Tolerance stackup: worst-case vs RSS

When several components assemble end-to-end, their individual tolerances stack up. The assembly gap or overall length can be anywhere within the combined tolerance range. Two methods quantify the risk: worst-case (conservative, 100% assembly guarantee) and RSS (root sum of squares, statistical, assumes tolerances are independent and normally distributed).

The formulas

RSS is smaller than worst-case. The trade-off: worst-case guarantees 100% good assemblies; RSS guarantees approximately 99.73% good assemblies when each component is centered and normally distributed at ±3σ.

Worked example

Four components are assembled in series. Their nominal lengths and bilateral tolerances are:

• c₁ = 10.0 ± 0.05
• c₂ = 5.0 ± 0.03
• c₃ = 8.0 ± 0.04
• c₄ = 3.0 ± 0.02

Comparison: worst-case reserves ±0.14 (total spread 0.28). RSS needs only ±0.0735—nearly half. If the assembly gap requirement is ≥ 0.10, worst-case says the design fails (26.0 − 14 could reach 25.86, which violates no constraint here, but for a gap problem you would check the minimum gap). The key takeaway: RSS allows tighter (and cheaper) individual tolerances for the same assembly yield.

Practice problem

Three shims are stacked. Their tolerances are ±0.10, ±0.10, ±0.10. What is the RSS tolerance of the stack?

Common exam traps

• RSS applies to bilateral tolerances. If tolerances are unilateral or asymmetric, convert them first.
• RSS is a statistical estimate, not a guarantee. Approximately 0.27% of assemblies will still exceed the RSS limits.
• Worst-case is the right choice for safety-critical applications. When even one bad assembly is unacceptable, use worst-case despite the cost.
• Dynamic tolerance stackup adds time-dependent drift. The CQE exam focuses on static (dimensional) stackup, but know that wear, creep, and thermal expansion can increase the effective tolerance over time.

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