 13.1 years. Each standard deviation has the following age ranges:

One standard deviation (u+- s), is (13.1 – 1.5) to (13.1+ 1.5), or 11.6–14.6

There are two standard deviations (u+- 2s), 13.1 – (2×1.5) to 13.1 + (2×1.5), or 10.1 to 16.1.

Three standard deviations (u+- 3s), 13.1 – (3×1.5) to 13.1+ (3×1.5), or 8.6 to 17.6

This problem requires that the person solve it to calculate the probability that the animal will live for at least 14.6 years. The empirical rule states that 68% of the distribution falls within one standard deviation. In this instance, it is 11.6 to 14.6. The remaining 32% of the distribution is outside this range. One-half lies above 14.6 while the other is below 11.6. The probability that the animal will live for more than one year is, therefore, 16%. This is 32% divided by 2.

Another example is to assume that an animal at the zoo will live for an average of 10 years, with a standard deviation of 1.4 years. Let’s say that the zookeeper attempts to figure out how likely it is for an animal to live more than 7.2. The following distribution is shown:

A standard deviation (u+- s) is 8.6 to 11.4 Years.

Two standard deviations (u+- 2s), 7.2 to 12.8 Years

Three standard deviations (u +-3s): 5.8-14.2 years

According to the empirical rule, 95% of the distribution falls within two standard deviations. The empirical rule states that 5% of the distribution lies outside the two standard deviations. Half the variance is above 12.8 years, and half below 7.2. The probability that you will live for longer than 7.2 years is, therefore:

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

What is the Empirical Rule?

The empirical statistic rule states that 99.7% of data is within three standard deviations from the mean within a normal distribution. This means that 68% of observed data will be within the first standard deviation, 95% in the second deviation, and 97.5% in the third standard deviation. The empirical rule predicts probability distribution for a given set of outcomes.

What is the Empirical Rule?

The empirical rule predicts possible outcomes in normal distributions. Statisticians use this to calculate the percentage of cases falling within each standard deviation. The standard deviation is 3.1, and the average equals 10. The first standard deviation would be between (10+3.2.= 13.2 or (10-3.2.= 6.8). The second deviation would be between 10 + (2X 3.22)= 16.4 or 10 – (2X 3.22)= 3.6, respectively.

What are the benefits of the Empirical Rule?

Because it can be used to forecast data, the empirical rule is a useful tool. This is particularly true for large datasets or those with unknown variables. The empirical rule applies to finance, specifically stock prices, price indexes, and log values for forex rates. These tend to fall within a bell curve or normal distribution.

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