Stack a chain of dimensions and see whether the assembly fits — the guaranteed worst-case range and the statistical RSS range side by side, checked against your spec limits, with the gap distribution drawn so you can see the margin and choose between a safe design and a cheaper one.
| # | Feature | Dir | Nominal | ± Tol | or asym | +Tol | −Tol |
|---|
For a clearance that must never interfere, set LSL = 0 (or a minimum required gap).
RSS assumes each dimension is normally distributed and centered. A factor >1 (e.g. 1.5) adds conservatism for non-ideal processes.
The nominal gap sits at center. The wide blue span is the worst-case range (every part at its worst at once). The narrow brass bell is the RSS statistical range. Red lines are your spec limits — any part of a range crossing them is out of spec.
The dimension chain. Trace a closed loop from one side of the critical gap to the other, listing every part dimension along the way. Give each a direction: + if increasing it opens the gap, − if increasing it closes the gap. The nominal gap is the signed sum: Σ(dir · nominal).
Worst-case (arithmetic). Assumes every part sits at its worst limit simultaneously. Total tolerance is the plain sum: T_wc = Σ|tol|. The gap ranges from nominal − T_wc to nominal + T_wc. If this whole range fits your limits, 100% of assemblies fit, guaranteed — but it often forces expensive tight tolerances.
RSS (statistical). Recognizes that all parts hitting their worst limit at once is astronomically unlikely. Combined tolerance is the root-sum-square: T_rss = √(Σ tol²). For many contributors this is far tighter than worst-case — roughly half for four equal parts — letting you loosen individual tolerances and cut cost. The trade: it predicts ~99.73% fit at ±3σ, not 100%, and assumes each dimension is normal and centered.
Process capability. RSS at ±3σ corresponds to Cpk 1.0 (barely capable). If your processes hold ±4σ (Cpk 1.33) or better, the real spread is tighter still. The safety factor lets you inflate T_rss for non-ideal, non-centered, or drifting processes (a common choice is 1.5).
How to use both. Start with worst-case. If it passes, you're done — the design is bulletproof. If only RSS passes, the design works statistically but demands capable, monitored processes; fine for high volume, risky for safety-critical or low-volume builds where one bad assembly matters.
Linear (1-D) stacks of independent, normally distributed dimensions. Geometric tolerances (GD&T), angular effects, and correlated variation need vector or Monte-Carlo methods.