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Empirical Rule Calculator

Table of contents

What is the empirical rule?The empirical rule - formulaAn example of how to use the empirical ruleWhere is the empirical rule used?FAQs

The empirical rule calculator (also a 68 95 99 rule calculator) is a tool for finding the ranges that are 1 standard deviation, 2 standard deviations, and 3 standard deviations from the mean, in which you'll find 68, 95, and 99.7% of the normally distributed data respectively. In the text below, you'll find the definition of the empirical rule, the formula for the empirical rule, and an example of how to use the empirical rule.

If you're into statistics, you may want to read about some related concepts in our other tools, such as the Z-score calculator or the point estimate calculator.

What is the empirical rule?

The empirical rule (also called the "three-sigma rule" or the "68-95-99.7 rule") is a statistical rule that states that, for normally distributed data, almost all the data points will fall within three standard deviations on either side of the mean.

More specifically, you'll find:

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

Let's explain the concepts used in this definition:

Standard deviation is a measure of spread; it tells how much the data varies from the average, i.e., how diverse the dataset is. The smaller value, the more narrow the range of data is. Our standard deviation calculator expands on this description.

Normal distribution is a distribution that is symmetric about the mean, with data near the mean being more frequent in occurrence than data far from the mean. In graphical form, normal distributions appear as a bell-shaped curve, as you can see below:

The graph of normal distribution

Of course, you can learn more by visiting the normal distribution calculator.

The empirical rule - formula

The algorithm below explains how to use the empirical rule:

1. Calculate the mean of your values:

μ=xin\mu = \frac{\sum x_i}{n}


  • xi\sum x_i — Sum of all values in the sample; and
  • nn — The size of the sample (number of data points).

You can also make your life easier by simply using the average calculator.

2. Calculate the standard deviation:

σ=(xiμ)2n1\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}}

3. Apply the empirical rule formula:

  • 68% of data falls within 1 standard deviation from the mean - that means between μσ\mu - \sigma and μ+σ\mu + \sigma.

  • 95% of data falls within 2 standard deviations from the mean - between μ2σ\mu - 2\sigma and μ+2σ\mu + 2\sigma.

  • 99.7% of data falls within 3 standard deviations from the mean - between μ3σ\mu - 3\sigma and μ+3σ\mu + 3\sigma.

    Enter the mean and standard deviation into the empirical rule calculator, and it will output the intervals for you.

An example of how to use the empirical rule

Intelligence quotient (IQ) scores are normally distributed with the mean of 100 and the standard deviation equal to 15. Let's have a look at the maths behind the 68 95 99 rule calculator:

  1. Mean: μ=100\mu = 100

  2. Standard deviation: σ=15\sigma = 15

  3. Empirical rule formula:

    μσ=10015=85\mu - \sigma = 100 - 15 = 85
    μ+σ=100+15=115\mu + \sigma = 100 + 15 = 115
    (68% of people have an IQ between 85 and 115)

    μ2σ=100215=70\mu - 2\sigma = 100 - 2 \cdot 15 = 70
    μ+2σ=100+215=130\mu + 2\sigma = 100 + 2 \cdot 15 = 130
    (95% of people have an IQ between 70 and 130)

    μ3σ=100315=55\mu - 3\sigma = 100 - 3 \cdot 15 = 55
    μ+3σ=100+315=145\mu + 3\sigma = 100 + 3 \cdot 15 = 145
    (99.7% of people have an IQ between 55 and 145)

For quicker and easier calculations, input the mean and standard deviation into this empirical rule calculator, and watch as it does the rest for you.

Where is the empirical rule used?

The rule is widely used in empirical research, such as when calculating the probability of a certain data point occurring, or for forecasting outcomes when some data is missing. It gives insight into the characteristics of a population without the need to test everyone, and helps to determine whether a given data set is normally distributed. It is also used to find outliers, which may be the result of experimental errors.


How do I calculate the empirical rule?

To calculate the empirical rule:

  1. Determine the mean m and standard deviation s of your data.
  2. Add and subtract the standard deviation to/from the mean: [m − s, m + s] is the interval that contains around 68% of data.
  3. Multiply the standard deviation by 2: the interval [m − 2s, m + 2s] contains around 95% of data.
  4. Multiply the standard deviation by 3. 99.7% of data falls in [m − 3s, m + 3s].

What is the empirical rule for data with variance 1?

Variance 1 means that the standard deviation equals 1 as well. The empirical rules says that:

  • 68% of your data lies at most one unit from the average;
  • 95% lies at most two units from the average; and
  • 99.7% lies at most three units from the average.
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