Tolerance Sensitivity Analysis: Finding the Dimensions That Matter
Not all tolerances contribute equally to your stack-up. Sensitivity analysis shows which dimensions drive your variation—so you know where to focus improvement efforts.
Key Insight
In RSS analysis, sensitivity follows the square of the tolerance. A dimension with ±0.2mm tolerance contributes 4× more variation than one with ±0.1mm—not 2×.
What Is Sensitivity?
Sensitivity (also called contribution percentage) measures how much each dimension contributes to the total RSS variation. It answers: "If I could tighten only one tolerance, which would help most?"
The Formula
For RSS analysis, sensitivity of dimension i is:
Where:
- t_i = tolerance of dimension i
- Σt_j² = sum of squares of all tolerances
Example Calculation
Consider a 4-dimension stack-up:
| Dimension | Tolerance | t² | Sensitivity |
|---|---|---|---|
| Housing bore | ±0.10 | 0.010 | 10% |
| Bearing OD | ±0.05 | 0.0025 | 2.5% |
| Shaft diameter | ±0.15 | 0.0225 | 22.5% |
| End plate thickness | ±0.25 | 0.0625 | 62.5% |
| Total | 0.0975 | 100% |
The end plate contributes 62.5% of the total variation—more than all other dimensions combined. This is your optimization target.
How to Use Sensitivity Analysis
1. Identify Critical Dimensions
Dimensions with >20% sensitivity are your "critical few." These drive most of your variation.
2. Focus Improvement Efforts
Tightening a 60% contributor has 12× more impact than tightening a 5% contributor.
3. Justify Tolerance Decisions
Show reviewers why you're spending money on one tight tolerance vs. another.
Common Sensitivity Patterns
The Dominant Dimension
One dimension contributes >50%. Common in assemblies with one large part and several small ones. Focus almost exclusively on the dominant dimension.
The Balanced Stack
All dimensions contribute 15–25%. Rare in practice, but means any tolerance improvement helps roughly equally. Look for the cheapest tolerance to tighten.
The Long Tail
Two dimensions contribute 70–80%, others contribute 2–5% each. Common in multi-part assemblies. Ignore the tail—focus on the top contributors.
Sensitivity vs. Worst-Case
Sensitivity is most meaningful for RSS analysis because RSS uses the sum of squares. In Worst-Case analysis, all tolerances add linearly, so a ±0.2 tolerance always contributes exactly 2× a ±0.1 tolerance.
However, even for WC-driven designs, sensitivity still shows where tightening tolerances will have the most impact on the RSS result—useful for cost-benefit analysis.
Practical Tips
Don't Chase Small Contributors
Tightening a 3% contributor from ±0.1 to ±0.05 costs machining time/money but barely moves your total. The math doesn't justify the effort.
Watch for Sensitivity Shifts
If you tighten the dominant dimension, the second-largest becomes the new dominant. Re-run sensitivity after major tolerance changes.
Consider Cost per Percent
A 40% contributor that's cheap to tighten (stock tolerance vs. precision) beats a 50% contributor that requires grinding.
Document in Design Reviews
Sensitivity charts justify your tolerance decisions. "We're holding ±0.05 on the shaft because it contributes 45% of variation" is defensible. "We always use ±0.05 on shafts" is not.
Summary
- Sensitivity = t² / Σt² × 100% — contribution to RSS variation
- Focus on >20% contributors — the "critical few"
- Square relationship — 2× tolerance = 4× sensitivity
- Re-run after changes — sensitivity shifts when you tighten tolerances
- Use in design reviews — quantified justification for tolerance decisions
Open TolReport → — sensitivity analysis is included in both free and Pro tiers.