The Empirical Rule, sometimes called the 68-95-99.7 rule, states that for a given dataset with a normal distribution:

• 68% of data values fall within one standard deviation of the mean.
• 95% of data values fall within two standard deviations of the mean.
• 99.7% of data values fall within three standard deviations of the mean.

In this tutorial, we explain how to apply the Empirical Rule in R to a given dataset.

Applying the Empirical Rule in R

The pnorm() function in R returns the value of the cumulative density function of the normal distribution.

This function uses the following basic syntax:

pnorm(q, mean, sd)

where:

• q: normally distributed random variable value
• mean: mean of distribution
• sd: standard deviation of distribution

We can use the following syntax to find the area under the normal distribution curve that lies in between various standard deviations:

#find area under normal curve within 1 standard deviation of mean
pnorm(1) - pnorm(-1)

[1] 0.6826895

#find area under normal curve within 2 standard deviations of mean 
pnorm(2) - pnorm(-2)

[1] 0.9544997

#find area under normal curve within 3 standard deviations of mean 
pnorm(3) - pnorm(-3)

[1] 0.9973002

From the output we can confirm:

• 68% of data values fall within one standard deviation of the mean.
• 95% of data values fall within two standard deviations of the mean.
• 99.7% of data values fall within three standard deviations of the mean.

The following examples show how to use the Empirical Rule with different datasets in practice.

Example 1: Applying the Empirical Rule to a Dataset in R

Suppose we have a normally distributed dataset with a mean of 7 and a standard deviation of 2.2.

We can use the following code to find which values contain 68%, 95%, and 99.7% of the data:

#define mean and standard deviation values
mean=7sd=2.2

#find which values contain 68% of data
mean-2.2; mean+2.2

[1] 4.8
[1] 9.2

#find which values contain 95% of data
mean-2*2.2; mean+2*2.2

[1] 2.6
[1] 11.4

#find which values contain 99.7% of data
mean-3*2.2; mean+3*2.2

[1] 0.4
[1] 13.6

From this output, we can see:

• 68% of the data falls between 4.8 and 9.2
• 95% of the data falls between 2.6 and 11.4
• 99.7% of the data falls between 0.4 and 13.6

Example 2: Finding What Percentage of Data Falls Between Certain Values

Imagine we have a normally distributed dataset with a mean of 100 and standard deviation of 5.

Suppose we want to know what percentage of the data falls between the values 99 and 105 in this distribution.

We can use the pnorm() function to find the answer:

#find area under normal curve between 99 and 105
pnorm(105, mean=100, sd=5) - pnorm(99, mean=100, sd=5)

[1] 0.4206045

We see that 42.06% of the data falls between the values 99 and 105 for this distribution.