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VerifyKit sample-size calculator: reference-value check report

Generated 2026-11-10 by tools/build.mjs (Node v24.20.0, 32.7 s) from the same reference data the automated tests use (reference-data.js). Calculator version 2.0.0.

SHA-256 stats.js: ba470c2e5db06266ae2a3d8604cbca94bc0c53a646e14fa30179f38d3d0a204d
SHA-256 rationale.js: 970159523a58d1084a75e970a852e904f8a5674a8f6263b55f62dbb28a621f85
SHA-256 reference-data.js: b6b20e8f46b446ffcfbdddb8aba65a6f2468b26fdd61618838c8c57a09f598a8

1. Comparison with published reference values

R is reliability (coverage proportion) and C is confidence. "exact" means the integers must match exactly. Other cases are compared within half a unit of the last published digit, plus 1e-9 to allow for a value that sits exactly on a rounding boundary. Where the source rounds in a known way, that rounding is applied first. Datasets for the normality and outlier cases are the public R example datasets (mtcars, ToothGrowth, airquality, iris) and the NIST uranium isotope example, embedded in reference-data.js.

2. Findings from the source checks

• Two-sided 95/95 factors in older tables are approximations. The values 3.379 (n = 10), 2.752 (n = 20) and 2.549 (n = 30) are reproduced by the Wald-Wolfowitz approximation. The exact factors are 3.3934, 2.7603 and 2.5549. VerifyKit uses the exact factor. It agrees with Odeh & Owen (1980) and R EnvStats.
• ISO 16269-6. The worked examples in the ISO 16269-6:2005 preview give k3(12; 0.95; 0.95) = 2.737 and k4(12; 0.90; 0.95) = 2.671. These equal the exact values (2.73634 and 2.67028) rounded up to 3 decimals. The Wald-Wolfowitz approximation would give 2.655. The 2014 edition's tables were not seen.
• Janiga & Garaj (2009) list the exact two-sided factors rounded up to 4 decimals.
• NIST Dataplot output for the Howe approximation (N = 195) is reproduced to all printed digits.
• NIST binomial example. NIST prints 0.401029. Direct summation gives 0.4010281, so the last digit differs by 1.
• NIST Grubbs example. NIST prints G = 2.4687. The computed value is 2.468765, which suggests NIST truncates. It is compared within 1e-4.
• Grubbs critical values (Harvey, LibreTexts). The table matches the two-sided formula in NIST 1.3.5.17.1 to within 1 unit in the last digit, but its rounding is inconsistent. Its n = 15, α = 0.01 entry repeats the n = 14 value, so that entry is excluded.
• Shapiro-Wilk, mtcars mpg with cyl = 6 (n = 7). Two pages on the same site print the same W but different p-values (0.3252 and 0.3278). VerifyKit gives 0.32518, so only W is used as a reference.
• Shapiro-Wilk, mtcars mpg with cyl = 4 (n = 11). Unresolved. The page prints W = 0.91243. VerifyKit's W = 0.9124392 would print as 0.91244, a difference of 9e-6. The p-value (0.2606) matches. No second source was found. Every other W, from n = 7 to n = 153, matches to all printed digits. The case is kept visible as a documented discrepancy.
• Distribution-free sample size. NIST's chi-square approximation gives 473 for 99%/95% two-sided, and the exact formula also gives 473.
• Notation. "95/90" is written as C/R in some places and R/C in others. The calculator always labels C and R explicitly.

3. Independent numerical checks and simulations

• One-sided k. For n in {2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30, 40, 50, 60, 80, 100, 150, 200, 300, 500, 750, 1000}, R in {0.9, 0.95, 0.99} and C in {0.9, 0.95, 0.99}, the confidence each factor achieves was recomputed with a different integral (over the normal variable, using the chi-square survival function). Worst |confidence - C| = 2.98e-12.
• Two-sided k. Same grid, recomputed by integrating over the SD first. Worst |confidence - C| = 1.03e-12.
• Attribute and success-run n. Each n was confirmed to be the smallest sufficient sample size by direct binomial sums, in 96 of 96 cases.
• Tolerance factors by Monte Carlo (seeded, 100,000 samples, 95/95): one-sided n = 5: 95.02%; one-sided n = 10: 95.02%; two-sided n = 5: 95.12%; two-sided n = 10: 95.08% (target 95%, standard error about 0.07%).
• Probability of passing (exact noncentral-t / integral compared with 40,000 simulated tests): one-sided LSL, n = 10, true margin 3.5 SD, 95/95: exact 81.58%, simulated 81.44%; two limits, whole-interval rule, n = 12, 95/95: exact 78.51%, simulated 78.45%; two limits, per-limit rule, n = 12, 95/95: exact 93.67%, simulated 93.53%.
• False-alarm rates for normal data (2,000 samples, alpha = 0.05): Anderson-Darling 4.3% (n = 20), Shapiro-Wilk 5.0% (n = 20), Grubbs 4.3% (n = 15). The target is about 5% for each.
• Upper 90% confidence bound on the SD (n = 10, 20,000 samples) covered the true SD in 90.0% of samples (target 90%).
• Transforms. The tests check that Box-Cox and its inverse round-trip, that λ = 0 gives the log, that the transform is increasing, and that the analysis after transforming the data and the limits matches a direct calculation. They also check that the chosen λ maximizes the normal probability plot correlation (NIST 1.3.3.6).

4. Two-sided approximations compared with the exact factor (R = 95%, C = 95%)

For information only. The calculator does not use these approximations for decisions.

5. Computed factor tables

These are VerifyKit outputs, not published values.

One-sided exact k

Two-sided exact k

6. Methods

• Success-run: n = ceil(ln(1 - C) / ln R). Attribute: the smallest n with P(X ≤ c | n, 1 - R) ≤ 1 - C, using an exact binomial via the regularized incomplete beta function.
• One-sided normal k: exact, from the noncentral t distribution (NIST 7.2.6.3), using Gauss-Legendre integration and Brent root finding. Two-sided: the exact Odeh-type integral.
• Two-sided decision rules. The default is "whole interval inside spec", using the two-sided k. The optional "one-sided bound at each limit" rule uses the one-sided k at each limit. No verified reference that prescribes the second rule was found, so the page explains its weaker guarantee instead of citing one.
• Probability of passing: P = E[max(0, Φ(√n(K_U - kV)) - Φ(√n(kV - K_L)))], with V = s/σ. For one limit this is the noncentral t expression. The SD upper bound uses the chi-square quantile (NIST 1.3.5.8).
• Anderson-Darling: case 3, mean and SD estimated. The modified statistic and the p-value follow Stephens (1986), as in R nortest. The p-value needs n ≥ 8. Shapiro-Wilk: Royston AS R94, as in R shapiro.test, for 3 ≤ n ≤ 5000.
• Grubbs: two-sided, with critical value ((n-1)/√n)√(t²/(n-2+t²)) and t at α/(2n) with n - 2 df (NIST 1.3.5.17.1). It flags values only and never removes them.
• Box-Cox: (x^λ - 1)/λ, or ln x at λ = 0. λ is chosen to maximize the normal probability plot correlation (NIST 1.3.3.6), and the spec limits are transformed the same way. Distribution-free: NIST 7.2.6.4.

7. Citation status

• Verified on the source page: the NIST/SEMATECH e-Handbook sections cited above, the FDA QMSR page (21 CFR Part 820 incorporates ISO 13485:2016 by reference, effective 2 February 2026), the R documentation and outputs listed in the sources, and the ISO 16269-6:2005 preview examples.
• Verified through catalogue or secondary sources only (paid standards): ISO 13485:2016 clause 7.3.6 content (paraphrased, never quoted), ISO 13485:2016 clause 4.1.6 (validation of QMS software), the titles and scope of ISO 16269-6:2014, ISO 2859-1:2026, ISO 3951-1:2022 and ISO/TR 80002-2:2017.
• Not verified from a primary table: the one-sided 95/95 values 2.911, 2.396 and 2.220 (a search index shows them in Illinois 35 IAC 742 Appendix A Table B, but the page could not be opened), and an exact two-sided value for n = 30 (2.5549 here). Both are supported by independent integrals and simulation. The one-sided method is also confirmed by ISO, NIST and EnvStats examples.

8. Sources

• NIST_TI: NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.3 "Tolerance intervals for a normal distribution" https://www.itl.nist.gov/div898/handbook/prc/section2/prc263.htm
• NIST_NP: NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.4 "Tolerance intervals based on the largest and smallest observations" https://www.itl.nist.gov/div898/handbook/prc/section2/prc264.htm
• NIST_BIN: NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.4.1 "Confidence intervals" (exact binomial limits for a proportion) https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm
• ENVSTATS_K: R package EnvStats, help page tolIntNormK (printed example output) https://rdrr.io/cran/EnvStats/man/tolIntNormK.html
• ENVSTATS_NP: R package EnvStats, help page tolIntNparN (printed example output) https://rdrr.io/cran/EnvStats/man/tolIntNparN.html
• SPC_ODEH: R package spc, help page tol.lim.fac, which reproduces Odeh & Owen (1980), "Tables for Normal Tolerance Limits, Sampling Plans and Screening", p. 98, Table 3.4.1 https://search.r-project.org/CRAN/refmans/spc/html/tol.lim.fact.html
• JANIGA: Janiga I., Garaj I. (2009), "On Two-Sided Statistical Tolerance Intervals for Normal Distributions with Unknown Parameters", MEASUREMENT 2009, Proc. 7th Int. Conf., Smolenice, Table 3 (m = 1) https://www.measurement.sk/M2009/proceedings/057_Janiga1.pdf
• SPEC_TABLE: VerifyKit spec reference table (classical 3-decimal 95/95 factors). One-sided values also appear in Illinois 35 IAC 742 Appendix A Table B according to a search-engine index; that page could not be opened, so treat as secondary
• CLOSED_FORM: Closed form n = ceil(ln(1 - C) / ln(R)); value checked by direct arithmetic
• ISO16269_2005: ISO 16269-6:2005 (since replaced by the 2014 edition), worked examples in clause 5.4 (Example 3, one-sided, unknown variance) and clause 5.5 (Example 4, two-sided, unknown variance), read in the free preview published by iTeh. The factors in the standard are rounded up https://cdn.standards.iteh.ai/samples/38772/9532c47232fc4eb8a52ed3c4c86281a5/ISO-16269-6-2005.pdf
• DATAPLOT_HOWE: NIST Dataplot Reference Manual, TOLERANCE LIMITS, example output (two-sided normal limits, Howe method, N = 195) https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/tolelimi.htm
• R_DATASETS: R built-in datasets (mtcars, ToothGrowth, airquality, iris), values taken from the Rdatasets CSV mirror https://vincentarelbundock.github.io/Rdatasets/datasets.html
• R_AD_MTCARS: R-bloggers, Anderson-Darling Test in R (Quick Normality Check): printed nortest::ad.test(mtcars$mpg) output https://www.r-bloggers.com/2021/11/anderson-darling-test-in-r-quick-normality-check/
• R_AD_IRIS: Statology, How to Conduct an Anderson-Darling Test in R: printed nortest::ad.test(iris$Petal.Width) output https://www.statology.org/anderson-darling-test-r/
• METRICGATE_AD: MetricGate Anderson-Darling documentation: iris Sepal.Length computed in R with nortest::ad.test (A2, modified A2*, p) https://metricgate.com/docs/fr/anderson-darling-test/
• NORTEST_SRC: R package nortest, ad.test source code (modified statistic and piecewise p-value constants), copy in package ufs https://rdrr.io/cran/ufs/src/R/ad.test_from_nortest.R
• R_SW_HOWTO: r-statistics.co, Shapiro-Wilk Test in R: printed shapiro.test output for mtcars$mpg (overall and by cyl) and iris setosa https://r-statistics.co/How-to-do-Shapiro-Wilk-Test-in-R.html
• R_SW_NORMVAR: r-statistics.co, Test Normality and Equal Variance in R: printed shapiro.test output for mtcars$wt, airquality$Temp, iris species https://r-statistics.co/Normality-and-Variance-Tests-in-R.html
• R_SW_TOOTH: R-snippets documentation, Statistical Tests: printed shapiro.test(ToothGrowth$len) output https://r-snippets.readthedocs.io/en/latest/stats/tests.html
• NIST_GRUBBS: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.17.1, Grubbs Test for Outliers (uranium isotope example, N = 8) https://www.itl.nist.gov/div898/handbook/eda/section3/eda35h1.htm
• LIBRETEXTS_GRUBBS: Harvey, Analytical Chemistry 2.1 (Chemistry LibreTexts), Appendix 16.7, Critical Values for Grubbs Test. The page does not say whether the values are one- or two-sided https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/16%3A_Appendix/16.07%3A_Critical_Values_for_Grubb%27s_Test
• NIST_CHISQ_SD: NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.8, Chi-Square Test for the Variance (chi-square critical values for 99 df) https://www.itl.nist.gov/div898/handbook/eda/section3/eda358.htm

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