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The Empirical Rule

The empirical rule is used in statistics to visualize and calculate values in the normal distribution.



The Empirical Rule shows that almost all data falls in the interval of three standard deviations (σ) from the mean with the following split data proportions:

  • 68% of data lies within 1σ from the mean
  • 95% of data lies within 2σ from the mean
  • 7% of data lies within 3σ from the mean

Therefore, The Empirical Rule is also called ‘The 68-95-97.5 Rule’ and the ‘Three Sigma Rule’.

Empirical rule. Carsten Grube. Statistics



Application of the Empirical Rule

With a well-defined mean and variance (standard deviation) the Empirical Rule can help calculate probabilities within the distribution.


An example: We are informed that the mean height of 12-year old boys in Spain is 143.5 cm with a standard deviation (σ) of 7.1 cm. What is the probability that a 12-year old Spanish boy is taller than 157.7 cm

When finding the z-score for the 157.7, we can approximate the probability value on the Empirical Rule. Z-score: 157.7 – 143.5 / 7.1 = 2. In this example, we get a z-score of an integer, so we can directly see from the Empirical rule that the probability for a Spanish 12-year old boy is = 2.5%.

We solve for 157.7 by calculating, or in this case from viewing, the area to the right of the 2 standard deviations, which is 2.5%.


The Empirical Rule to get a quick overview

With the Empirical Rule we can, like in the example above, get a quick overview of probabilities in the normal distribution by finding the z-score and viewing or calculating the corresponding values to the left, right or for an interval between two z-scores.

The Empirical Rule is applied for the normal distribution and can only be applied for other distributions through the ‘Chebyshev’s Theorem’.


The Empirical rule in MS Excel

You can use the =NORM.DIST and the =NORM.S.DIST Excel functions to help you construct the values in the The Empirical Rule.

Emperical rule in Excel

Thanks to the ExcelsFun video:  Empirical Rule, Calculating Probability NORM.DIST & NORM.S.DIST


Learning resources


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