Empirical Rule Calculator (68-95-99 Rule)

Enter the mean and standard deviation for a standard normal distribution to calculate the amount of data that will fall within 68%, 95%, and 99.7% of the mean.

Results:

68% of the data is between

 
and
 

95% of the data is between

 
and
 

99.7% of the data is between

 
and
 

Steps to Solve

Part 1: 68% of the data falls within 1 standard deviation of the mean

μ - σ = ?

μ + σ = ?

Part 2: 95% of the data falls within 2 standard deviations of the mean

μ - (2 × σ) = ?

μ + (2 × σ) = ?

Part 3: 99.7% of the data falls within 3 standard deviations of the mean

μ - (3 × σ) = ?

μ + (3 × σ) = ?



What is the Empirical Rule?

In statistics, the empirical rule, sometimes also called the 68 95 99.7 rule, states that 99.7% of data that is normally distributed will fall within three standard deviations of the mean.

In addition, the rule states that 68% of the data will fall within one standard deviation of the mean, and 95% will fall within two standard deviations of the mean.

graph of a standard normal distribution illustrating the empirical rule where 68% of the data is within 1 SD of the mean, 95% is within 2 SD of the mean, and 99.7% is within 3 SD of the mean

In a normally distributed data set, the distance from the mean in standard deviations is the z-score. For instance, a z-score of 2.0 is a 2σ distance from the mean.

Thus, the empirical rule can be used to estimate the percentage of the data between z-scores.

Key Takeaway

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

If you’re just getting started with statistics, the mean is the average value of the data set and is often represented using the greek letter mu μ. The standard deviation is a measure of the variability within the data and is represented using the greek letter sigma σ.

Empirical Rule Formula

The empirical rule can be represented using the following formulas:

    68% of the data falls between these two values:

    z = μ ± σ

    Thus, 68% of the data will fall between the mean μ plus or minus the standard deviation σ.

    95% of the data falls between these two values:

    z = μ ± (2 × σ)

    So, 95% of the data will fall between the mean μ plus or minus 2 times the standard deviation σ.

    99.7% of the data falls between these two values:

    z = μ ± (3 × σ)

    So, 99.7% of the data will fall between the mean μ plus or minus 3 times the standard deviation σ.

You might also be interested in our p-value calculator, which can find the percentage of values above or below a score rather than between scores.

For example, given a data set with a mean of 48 and a standard deviation of 13, let’s find the range of the values above and below the mean that will contain 95% of the data.

Using the formulas above, the range that will contain 95% of the data is:

z = μ ± (2 × σ)

Substituting the mean of 48 and standard deviation in the formula is:

z = 48 ± (2 × 13)
z = 48 ± 26

This can be rewritten as the following formulas:

z = 48 – 26 = 22
z = 48 + 26 = 74

Thus, 95% of the data in this set will be between 22 and 74.