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.

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.

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.

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.