Statistics & Probability

Instant, private, and free

Empirical Rule Calculator.

Calculate one-, two-, and three-standard-deviation bounds.

On-device calculationNo signup
01

Set your values

Results update as you type.

1σ lower: 85

1σ lower

85.000000
1σ upper: 115

1σ upper

115.000000
2σ lower: 70

2σ lower

70.000000
2σ upper: 130

2σ upper

130.000000
3σ lower: 55

3σ lower

55.000000
3σ upper: 145

3σ upper

145.000000

Results update automatically as you type.

Use Cases

Quality control in manufacturing

Use the empirical rule to determine the expected range of product measurements (e.g., dimensions, weight) if the process is normally distributed. This helps identify when a process is out of control.

Example: If a part's length has mean 10 cm and standard deviation 0.2 cm, 95% of parts should be between 9.6 and 10.4 cm.

Understanding test scores

For standardized tests with a normal distribution, the empirical rule helps interpret an individual's score relative to the population.

Example: If the mean score is 100 and standard deviation is 15, 68% of students score between 85 and 115.

Frequently Asked Questions

What is the empirical rule?
The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
How do I use this calculator?
Simply enter the mean and standard deviation of your normally distributed data. The calculator instantly shows the ranges for one, two, and three standard deviations from the mean, based on the empirical rule.
What if my data is not normally distributed?
The empirical rule is only accurate for data that follows a normal (bell-shaped) distribution. If your data is skewed or has outliers, the rule may not apply, and you should use other methods like Chebyshev's theorem.

Tips & Common Mistakes

Tips

  • Ensure your data is approximately normally distributed before applying the empirical rule.
  • Use the mean and standard deviation from your sample or population, but be consistent.
  • The empirical rule is a quick estimate; for exact probabilities, use a z-table or statistical software.
  • Remember that the rule applies to the percentage of data within k standard deviations, not to individual data points.

Common Mistakes to Avoid

  • Applying the empirical rule to data that is not normally distributed.
  • Using the sample standard deviation when the population standard deviation is known, or vice versa.
  • Misinterpreting the ranges: 68% of data falls within one standard deviation, not 68% of the range.

Last updated: August 13, 2026