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Manufacturing & Production Engineering DISCUSSION
Five machined spacers are stacked on a shaft inside a housing, and the design needs an end gap between 0.05 and 0.75 mm. The nominal gap is 0.40 mm. Each of the five spacers and the housing length has a tolerance of ±0.10 mm.
A worst-case stack says the gap can go negative, so the parts could jam, but a colleague says a root-sum-square analysis shows it is fine. Which one do I believe, and what would I have to control in production for the statistical answer to be valid?
Both calculations are correct; they answer different questions. Worst case adds the tolerances directly: six contributors at ±0.10 mm give ±0.60 mm, so the gap could be anywhere from -0.20 to +1.00 mm. It guarantees assembly only if that whole range is acceptable, which here it is not.
RSS adds them as independent random variables: √(6 × 0.10²) = ±0.245 mm, giving a gap of 0.155 to 0.645 mm, inside your 0.05 to 0.75 mm limits. The worst case needs all six parts at the wrong extreme together, which becomes very unlikely as the number of parts grows.
RSS is only as good as its assumptions: each dimension is independent, roughly normally distributed, centred on nominal, and the ±0.10 mm tolerance corresponds to about ±3σ (Cp and Cpk near 1). If those hold, ±0.245 mm is the ±3σ range of the gap and about 99.73 percent of assemblies fall inside it.
Here σ of the gap is 0.245 / 3 = 0.082 mm. Each gap limit is 0.35 mm from nominal, which is 4.3σ, so the predicted reject rate is on the order of 10 parts per million at each limit. The statistical answer is comfortable, provided production really behaves that way.
Where it goes wrong in practice is mean shift. Operators turning an outside dimension tend to aim near the upper limit so an oversize part can be reworked, and tools wear in one direction. If five spacers from the same machine all run 0.06 mm thick, the shifts add directly, 0.30 mm in total, and RSS does not see that at all.
Two protections: require capability data (Cpk at or above 1.33 with the mean near nominal) on the contributing dimensions, or use a modified RSS with a correction factor, commonly 1.5, which here gives ±0.37 mm and a gap of 0.03 to 0.77 mm. That just fails, which tells you the design is relying on well-centred processes.
If you want more margin, change the design rather than tightening everything. In RSS the largest contributors dominate because the terms are squared, so look at where the variation really comes from. Replacing five separate spacers with one sleeve cuts the stack from six contributors to two: √(2 × 0.10²) = ±0.14 mm even at the same tolerance, and worst case becomes ±0.20 mm, which passes outright.
Other options are a selective shim or an adjustable end nut that absorbs the variation at assembly. For a safety-critical fit or a low-volume build, stay with worst case, because statistics need numbers behind them.
