How to Calculate the Empirical Rule (68-95-99.7)

This page applies the empirical rule to a normal distribution. Enter the mean and the standard deviation; you get the range that holds about 68% of the values (one standard deviation either side) and the range that holds about 95% (two standard deviations). The 99.7% range is in the worked example.

Formula

About 68% of values lie in μ ± σ, about 95% in μ ± 2σ and about 99.7% in μ ± 3σ.

• μ: the mean; σ: the standard deviation

Worked Example

A normal distribution with mean 100 and standard deviation 15.

1. 68%: 100 − 15 = 85.00 to 100 + 15 = 115.00
2. 95%: 100 − 30 = 70.00 to 100 + 30 = 130.00
3. 99.7%: 100 − 45 = 55 to 145

How to read the result

Values outside μ ± 2σ happen about 1 time in 20; outside μ ± 3σ about 3 times in 1,000. The exact shares are 68.27%, 95.45% and 99.73%.

Limits and common mistakes

• The rule only holds for roughly normal (bell-shaped) data. Skewed data do not follow it.
• Use the population or sample standard deviation as your course says.

Frequently Asked Questions

What does 68-95-99.7 mean?

The share of values within 1, 2 and 3 standard deviations of the mean.

Does it work for any data?

Only for data that are roughly normal.

What share lies above μ + σ?

About 16%.

Is 95% exactly 2 standard deviations?

Very nearly; exactly 95% is 1.96 standard deviations.

in m | Mean | - | 100 | -
in s | Standard deviation | - | 15 | 0
out lo1 | 68% range from | - | m - s | 2
out hi1 | 68% range to | - | m + s | 2
out lo2 | 95% range from | - | m - 2 * s | 2
out hi2 | 95% range to | - | m + 2 * s | 2