Empirical Rule Calculator (68-95-99.7 Rule)

How to use the empirical rule calculator

Enter the mean and standard deviation of your data. If you only have raw numbers, find them with the standard deviation calculator.
Optionally enter a value to check, such as one test score, and the units of your data.
Click Calculate to see the three ranges, highlighted on a bell curve.
Read where your value falls and whether it counts as typical, unusual or very unusual.
Download the chart for a report or class notes.

The empirical rule formula

For a normal distribution with mean μ and standard deviation σ:

About 68% of values lie in μ ± 1σ

About 95% of values lie in μ ± 2σ

About 99.7% of values lie in μ ± 3σ

μ: The mean, the center of the bell curve
σ: The standard deviation, the typical distance of values from the mean
68%, 95%, 99.7%: Rounded from the exact normal-curve areas 68.27%, 95.45% and 99.73%

Because the curve is symmetric, each range splits evenly: 34% of values lie between the mean and 1σ above it, 13.5% between 1σ and 2σ, 2.35% between 2σ and 3σ, and about 0.15% beyond 3σ on each side.

Worked example

Adult IQ scores have a mean of 100 and a standard deviation of 15. What ranges does the empirical rule give?

One standard deviation either side
100 ± 15 = 85 to 115 (68%)
Two standard deviations either side
100 ± 30 = 70 to 130 (95%)
Three standard deviations either side
100 ± 45 = 55 to 145 (99.7%)

What it means: about 95% of adults score between 70 and 130. A score of 135 is more than 2 standard deviations above the mean, so it's unusual: only about 2.5% of people score higher than 130.

How to interpret the empirical rule

Within 1 SD: typical. About two-thirds of all values are this close to the mean.
Between 1 and 2 SD: still common; about 27% of values fall here.
Beyond 2 SD: unusual. Only about 5% of values (2.5% on each side) are this far out.
Beyond 3 SD: very unusual, about 3 in 1,000. Worth double-checking for errors or special causes; quality control charts use this as an alarm limit.

The rule is an approximation for bell-shaped data. For exact areas between any two values, use the normal distribution calculator or look them up in the z-table.

Frequently asked questions

What is the empirical rule?

The empirical rule, or 68-95-99.7 rule, says that for data with a bell-shaped (normal) distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. It is a quick way to judge what counts as typical or unusual without any tables.

What are the exact percentages?

For a normal distribution the exact figures are 68.27% within 1 standard deviation, 95.45% within 2 and 99.73% within 3. The rule rounds them to 68, 95 and 99.7. (The often-quoted 1.96 standard deviations is what gives exactly 95%.)

When can I use the empirical rule?

Only when the data are roughly symmetric and bell-shaped. Check with a histogram first. For skewed data, such as incomes or waiting times, the percentages can be badly wrong; Chebyshev’s theorem gives a weaker guarantee that works for any shape (at least 75% within 2 SD and 89% within 3 SD).

How do I find the percentage between two values?

If both values sit exactly on whole standard deviations, combine the rule’s pieces: for example, from 1 SD below to 2 SD above the mean is 34% + 34% + 13.5% = 81.5%. For values that aren’t whole standard deviations, use the normal distribution calculator, which handles any values exactly.

What counts as unusual under the empirical rule?

A common convention calls a value unusual if it is more than 2 standard deviations from the mean (only about 5% of values are that far out) and very unusual or a possible outlier beyond 3 standard deviations (about 0.3%).

How is the empirical rule related to z-scores?

A z-score counts how many standard deviations a value is from the mean, so the rule can be read in z terms: 68% of values have |z| < 1, 95% have |z| < 2 and 99.7% have |z| < 3. The z-score calculator finds the exact percentile for any value.