MasterTool

Empirical Rule Calculator

Calculate empirical rule using your data.

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What this tool does

Calculate empirical rule using your data. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.

How to use the Empirical Rule Calculator

  1. Enter or select mean (μ).
  2. Enter or select standard deviation (σ).
  3. Read the calculated result; change any measurement to compare alternatives.

Formula

empirical rule: about 68% of normal data lie within μ±1σ, 95% within μ±2σ, 99.7% within μ±3σ
mu
Mean (μ)
sigma
Standard deviation (σ)

The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.

Worked example

For empirical rule calculator, the following measurements illustrate the exact method: Mean (μ): 100; Standard deviation (σ): 15.

Inputs

  • Mean (μ)100
  • Standard deviation (σ)15

Result

  • 68% range — lower (μ−σ)85
  • 68% range — upper (μ+σ)115
  • 95% range — lower (μ−2σ)70
  • 95% range — upper (μ+2σ)130
  • 99.7% range — lower (μ−3σ)55
  • 99.7% range — upper (μ+3σ)145

Results explained

68% range — lower (μ−σ)
68% range — lower (μ−σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.
68% range — upper (μ+σ)
68% range — upper (μ+σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.
95% range — lower (μ−2σ)
95% range — lower (μ−2σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.
95% range — upper (μ+2σ)
95% range — upper (μ+2σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.
99.7% range — lower (μ−3σ)
99.7% range — lower (μ−3σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.
99.7% range — upper (μ+3σ)
99.7% range — upper (μ+3σ) from the formula above. The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.

Frequently asked questions

empirical rule: about 68% of normal data lie within μ±1σ, 95% within μ±2σ, 99.7% within μ±3σ

The 68–95–99.7 percentages are exact only for a perfectly normal distribution; they are approximations for real, roughly bell-shaped data.

Enter numbers only, in the units and format each label describes. Remove missing values rather than substituting zero, unless zero is a real observation.

No. Results describe the numbers you entered. Statistical inference also depends on sampling design, independence, model fit and interpretation; a p-value is not the probability that a hypothesis is true.

No. Every calculation, including the distribution algorithms, runs entirely in your browser.