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Statistical Tolerance Analysis

An accumulation model — instantiates Tolerance Stack Management

Models each contributor as a distribution with a known process capability and propagates those distributions analytically, predicting the assembly's yield and how sensitively it responds to each contributor's spread and centering.

Root-sum-square gives a single combined tolerance; Statistical Tolerance Analysis gives a distribution and a yield. It models each contributor not as a symmetric ± band but as a real statistical process — with a spread and a centering captured in a capability index — and propagates those distributions through the stack analytically, using sensitivity coefficients that weight each contributor by how strongly the output responds to it. Its defining outputs are the predicted defect rate of the assembly (what fraction of builds will fall outside the fit limit) and a sensitivity ranking of contributors, including the effect of a process running off-center. Where RSS answers "how wide is the combined band?", this answers "what yield will we get, and which contributor's capability most controls it?" — connecting the tolerance stack to process capability and cost of quality.

Example

A resistor divider must hold an output voltage inside a window for a sensor to read correctly — roughly ±2% of nominal. Two resistors set the ratio, each specified at ±1% but actually produced by processes with known capability: one supplier runs a tight, well-centered process (a high capability index), the other a wider process that also sits slightly high of nominal. Statistical Tolerance Analysis models each resistor as its real distribution, applies the sensitivity coefficient of the voltage to each (the divider makes the output more sensitive to one resistor than the other), and propagates them to a predicted output distribution. It reports an expected out-of-window rate of roughly 300 parts per million and — decisively — that ~80% of that risk traces to the second supplier's off-center, wide process, not to the ±1% spec both share. The fix is not tightening both resistors but re-centering one process: a targeted move the analysis identifies and a worst-case sum never could.

How it works

Each contributor is characterized by a distribution — spread and mean offset, usually summarized in a process-capability index — rather than a bare tolerance. These are propagated through the stack using each contributor's sensitivity coefficient (how much the output moves per unit of that input), which lets the method weight contributors unequally and account for centering, not just width. The result is a predicted output distribution, from which a defect rate against the fit limit is read directly. What distinguishes it from RSS is that it carries capability and centering through the math and yields a yield and a ranking rather than a lumped combined tolerance — and unlike Monte Carlo, it does so analytically rather than by sampling.

Tuning parameters

  • Capability model — whether contributors are described by short-term spread only or by a longer-term capability index that also captures drift and centering. The latter predicts field yield; the former flatters it.
  • Sensitivity coefficients — how the output's response to each contributor is derived (analytic derivative versus linearized estimate). Wrong coefficients mis-rank the contributors and misdirect the fix.
  • Centering assumption — whether processes are taken as centered or allowed a modelled mean offset. Assuming centering is the most common source of optimism.
  • Defect-rate tail — how far into the distribution tails the yield prediction is trusted, which depends on how well the assumed distribution shape holds far from the mean.
  • Linearity scope — whether the analytic propagation is valid, or the relationship is non-linear enough that the method must hand off to a simulation.

When it helps, and when it misleads

Its strength is turning a stack into a business-legible prediction — a yield and a ranked list of which contributors' capability drives it — so tolerance decisions connect to defect rate and cost rather than abstract bands. Framing contributors through process capability (Cpk) is what lets it separate a spread problem from a centering problem and target the real driver.[1]

It misleads when the assumed distributions and capability numbers are wishful: a tail-defect prediction is exquisitely sensitive to distribution shape far from the mean, where data is thinnest, so a confident parts-per-million figure can rest on almost nothing. It shares RSS's independence blind spot — correlated contributors break the analytic propagation — and its linear sensitivity coefficients quietly fail on non-linear relationships. The discipline is to ground capability numbers in real process data, state the independence and linearity assumptions as conditions to be checked, and escalate to Monte Carlo Stack Simulation when the relationship is non-linear or the inputs are non-normal or correlated.

How it implements the components

  • accumulation_model — it combines contributors as propagated distributions weighted by sensitivity, yielding a predicted assembly distribution and defect rate.
  • sensitivity_analysis — its sensitivity coefficients quantify how strongly the integrated result responds to each contributor's spread and centering, ranking where capability matters most.

It does not draw the contributor loop or check the built article — the map is Dimensional Chain Diagram, the empirical gate is Integration Acceptance Test — and it hands non-linear or correlated cases to Monte Carlo Stack Simulation.

Notes

Root-sum-square is really the special case of this method with every contributor assumed equally weighted, centered, and normal; Statistical Tolerance Analysis earns its keep exactly where those assumptions loosen — unequal sensitivities, off-center processes, differing capability. When the relationship itself goes non-linear, though, the analytic propagation breaks and the honest move is to switch to sampling rather than push the closed form past where it holds.

References

[1] The process-capability index Cpk measures how well a process fits inside a tolerance while accounting for how far its mean sits from center — so it distinguishes a process that is wide from one that is off-center, two problems with different fixes that a symmetric ± tolerance blurs together. ↩