# The Empirical Rule

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

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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’*.** **

** **

## 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.

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

## Learning resources

- Khan Academy video: Normal distribution problems: Empirical rule (from ck12.org)

#### Carsten Grube

Freelance Data Analyst

##### Normal distribution

##### Two-sample inference

##### Confidence intervals

##### Simple linear regression, fundamentals

##### ANOVA & the F-distribution

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