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Mean ± Sigma: Upper & Lower Control Limits

Calculate symmetric limits around a supplied mean using a nonnegative standard deviation and sigma multiplier.

  • Formula & worked example
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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

For mean 100, standard deviation 5 and multiplier 3, the upper limit is 115 and the lower limit is 85.

Upper limit = mean + k × SD; lower limit = mean − k × SD

Worked example

Enter these known values and leave the other values blank.

Center mean
100
Standard deviation for this chart statistic
5
Sigma multiplier
3
Upper statistical limit
115
Lower statistical limit
85

Assumptions and limitations

  • Supply the standard deviation appropriate to the statistic being charted. This tool does not estimate it or select chart-specific constants.
  • Standard deviation and multiplier are nonnegative. Zero collapses both limits to the mean; inverse calculations can then be underdetermined.
  • Statistical control limits differ from engineering specification limits. Negative lower limits are retained, with no count-chart clipping or process-capability conclusion.

Common questions

Are three-sigma limits always the product tolerance?

No. Statistical limits describe a process model; tolerances or specifications are set by separate requirements.

Does a point inside these limits prove the process is stable?

No. Chart selection, model assumptions, measurement quality and pattern-based checks also matter.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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