BIZOMAT 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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Process mean
- What it is
- The average value of the measured characteristic from a stable, representative sample of the process, in the drawing’s units.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.02 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moving the mean towards one limit raises the share out of tolerance on that side.
- Get it right
- Use the same units as the specification limits and standard deviation.
What this means: Standard deviation
- What it is
- The spread of the characteristic around its mean, from the same sample, in the same units as the limits.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.03 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger spread raises the share out of tolerance on both sides.
- Get it right
- Use within-process spread for short-term capability, overall spread for long-term performance.
What this means: Lower specification limit
- What it is
- The smallest acceptable value of the characteristic, from the drawing or specification.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 9.9 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raising it increases the share below the lower limit.
- Get it right
- Use the limit itself, not the nominal minus a tolerance you have tightened.
What this means: Upper specification limit
- What it is
- The largest acceptable value of the characteristic, from the drawing or specification.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10.1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lowering it increases the share above the upper limit.
- Get it right
- It must be above the lower limit.
What this means: Production quantity
- What it is
- The number of parts in the run or period you want to estimate, used to turn the out-of-tolerance fraction into a count of parts.
- Units and format
- Enter a number in parts: parts. The form accepts 0 parts or more. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10000 parts as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The expected count out of tolerance rises in direct proportion.
- Get it right
- It only scales the count; the fraction and ppm do not depend on it.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Inspection as inference
In Accept or Reject by the Numbers
What it covers: Treat an inspection result as evidence: false accepts and false rejects, the base rate from process capability, Bayes' rule in odds form, and why reject bins hold mostly good parts.
Method explained here · BIZOMAT original book
Variation is a random variable
In Control Charts by the Numbers
What it covers: A measured dimension as a random variable: mean, standard deviation and the normal distribution, how independent sources of variation add, subgroup means, and predicting parts out of tolerance.
Method explained here · BIZOMAT original book
Random numbers you can trust
In Simulating Noisy Systems by the Numbers
What it covers: How pseudo-random generators work, seeds and reproducible streams, turning uniform numbers into Gaussian ones with Box–Muller, testing a generator's spread and tails, and the shortcuts that fail.
Method explained here · BIZOMAT original book
Margins in standard deviations
In Trips and Alarms by the Numbers
What it covers: Measuring a signal's noise from a log, splitting it into fast and slow parts with the autocorrelation, expressing a threshold as a margin in standard deviations, and reading tail probabilities.
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