All You Need to Know About the Empirical Rule in Statistics

The empirical rule came into existence because of the repetition of distribution curves that continued to appear repeatedly to statisticians. The empirical rule is associated with a normal distribution. In this tutorial, you will explore the empirical Rule in Statistics in depth.

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What Is an Empirical Rule in Statistics?

The empirical rule, also known as the three-sigma rule or the 68-95-99.7 rule, is a statistical rule that states that almost all observed data for a normal distribution will fall within three standard deviations (denoted by σ) of the mean or average (denoted by µ).

According to this rule, 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean.

When you reasonably expect your data to approximate a normal distribution, the mean and standard deviation become even more valuable, thanks to the empirical rule. You can calculate probabilities and percentages for various outcomes simply by knowing these two statistics.

Graphical Explanation


  • The normal distribution is associated with the 68-95-99.7 rule which is shown in the image above.
  • 68% of the data is within 1 standard deviation (σ) of the mean (μ).
  • 95% of the data is within 2 standard deviations (σ) of the mean (μ).
  • 99.7% of the data is within 3 standard deviations (σ) of the mean (μ).

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The formula for Empirical Rule is: 


µ = Mean

σ  = Standard deviation

m  = Multiplier 


Suppose the pulse rates of 100 students are bell-shaped with a mean of 75 and a standard deviation of 4.

  • About 68% of the men have pulse rates in the interval  75 土 1(4) = [71, 79]
  • About 95% of the men have pulse rates in the interval  75 土  2(4) = [67, 83]
  • About 99.7% of the men have pulse rates in the interval  75 土 3(4) = [63, 87]


The Empirical Rule or the 68–95–99.7 can only be applied to a symmetric and unimodal distribution because it is only applicable to Normal Statistical Distributions.

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Empirical Rule is a statistical concept that aids in showing the probability of observations and is particularly useful when approximating a large population. It's important to remember that these are only estimates. There is always the possibility of outliers who do not fit into the distribution. As a result, the findings are inaccurate, and you should exercise caution when acting on the forecast.

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If you need any clarification or have any doubts, please mention them in the comments section of this tutorial page. We'll have our experts look into them and respond with answers promptly.

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Kartik is an experienced content strategist and an accomplished technology marketing specialist passionate about designing engaging user experiences with integrated marketing and communication solutions.

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