Empirical Rule Calculator (68-95-99.7)

Apply the 68-95-99.7 rule to any normal distribution. Enter the mean and standard deviation to get the ranges that contain about 68%, 95% and 99.7% of values, plus the exact percentages, the expected number of observations and the z-score and percentile of any value you choose. Example: for a mean of 100 and a standard deviation of 15, about 68% of values fall between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145.

How to use the Empirical Rule Calculator (68-95-99.7)

1. Enter the mean (μ) and the standard deviation (σ) of your data.
2. Optional: enter a sample size to see how many observations are expected in each range.
3. Optional: enter a value to see its z-score, percentile and band.
4. Press Calculate. Open Show steps to see each range worked out.

Formula and method

Range = mean ± k × standard deviation, for k = 1, 2, 3

In a normal distribution about 68.27% of values lie within 1 standard deviation of the mean, 95.45% within 2, and 99.73% within 3. The rule is often written 68-95-99.7.

Example: mean 100, SD 15. 1σ: 85 to 115. 2σ: 70 to 130. 3σ: 55 to 145.

Z-score = (x − mean) ÷ SD. The value 130 has z = (130 − 100) ÷ 15 = 2.0, which is about the 97.7th percentile.

What is the empirical rule?

The empirical rule, also called the 68-95-99.7 rule or three-sigma rule, says that in a normal (bell-shaped) distribution nearly all values fall within three standard deviations of the mean. It is a fast way to describe how spread out data is without a z-table.

Exact percentages

The rounded numbers in the rule are close but not exact. This calculator reports both.

Worked example: IQ scores

IQ scores are scaled to a mean of 100 and an SD of 15. About 68% of people score between 85 and 115, 95% between 70 and 130, and 99.7% between 55 and 145. A score of 130 is 2 SD above the mean, putting it at roughly the 97.7th percentile.

Worked example: test scores

A class has a mean of 72 and an SD of 8. About 95% of students score between 56 and 88. A score of 90 is 2.25 SD above the mean, so it is rarer than 1 in 80.

When the rule does not apply

The rule assumes a bell-shaped, symmetric distribution. Income, house prices and waiting times are skewed. For those, use percentiles from the data, or Chebyshev’s inequality: at least 75% of values lie within 2 SD and at least 88.9% within 3 SD for any distribution with finite variance.

Using the rule to spot outliers

A value more than 3 SD from the mean is rare (about 3 in 1,000 for a normal distribution). Many quality control charts use the three-sigma limits to flag unusual measurements.

Empirical rule vs z-table

A z-table gives exact probabilities for any z. The empirical rule gives three quick landmarks. The calculator’s optional value field uses the exact normal distribution, so you get both.

Accuracy and limits

Normal-distribution percentages use a numerical approximation of the normal CDF accurate to about 7 decimal places. If your data are not normal, treat the results as approximations.

Frequently asked questions

What is the 68-95-99.7 rule?

In a normal distribution, about 68% of values are within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.

How do I use the empirical rule?

Compute mean ± 1, 2 and 3 standard deviations. The ranges contain about 68%, 95% and 99.7% of the data.

Is the empirical rule exact?

No, the numbers are rounded. The exact values are 68.27%, 95.45% and 99.73% for a normal distribution.

What if my data is not normal?

The rule may not hold. Use Chebyshev’s inequality or percentiles from the data.

How is it related to the z-score?

A z-score counts how many standard deviations a value is from the mean. The three bands of the rule correspond to z = ±1, ±2, ±3.

Sources

• 68-95-99.7 rule (Wikipedia)
• NIST/SEMATECH e-Handbook: Normal distribution
• Standard score (Wikipedia)

Related calculators

• Normal Distribution Calculator
• Z-Score Calculator
• Standard Deviation Calculator
• Percentile Calculator
• Confidence Interval Calculator

Browse all calculators