+34 616 71 29 85 carsten@dataz4s.com
Select Page

# 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’.

## 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

#### Carsten Grube

Freelance Data Analyst

p
p
p
##### ANOVA & the F-distribution

+34 616 71 29 85

Call me

Spain: Ctra. 404, km 2, 29100 Coín, Malaga

...........

Denmark: c/o Musvitvej 4, 3660 Stenløse

Drop me a line

What are you working on just now? Can I help you, and can you help me?