26 Empirical Rule
Learning Objectives
Empirical Rule
The empirical rule states that almost all observed natural data, follow a bell-shaped (also called “normal”) distribution with the following properties:
- About 68% of all values fall within 1 standard deviation of the mean.
- About 95% of all values fall within 2 standard deviations of the mean.
- About 99.7% of all values fall within 3 standard deviations of the mean
The empirical rule is very visual! Please watch this video for a great description with examples.
Typically we refer to a population mean with the symbol μ (“mu”) and the standard deviation σ (“sigma”). Below is an image of the normal distribution with μ=0 and σ=1 and the the percentages for all the areas between standard deviations.
Example: Using the Empirical Rule
For each problem, draw and label the graph, and then answer the question.
The weights of adorable one-month-old goat babies are normally distributed with a mean of 13.2 pounds and a standard deviation of 1.3 pounds.
- Using the empirical rule, what percent of one-month-old kids weigh between 10.6 and 15.8 pounds?
Solution: Notice that 10.6 is two standard deviations below the mean and 15.8 is two standard deviations above the mean.
Adding up the percentages in the graph tells us that 95 % are between 10.6 and 15.8 pounds
2. What percent are between 11.9 and 14.5 pounds?
Solution:
68% are between 11.9 and 14.5 pounds
3. Using the empirical rule, find the percent of one-month-old kids weigh more than 14.5 pounds.
Solution:
4. In a random sample of 75 one-month-old kids, how many would be expected to weigh more than 14.5 pounds?
Solution: 
5. Using the empirical rule, find the percent of month- old kids that weigh less than 9.3 pounds and greater than 15.8 pounds.
Solution:
Attributions
- Content and structure adapted from RSCC Math 1410/1420 OER Team, 2022, CC BY 4.0.
- Portions of this content adopted form Openstax ‘Introductory Statistics’: The Standard Normal Distribution (https://openstax.org/books/introductory-business-statistics/pages/6-1-the-standard-normal-distribution?query=empirical%20rule)
