Tolerance Stack-Up Calculator: Worst Case, RSS & Monte Carlo

Let $d_i$ be each signed nominal contribution and $t_i\ge0$ its scaled bilateral half-width. These equations are derived for a linear sum and implement the arithmetic and RSS methods in Scholz (1995), printed pp. 12–15 (attached pp. 20–23).

The classic geometric RSS half-width is $T_{RSS}=\sqrt{\sum_i t_i^2}$. For the selected distributions, use $\sigma_i=t_i/k$ for Normal, $t_i/\sqrt3$ for Uniform, and $t_i/\sqrt6$ for symmetric Triangular variation (NIST §2.5.4.1). Independence gives

Thus $T_{RSS}=3\sigma_Y$ when all varying contributors are Normal with $k=3$. A ±3σ interval has 99.73% coverage for an all-Normal chain; this coverage is not assigned to bounded or mixed distributions. Normal-model yield is the difference of Normal CDFs at the requirement limits, using $Y_0$ and $\sigma_Y$. Zero spread is evaluated as a point mass.

The worst-case margin is the smaller of the lower and upper clearance to the functional limits. A nonnegative margin means all in-tolerance combinations meet the requirements.

Monte Carlo uses the Park–Miller recurrence (multiplier 16807, modulus 2147483647) with exact modular block products and the chosen seed. Box–Muller transforms two uniforms to a Normal variate; $2U-1$ gives Uniform variation and $U+V-1$ gives symmetric Triangular variation. Add the independent sampled contributions to $Y_0$. Sample standard deviation uses the $N-1$ denominator. The central interval uses the 2.5th and 97.5th percentiles from a sorted copy of the samples. For probability $p$, R7 interpolation uses rank $r=1+p(N-1)$ and linearly interpolates between the observations at $\lfloor r\rfloor$ and $\lceil r\rceil$ (NIST §7.2.6.2). The interpolation is explicit so browser and server runtimes follow the same algorithm. Histogram bins report percentages of all assemblies, including their out-of-spec subset.

The 95% Wilson score interval quantifies sampling uncertainty in the simulated yield (NIST §7.2.4.1). Its width remains nonzero when no rejects are observed. It concerns the simulation model, not confidence in the manufacturing assumptions.

For unit verification, simulation runs on dimensionless deviations normalized by the worst-case half-width (a 1 mm reference is used only when all tolerances are zero). Named steps restore mean, standard deviation, percentiles and sample extremes to physical units. The function returns compact statistics and histogram counts; full sample vectors remain local.

• One linear, one-dimensional chain of one to eight contributors; coefficients +1 and −1 are encoded in the signed nominals. Symmetric tolerances affect spread equally for either sign.
• Each contributor has a symmetric distribution centered on its signed nominal. The shared Normal factor maps tolerance to standard deviation; Uniform and Triangular endpoints are the entered limits.
• Contributors vary independently. Parts from the same drifting process, or a dimension reused twice in a loop, can violate independence.
• Normal distributions are untruncated. Their samples can exceed drawing limits and fall outside the worst-case interval. Worst-case bounds apply only when every actual dimension is within tolerance; they are not absolute support limits of a Normal model.
• Functional limits are inclusive, and the upper limit must exceed the lower limit. A negative result represents overlap or interference; it is acceptable only if the requirements allow it.
• Lengths remain in mm internally. The viewer converts physical inputs, cards, chart axes and table columns to metric or US units. The schematic explicitly retains mm labels; its vertical row spacing is illustrative.

Use for axial lengths, gaps and clearances that are algebraic sums. This model does not include angular errors, 2D/3D geometry, GD&T, contact, compliance, thermal expansion or deformation. It does not model mean shifts, covariance, truncation, sorting or selective assembly. For asymmetric bounds, enter their midpoint as the signed nominal and half their width as the symmetric tolerance; this changes the assumed center and should be justified.

Choose distributions and the Normal tolerance/σ factor from process evidence. RSS variance propagation is exact for independent contributors. Normal-model yield is exact for an all-Normal chain (or zero spread), and an approximation otherwise. Use Monte Carlo for the selected bounded or mixed distributions.

The central 95% Monte Carlo interval differs from a ±3σ interval and from observed sample extremes. Counts of 1,000–50,000 trials produce estimates, not reliable parts-per-million certification. Wilson limits cover sampling uncertainty only, not model or process uncertainty. Change the seed and increase trials when a decision is close.

NASA RP-1253's three-dimension example (printed p.69, attached p.74) reports 0.172 mil RSS after rounding each intermediate σ to 0.033 mil. Full-precision propagation gives 0.1732050808 mil. The reference check discloses and allows that rounding difference; the calculator retains full precision.