Vose SoftwareModelRisk example · Manufacturing
Tolerance Stack-Up and Process Capability
Will the assembly's gap stay within specification? Worst case, root-sum-square and simulation compared.
From the model’s saved run
Will the gap stay in specification?
The gap's range (mm), capability and defects per million, by method and design.
| Method | Low mm | High mm | Cp | Cpk | Out of spec ppm |
|---|---|---|---|---|---|
| Worst case (tolerances add up) | 0.000 | 0.600 | – | – | – |
| Root-sum-square (all Normal, centred) | 0.158 | 0.442 | 1.06 | 1.06 | 1,545 |
| Simulation: as made today | 0.125 | 0.440 | 0.90 | 0.79 | 8,799 |
| Simulation: capable bearing supply | 0.134 | 0.421 | 1.06 | 0.92 | 3,700 |
| Simulation: capable bearing, centred spacer | 0.156 | 0.441 | 1.06 | 1.06 | 1,400 |
Low and high: for the methods, worst case and ±3 standard deviations; for the simulation, the 0.135th and 99.865th percentiles (the same ±3σ coverage). Specification: 0.15 to 0.45 mm.
S-curve (cumulative probability) of As made today, Capable bearing, … and centred spacer, Lower specification, Upper specification from the saved simulation.
Histogram of Gap as made today from the saved simulation.
The business question
A gearbox's shaft assembly leaves an axial gap: the housing length minus the shaft shoulder, the bearing, the gear hub and a spacer. Too small a gap and the bearing is preloaded. Too large and the gear rattles — the drawing requires 0.15 to 0.45 mm. Each part is made within its tolerance, but tolerances add up.
A worst-case check says the gap could be anything from zero to 0.6 mm; the textbook root-sum-square check says it is fine. Neither knows that the bought-in bearings are sorted to tolerance — any width in the band is as likely as any other — or that the spacer grinder runs a little long.
The model simulates the stack-up as it really is made, measures the assembly's capability, and tests two improvements.
- Who it is for
- Design and quality engineers, Six Sigma practitioners, supplier quality teams.
- Main outputs
- The gap's range by worst case, root-sum-square and Monte Carlo · its capability (Cp, Cpk) and defects per million for the current design and two improvements · each part's share of the gap's variation.
How the model works
Three sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
The nominal dimension and ± tolerance of the housing, shaft shoulder, bearing, gear hub and spacer. How far the spacer process runs off centre; the lower and upper specification of the gap.
Stack-up sheet
Each part is drawn from its real distribution — Normal with the tolerance at three standard deviations for parts made in-house, Uniform across the tolerance for the sorted bearings — and the gap is the housing minus the four parts, for the design as made today and for two improvements. Worst-case and root-sum-square ranges are calculated alongside.
Results sheet
The gap's range by each method; Cp, Cpk and defects per million from the simulation (VoseSixSigma functions). Each part's share of the variation of the gap.
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The results in detail
Which tolerance to tighten?
Each part's share of the variance of the gap, as made today.
| Part | Std dev mm | Share of variance |
|---|---|---|
| Housing | 0.0333 | 36% |
| Shaft shoulder | 0.0167 | 9% |
| Bearing width | 0.0346 | 39% |
| Gear hub | 0.0167 | 9% |
| Spacer | 0.0133 | 6% |
How to read these results
- The worst case adds all the tolerances: the gap could be anywhere from 0 to 0.6 mm, so the design fails. It assumes every part sits at its worst limit at once — almost never true — and cannot say how often the gap falls outside.
- Root-sum-square treats every part as Normal and centred: it predicts a Cpk of 1.06 and about 1,500 out-of-specification assemblies per million. Simulating the parts as they are really made gives a Cpk of 0.79 and nearly six times as many defects: the sorted bearings, flat across their tolerance, and the spacer running long make the difference.
- The bearing and the housing bring three-quarters of the variation (39% and 36%). A bearing supply with a Normal spread inside the same tolerance cuts the defects to about 3,700 per million. Re-centring the spacer as well brings the design to the root-sum-square figure. To reach the usual Cpk of 1.33, tighten the housing and the bearing next.
- Capability from 10,000 simulated assemblies resolves about 100 parts per million. For designs with fewer defects, run more samples.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Parts 10 inputs
The housing adds to the gap; the four parts inside take from it.
| Assumption | Unit | Value |
|---|---|---|
| Housing: nominalMachined in-house. | mm | 120.00 |
| Housing: tolerance | ± mm | 0.100 |
| Shaft shoulder: nominalTurned in-house. | mm | 40.00 |
| Shaft shoulder: tolerance | ± mm | 0.050 |
| Bearing width: nominalBought in: the supplier sorts to tolerance, so every width in it is equally likely. | mm | 25.00 |
| Bearing width: tolerance | ± mm | 0.060 |
| Gear hub: nominalMachined in-house. | mm | 30.00 |
| Gear hub: tolerance | ± mm | 0.050 |
| Spacer: nominalGround in-house; the process runs slightly long (see the shift). | mm | 24.70 |
| Spacer: tolerance | ± mm | 0.040 |
Process and specification 4 inputs
| Assumption | Unit | Value |
|---|---|---|
| Spacer process runs long byThe grinder's average is above nominal. | mm | 0.020 |
| A tolerance spansFor a Normal process made to its tolerance: ± tolerance = ± this many standard deviations (3: Cp = 1). | standard deviations | 3.0 |
| Gap: lower specificationLess and the bearing is preloaded. | mm | 0.15 |
| Gap: upper specificationMore and the gear rattles. | mm | 0.45 |
Units and conventions
- Units
- Millimetres (mm); defects in parts per million (ppm).
- Gap
- Housing − (shaft shoulder + bearing + gear hub + spacer).
- Cp
- Specification width ÷ 6 standard deviations: what the process could do if centred.
- Cpk
- Distance from the mean to the nearer limit ÷ 3 standard deviations: what it does. 1.33 is a common minimum.
- Root-sum-square
- The gap's standard deviation as √(Σ of the parts' variances), each part's tolerance taken as 3 standard deviations.
Key assumptions and limitations
- Parts
- In-house parts are Normal with the tolerance at ±3 standard deviations; sorted bearings are Uniform across the tolerance.
- Assembly
- Dimensions are independent and add exactly; no deformation or temperature effect.
- Simulation
- 10,000 assemblies resolve about 100 ppm: use more samples for very capable designs.
- Data
- The gearbox and all figures are invented for this example.
How to use it
- Enter each part's nominal and tolerance, and the gap's specification.
- Set how far any process runs off centre.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Tighten first the parts with the largest share of the variation.
Simulate this model with ModelRisk’s classic engine: it does not run in Turbo mode yet.
ModelRisk functions used
- VoseNormal, VoseUniform
- Each part's dimension.
- VoseSixSigmaCp, VoseSixSigmaCpk
- Process capability of the simulated gap.
- VoseSixSigmaDefectPPM
- The share of simulated gaps outside the specification, per million.
- VoseSimPercentile
- The gap's natural range: its 0.135th and 99.865th percentiles.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Fischer, B. R. (2011). Mechanical Tolerance Stackup and Analysis, 2nd edition. CRC Press.
- Montgomery, D. C. (2019). Introduction to Statistical Quality Control, 8th edition. Wiley.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.