Free · Probability and statistical models

Normal Distribution One to Three Sigma Bands

Find the familiar 68%, 95% and 99.7% bands around a normally distributed mean.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Mean 100 and deviation 10 give bands 90–110, 80–120 and 70–130.

Bands = μ ± σ, μ ± 2σ, μ ± 3σ

Worked example

Enter these known values and leave the other values blank.

Mean
100
Standard deviation
10
Result
Approximate normal-distribution bands: 90 ; 110 ; 80 ; 120 ; 70 ; 130

Assumptions and limitations

  • The percentages describe a normal distribution approximately. They are not guaranteed for arbitrary data.
  • Standard deviation must be nonnegative. A zero deviation is a degenerate constant distribution.

Common questions

What should I check before using this result?

The percentages describe a normal distribution approximately. They are not guaranteed for arbitrary data.

References

Enforce the mathematical input domain and include valid zero boundaries while preserving the captured arithmetic and control graph.

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