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Scholz (1995) methods on the tool's five-part shaft-and-housing chain: worst case, RSS, Bender and Cpk

Tool	Tolerance Stack-Up
Source	Scholz, F. W. (1995). Tolerance Stack Analysis Methods: A Critical Review. Boeing Information and Support Services ISSTECH-95-030, The Boeing Company, Seattle, WA. Sections 3–5, Table 1.
Level	Professional

Problem

A housing of 3.000 ± 0.010 in encloses bearing A 0.500 ± 0.004, a spacer 1.250 ± 0.005, a gear hub 0.750 ± 0.004 and bearing B 0.480 ± 0.004 in. The end gap must lie between 0.002 and 0.040 in. Analyze the gap by worst case, root-sum-square, Bender's modified RSS and the mean-shift model, and estimate Cp, Cpk and the fraction of assemblies out of limits.

Inputs to Enter

Five dimensions as loaded by default (housing direction +, the four parts −), all normal with k = 3
Gap limits 0.002 and 0.040 in; Bender factor 1.5; mean-shift fraction η 0.2 when the Mean Shift box is on
Monte Carlo 20,000 trials, seed 2026

Open the Tolerance Stack-Up tool (loaded with this example)

Answers

Nominal gap 0.020 in; worst case ±0.027 → −0.007 to 0.047 (fails)
RSS: √ΣT² = 0.013153, σ_G = 0.0043843, band 0.006847 to 0.033153 (passes); RSS/worst case 0.4871; housing carries 37.0 % of the worst case and 57.8 % of the variance
Bender 1.5 √ΣT² = 0.019729 → 0.000271 to 0.039729 (fails the 0.002 limit)
Mean shift η = 0.2 (Scholz eq. 7 and 11): T* = 0.0054 + 0.8 × 0.013153 = 0.015922, band 0.004078 to 0.035922 (passes)
Cp = 1.4446, Cpk = 1.3685, 22.7 ppm out of limits; numpy Monte Carlo of 2 × 10⁶ assemblies: sd 0.004381, 15.5 ppm

Notes

This is the tool's own illustrative chain (its default inputs), not a printed worked example: Scholz's report supplies the methods and the Table 1 factors (uniform √3, triangular √1.5, Bender 1.5), and the numbers above are the test script's closed-form values confirmed by an independent numpy Monte Carlo. No printed stack-up example could be obtained (Fischer and Drake are not open).