Empirical Rule Calculator

What is the Empirical Rule?

The Empirical Rule, also known as the 68-95-99.7 rule or the three-sigma rule, is a statistical rule which states that for a normal distribution (bell-shaped curve), nearly all observed data will fall within three standard deviations (σ) of the mean (μ).

Specifically, the rule states:

• Approximately 68% of the data falls within one standard deviation of the mean (μ ± 1σ).
• Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).
• Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).

How to Use the Calculator

1. Enter the Mean (μ): Input the average value of your dataset.
2. Enter the Standard Deviation (σ): Input the standard deviation of your dataset. This value must be non-negative.
3. View the Results: The calculator will automatically display the data ranges corresponding to approximately 68%, 95%, and 99.7% of the data based on the Empirical Rule.

Important Considerations

• Normal Distribution: The Empirical Rule is only an approximation and works best for datasets that closely follow a normal distribution. It may not be accurate for skewed or non-normal distributions.
• Sample vs. Population: The rule applies whether μ and σ represent population parameters or sample statistics, although the accuracy increases with larger sample sizes that better approximate a normal distribution.

What is the empirical rule?

The empirical rule, also known as the three-sigma rule or 68–95–99.7 rule, is a statistical rule that states that for a normal distribution, almost all data will fall within three standard deviations of the mean.

This rule is also known as the 68–95–99.7 rule because if we take a normal distribution and mark off two standard deviations above and below the mean (which covers approximately 95% of the data), we are left with approximately 68% of the data in between.

And if we take one more standard deviation above and below the mean (which covers approximately 99.7% of the data), we are left with approximately 95% of the data in between.

To summarize:

• 68% of data within 1 standard deviation
• 95% of data within 2 standard deviations
• 99.7% of data within 3 standard deviations

The emprical rule is calculated from two numbers:

• Mean
• Standard deviation

And it is based on the normal distribution. Let's look at each of these concepts.

What is the mean?

The mean is the average of a set of numbers. To find the mean, add up all the numbers in the set and then divide by the number of items in the set. The mean is also known as the arithmetic mean or simply the average.

What is the standard deviation?

In statistics, the standard deviation is a measure of the dispersion of a dataset relative to its mean. It is calculated as the square root of the variance. The standard deviation is a popular measure of variability because it is easy to compute and has an intuitive interpretation.

The standard deviation has some interesting properties. First, it is always positive. Second, it is unaffected by changes in location or scale. Third, it is sensitive to outliers. Fourth, it satisfies the triangle inequality.

The standard deviation can be computed for any dataset that has a mean and variance. It is most commonly used with normally distributed data, but can be used with any type of data.

What is the normal distribution?

A normal distribution is a statistical distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In a normal distribution, there are no outliers.

Normal distributions appear as a bell shaped curve.

Empirical rule formula

Calculate the mean: First, calculate the mean of your data. The mean is the sum of all the datapoints, divided by the number of datapoints.

• $\mu$ is the mean
• $\sum{}$ is the sum
• $x_i$ is each individual value from your data
• $n$ is is the number of datapoints

Calculate the standard deviation:

• $\sigma$ is the standard deviation
• $\sqrt{}$ is the square root

Apply the empirical rule formula:

• 68% of the data will fall within 1 standard deviation, or between $\mu - \sigma$ and $\mu + \sigma$
• 95% of the data will fall within 2 standard deviation, or between $\mu - 2 \sigma$ and $\mu + 2 \sigma$
• 99.7% of the data will fall within 3 standard deviation, or between $\mu - 3 \sigma$ and $\mu + 3 \sigma$

How is the empirical rule used?

The empirical rule is a valuable tool for statisticians and researchers as it allows them to quickly and easily determine if a dataset is normally distributed. Additionally, the empirical rule can be used to estimate the percentage of data that will fall within a certain range.

For example, if you know that the mean height of adult males in the United States is 70 inches and that the standard deviation is 3 inches, you can use the empirical rule to estimate that 68% of adult males in the United States are between 67 and 73 inches tall (one standard deviation above and below the mean), 95% are between 64 and 76 inches tall (two standard deviations above and below the mean), and 99.7% are between 61 and 79 inches tall (three standard deviations above and below the mean).

Limitations of the empirical rule

One limitation of the empirical rule is that it only applies to data that is normally distributed. This means that if your data is not normally distributed, then using the empirical rule will not give you accurate results.

Another limitation is that the empirical rule only applies to data that has been collected in a large enough sample size. If you have a small sample size, then using the empirical rule will again not give you accurate results.

While the empirical rule is a valuable tool, it is important to keep in mind that it only applies to normal distributions. If a dataset is not normally distributed, then using the empirical rule will not give you accurate results.