KEVOS® MAKEOS · Quality
Normal Distribution Out-of-Tolerance
What fraction of parts will fall outside these limits for this mean and standard deviation?
Fraction and parts per million below the lower and above the upper specification limit for a normally distributed characteristic, with Z values and expected yield.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Process mean | Average of the characteristic from a stable, representative sample. | |
| Standard deviation | Standard deviation (within-process for short-term, overall for long-term performance). | |
| Lower specification limit | Lower limit from the drawing or specification. | |
| Upper specification limit | Upper limit from the drawing or specification. | |
| Production quantity | parts | Number of parts to estimate the count out of tolerance. |
Method and formulas
- Z_L = (mean − LSL) ÷ σ; Z_U = (USL − mean) ÷ σ.
- Fraction below = Φ(−Z_L); fraction above = Φ(−Z_U), with Φ the standard normal distribution.
- ppm = total fraction × 10⁶.
- Example: mean 10.02, σ 0.03 within 9.9–10.1 gives about 3,862.1 ppm out of tolerance.
Limitations
- Assumes a stable, normal distribution; tails of real processes often differ from the normal model.
- Uses the standard deviation as entered; short-term σ understates long-term ppm.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Predict scrap from a process mean and spread
- Compare centring and spread improvements
- Estimate ppm for a customer quality agreement
Who it is for
- Quality engineers
- Process engineers
- Test engineers
What you get
- Z values to each limit
- Fraction below, above and total out of tolerance
- Parts per million and expected count
- Yield
Interpreting the result
- Centring the process usually cuts ppm faster than reducing spread when the mean is off-centre.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 128–133. Topic reference; no source tables or recommended settings are reproduced.
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