KEVOS® MAKEOS · Quality
Normal Tolerance Interval Check (Coverage, Confidence, Tolerance Factor, Against Specification)
From a sample mean and standard deviation, between what limits can I say a stated proportion of all parts lie, at my confidence level, and do those limits sit inside the specification?
Tolerance interval for a normally distributed characteristic from a sample: the tolerance factor for your sample size, coverage and confidence (two-sided by Howe’s method, one-sided exactly from the non-central t distribution, with the Natrella approximation alongside), the lower and upper tolerance limits, and whether they lie inside your specification limits.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Interval | choice | Two-sided for a characteristic with both limits; one-sided when only a minimum or only a maximum matters. |
| Sample mean | Average of the sample measurements. | |
| Sample standard deviation | Standard deviation of the sample measurements (divisor n − 1). | |
| Sample size | parts | Number of parts measured. |
| Coverage (proportion of all parts) | % | Proportion of the whole population the interval must contain. |
| Confidence | % | Confidence that the interval really contains that proportion. |
| Specification | ||
| Lower specification limit | Lowest value the drawing or specification allows. | |
| Upper specification limit | Highest value the drawing or specification allows. | |
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: Interval
- What it is
- Whether the interval has both a lower and an upper limit, or only one limit for a characteristic with only a minimum or only a maximum.
- Units and format
- Choose one of 3 options from the list: Two-sided (lower and upper limits); One-sided lower limit; One-sided upper limit.
- Example
- The form opens with “Two-sided (lower and upper limits)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- A one-sided interval uses a smaller factor, so its single limit sits closer to the mean.
- Get it right
- Match the specification: a bore diameter with ± tolerance is two-sided; a minimum strength is one-sided lower.
What this means: Sample mean
- What it is
- The average of the sample measurements of the characteristic, in the same units as the specification limits.
- 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 100 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Moves the whole tolerance interval up or down.
- Get it right
- Use parts made under the same conditions as production, measured with a capable gauge.
What this means: Sample standard deviation
- What it is
- The standard deviation of the sample measurements, calculated with the divisor n − 1 as spreadsheets do with STDEV.S.
- 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 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The interval width grows in proportion to it.
- Get it right
- Calculate it from the same sample as the mean; one outlier inflates it, so check the data first.
What this means: Sample size
- What it is
- The number of parts measured in the sample.
- Units and format
- Enter a number in parts: parts. The form accepts 3 parts to 10000 parts. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 43 parts as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger sample lowers the tolerance factor towards the normal value z.
- Get it right
- More parts shrink the factor: with fewer than about 10 the factor, and so the interval, becomes very wide.
What this means: Coverage (proportion of all parts)
- What it is
- The proportion of the whole population of parts the interval must contain, for example 99 % of all parts made.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 60% to 99.99%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher coverage widens the interval.
- Get it right
- Choose it from the risk of a part out of tolerance you can accept; 90 %, 95 % and 99 % are common.
What this means: Confidence
- What it is
- The confidence that the interval really contains at least the stated proportion of all parts, given only a sample.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 60% to 99.99%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 99% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher confidence widens the interval, more so for small samples.
- Get it right
- Use 95 % for most qualification work and 99 % for critical characteristics.
Specification
What this means: Lower specification limit
- What it is
- The lowest value the drawing or specification allows for the characteristic.
- 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 94 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pass or fail line for the lower tolerance limit.
- Get it right
- Take it from the drawing tolerance or material specification in the same units as the mean.
What this means: Upper specification limit
- What it is
- The highest value the drawing or specification allows for the characteristic.
- 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 106 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pass or fail line for the upper tolerance limit.
- Get it right
- Take it from the drawing tolerance or material specification in the same units as the mean.
Method and formulas
- Two-sided (Howe): k₂ = z₍₁₊ₚ₎/₂ √(ν (1 + 1/N) ÷ χ²), with ν = N − 1 and χ² the chi-square value with ν degrees of freedom exceeded with probability equal to the confidence (NIST e-Handbook 7.2.6.3).
- One-sided: k₁ = t ÷ √N, with t the confidence quantile of the non-central t distribution with N − 1 degrees of freedom and non-centrality z_p √N; the Natrella approximation k₁ = (z_p + √(z_p² − ab)) ÷ a, a = 1 − z²_conf ÷ (2(N − 1)), b = z_p² − z²_conf ÷ N, is shown for comparison.
- Tolerance limits = mean ± k s; the interval passes when it lies inside the specification limits.
- Example: tolerance factor 2.2173; tolerance interval 95.565 to 104.43.
Limitations
- Valid only for a normally distributed characteristic: check the data with a normal probability plot first; skewed data need a distribution-free interval.
- Howe’s two-sided factor is an approximation, very close to the exact value for usual sample sizes; the one-sided factor is exact.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Show that a stated share of production meets a drawing tolerance
- Set acceptance limits from a qualification sample
- Compare a supplier sample against a specification
Who it is for
- Quality engineers
- Design and test engineers
- Supplier quality staff
What you get
- Tolerance factor k
- Lower and upper tolerance limits
- Limits compared with the specification
Interpreting the result
- A tolerance interval covers individual parts, unlike a confidence interval for the mean: it does not shrink to zero as the sample grows, but approaches mean ± z s.
References
- NIST/SEMATECH e-Handbook of Statistical Methods, 7.2.6.3 Tolerance intervals for a normal distribution — https://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Coating thickness by lift-off
In Eddy-Current Inspection by the Numbers
What it covers: Measuring anodising on aluminium as lift-off, why coating thickness is not dimensional growth, calibrating with foils, a wrong zero that drove the process off target, and tolerance intervals.
Method explained here · KEVOS® original book
Variation and how many specimens
In Testing Welded Joints by the Numbers
What it covers: Describe scatter in weld test results with a mean and standard deviation, estimate a low value and a confidence interval, and judge how many specimens a question really needs.
Useful alongside · Library collection
Proving a process is ready for production: measurement, stability, capability and PPAP-style evidence
In Product Development and Launch
What it covers: Why good samples do not prove a process is ready to scale, and how measurement checks, control charts, capability indices and a traceable evidence chain show it can repeat.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.