What is Tolerance Stack-Up?
When multiple parts assemble together, each dimension has a tolerance. Those tolerances accumulate. Tolerance stack-up analysis tells you whether your assembly will work — or whether you’ll be forcing parts together on the shop floor.
There are two fundamental approaches: Worst Case (everything goes wrong simultaneously) and RSS/Statistical (realistic probability-based analysis).
Worst Case Analysis
Add up all the tolerances. If the total exceeds your gap or clearance requirement, the assembly might not work.
Formula: Total tolerance = Σ |individual tolerances|
Example: Three parts stack up to create a gap:
- Part A: 2.000 ± 0.005
- Part B: 1.500 ± 0.003
- Part C: 0.500 ± 0.002
- Housing: 4.020 ± 0.005
| Calculation | Nominal | Worst Case |
|---|---|---|
| Stack (A + B + C) | 4.000 | 4.000 ± 0.010 |
| Housing | 4.020 | 4.020 ± 0.005 |
| Gap | 0.020 | 0.020 ± 0.015 |
| Gap range | — | 0.005 to 0.035 |
Worst case guarantees 100% of assemblies will work — but it’s conservative. In practice, the chance that every dimension hits its worst limit simultaneously is vanishingly small.
RSS (Root Sum of Squares) — Statistical Method
Instead of adding tolerances, take the square root of the sum of squares. This accounts for the statistical reality that dimensions are normally distributed around their nominal values.
Formula: Total tolerance = √(t₁² + t₂² + t₃² + … + tₙ²)
Same example:
| Calculation | RSS Result |
|---|---|
| Stack tolerance | √(0.005² + 0.003² + 0.002²) = ±0.00616 |
| Total (incl. housing) | √(0.00616² + 0.005²) = ±0.00794 |
| Gap range (3σ) | 0.012 to 0.028 |
RSS predicts that 99.73% (3σ) of assemblies will fall within these limits. The range is tighter because it’s statistically unlikely that everything goes wrong at once.
When to Use Each Method
| Method | Best For | Trade-off |
|---|---|---|
| Worst Case | Safety-critical, low volume, can’t afford rejects | Tighter tolerances required → higher cost |
| RSS (3σ) | High volume production, cost-sensitive | ~0.27% reject rate acceptable |
| RSS (6σ) | Automotive, medical devices, high reliability | ~3.4 ppm reject rate |
| Monte Carlo | Complex assemblies, non-normal distributions | Requires simulation software |
Practical Tips
- Start with RSS, validate with worst case. If worst case passes, you’re golden. If only RSS passes, make sure your process is capable of maintaining normal distributions (Cpk ≥ 1.33).
- Watch out for systematic errors. RSS assumes random variation. Tool wear, thermal expansion, and setup offsets are systematic — they shift the mean, not just the spread. Add these as bias, not tolerance.
- Tolerance stack-up drives cost. A ±0.001″ tolerance costs 5-10× more than ±0.005″. Only put tight tolerances where the stack-up analysis shows you need them.
- Document your analysis. A tolerance stack-up is a living document. When someone changes a tolerance on one part, the stack needs to be re-checked.
Common Assembly Tolerance Capabilities
| Process | Typical Tolerance (in) |
|---|---|
| CNC Milling | ±0.002 to ±0.005 |
| CNC Turning | ±0.001 to ±0.003 |
| Grinding | ±0.0002 to ±0.001 |
| Sheet Metal (bending) | ±0.010 to ±0.030 |
| Die Casting | ±0.005 to ±0.015 |
| Injection Molding | ±0.003 to ±0.010 |
| 3D Printing (FDM) | ±0.010 to ±0.020 |
| 3D Printing (SLA) | ±0.002 to ±0.005 |
Related: GD&T Reference Guide | Surface Finish Guide