Just like Chebyshev’s theorem, the empirical rule can also be used to find the percentage of total observations that are within a certain interval around the mean.
Here it is empirical rule::
About 68% of all values are within one standard deviation of the mean.
About 95% of all values are within 2 standard deviations from the mean.
About 99.7% of all values are within 3 standard deviations from the mean.
A picture is shown below:
Note that 34% + 34% = 68% and 34% + 34% + 13.5% + 13.5% = 95%
However, like Chebyshev’s theorem, the empirical rule has its limits. What is the difference between Chebyshev’s theorem and the empirical rule?
Using Chebyshev’s theorem, you cannot find the percentage of values in a data set that are within one standard deviation of the mean.
However, you can apply the theorem to any type of distribution.
You can only apply the empirical rule to a specific type of distribution called a bell distribution or normal curve.
You can also use the rule to find the percentage of a record’s values that are within. lies a standard deviation the mean.
How do you apply the empirical rule?
For example, suppose the mean is 89 and the standard deviation is 14. Find the approximate percentage of the values that are between 47 and 131 of the mean.
The key to solving this problem is figuring out how many standard deviations 47 and 131 are from the mean.
In other words, we’re just looking for k.
My strategy is to put the mean, or 89 in the middle, 47 on the left side of the mean, and 131 on the right side of the mean.
47 ———————- 89 ———————- – 131
Recall that in the lesson on Chebyshev’s Theorem we showed that the distance between 89 and 131 is equal to kσ
kσ = 131 – 89
kσ = 42
Substitute 14 for σ.
k × 14 = 42
Divide both sides by 14.
We get k = 3.
About 99.7% of all values are between 47 and 131.