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Tolerance Stack-Up Calculator: Worst Case and RSS
Chain dimensions around a loop, mark each one as adding to or subtracting from the gap, and get the worst-case and root-sum-square (statistical) limits, with each dimension's share of the variation.
Tolerance stack-up calculator
Add every dimension in the loop. Mark it Adds if it makes the gap bigger, Subtracts if it makes the gap smaller. Enter the minus tolerance as a positive number.
| Dimension | Mean | ± Tolerance | RSS share |
|---|---|---|---|
| + Housing bore depth | 50.0000 | 0.1000 | 76.2% |
| − Bearing width | 20.0000 | 0.0500 | 19.0% |
| − Spacer | 29.8250 | 0.0250 | 4.8% |
| Step | Formula | Value |
|---|---|---|
| Mean and ± tolerance of each dimension | mᵢ = nominalᵢ + (plusᵢ − minusᵢ) / 2; tᵢ = (plusᵢ + minusᵢ) / 2 | Housing bore depth: 50.0000 ± 0.1000; Bearing width: 20.0000 ± 0.0500; Spacer: 29.8250 ± 0.0250 |
| Nominal gap | Σ ±nominalᵢ = 50.0000 − 20.0000 − 29.8000 | 0.2000 |
| Mean gap | Σ ±mᵢ = 50.0000 − 20.0000 − 29.8250 | 0.1750 |
| Worst-case tolerance | T_wc = Σ tᵢ = 0.1000 + 0.0500 + 0.0250 | 0.1750 |
| RSS tolerance | T_rss = √(Σ tᵢ²) = √(0.1000² + 0.0500² + 0.0250²) | 0.1146 |
| RSS as a fraction of worst case | T_rss / T_wc | 0.6547 |
Formulas used
- Mean of each dimension
- mᵢ = nominalᵢ + (plusᵢ − minusᵢ) / 2
- Equal-bilateral tolerance
- tᵢ = (plusᵢ + minusᵢ) / 2
- Mean of the gap
- M = Σ sᵢ·mᵢ (sᵢ = +1 adds, −1 subtracts)
- Worst case
- T_wc = Σ tᵢ; gap = M ± T_wc
- Root sum square
- T_rss = √(Σ tᵢ²); gap = M ± T_rss
- Contribution
- share_i = tᵢ² / Σ tᵢ²
Assumptions and limits
- The stack is one-dimensional: all dimensions are parallel to the direction of the gap. Angular and geometric (GD&T) effects need a 2D or 3D analysis.
- Unequal tolerances such as +0.05 / −0 are converted to a mean value with an equal ± tolerance, so the result is not biased.
- RSS assumes the dimensions are independent, roughly normal, centred on their mean, and that each tolerance spans ±3σ. Then about 99.73% of assemblies fall within the RSS limits.
- Worst case assumes every part can be at its extreme at once. It guarantees 100% interchangeability but is conservative.
Worked example
A bearing and a spacer sit in a housing bore 50 ±0.10 mm deep. The bearing is 20 ±0.05 mm wide and the spacer is 29.8 +0.05 / −0 mm. What axial gap is left?
| Step | Formula | Value |
|---|---|---|
| Mean and ± tolerance of each dimension | mᵢ = nominalᵢ + (plusᵢ − minusᵢ) / 2; tᵢ = (plusᵢ + minusᵢ) / 2 | Housing bore depth: 50.0000 ± 0.1000; Bearing width: 20.0000 ± 0.0500; Spacer: 29.8250 ± 0.0250 |
| Nominal gap | Σ ±nominalᵢ = 50.0000 − 20.0000 − 29.8000 | 0.2000 |
| Mean gap | Σ ±mᵢ = 50.0000 − 20.0000 − 29.8250 | 0.1750 |
| Worst-case tolerance | T_wc = Σ tᵢ = 0.1000 + 0.0500 + 0.0250 | 0.1750 |
| RSS tolerance | T_rss = √(Σ tᵢ²) = √(0.1000² + 0.0500² + 0.0250²) | 0.1146 |
| RSS as a fraction of worst case | T_rss / T_wc | 0.6547 |
Result: Worst-case gap 0.0000 to 0.3500 mm; RSS gap 0.0604 to 0.2896 mm. Worst case can reach zero, so the design should be checked if zero clearance is not acceptable.
How to set up a tolerance stack-up
1. Define the gap and draw the loop
Start at one side of the gap you care about and walk through the parts to the other side. Every dimension you cross becomes a row. Dimensions that make the gap bigger are added, and dimensions that make it smaller are subtracted.
A clear loop diagram prevents the most common error: getting a direction wrong.
2. Enter dimensions as drawn
Enter the nominal and the plus and minus tolerances exactly as they appear on the drawing, with the minus tolerance as a positive number. The calculator converts unequal tolerances to a mean ± tolerance before stacking.
3. Read worst case and RSS together
If the worst-case range meets the requirement, the design works whatever the parts do. If only the RSS range meets it, the design relies on the process being centred and in statistical control. That is a reasonable bet for capable, high-volume processes, and a risky one for low volume.
The contribution column shows where tightening one tolerance helps most. Under RSS, a tolerance's influence grows with its square.
4. Decide and document
Record the loop, the inputs, and the method used in the design file. The CSV export carries every step so a reviewer can follow the arithmetic.
Common mistakes
- Wrong direction on one dimension. One sign error can move the result by twice that dimension's nominal. Check the loop before trusting the numbers.
- Averaging unequal tolerances away. Treating +0.05 / −0 as ±0.025 without shifting the nominal biases the answer. Enter the tolerances as drawn.
- Using RSS for a few parts or uncontrolled processes. RSS relies on statistical averaging. With two or three contributors, or processes that run off-centre, it can be optimistic.
- Mixing units. All rows share the selected unit. Convert inch callouts to millimetres, or the reverse, before entering them.
- Leaving out geometric tolerances. Position, flatness, or runout can add to the stack. Include them as ± contributions where they act along the loop.
Frequently asked questions
What is the difference between worst-case and RSS tolerance analysis?
Worst case adds every tolerance, assuming all parts are at their limits at once. RSS adds tolerances as the square root of the sum of squares, assuming independent random variation. RSS predicts a narrower range that most assemblies will fall within.
When should I use RSS instead of worst case?
When there are several contributors, the processes are capable and centred, and an occasional out-of-range assembly can be tolerated or screened. Safety-critical fits and low-volume builds usually use worst case.
How do I enter a tolerance like +0.10 / −0.00?
Enter 0.10 in the plus field and 0 in the minus field. The calculator uses a mean of nominal + 0.05 and a tolerance of ±0.05 for that dimension.
What does the contribution percentage mean?
It is each dimension's share of the total RSS variance, tᵢ² / Σ tᵢ². A dimension with 60% contribution is the first place to look if the stack needs tightening.
Can I do an inch tolerance stack-up?
Yes. Switch the unit to inches and enter all dimensions in inches. Results are shown in the same unit.
Sources
- E. T. Fortini, Dimensioning for Interchangeable Manufacture, Industrial Press, 1967 — worst-case and statistical tolerance accumulation.
- K. W. Chase and W. H. Greenwood, "Design Issues in Mechanical Tolerance Analysis", Manufacturing Review, 1988 — worst-case and root-sum-square models and their assumptions.
Results are computed in your browser from the published formulas named above. They are an engineering aid, not a substitute for the governing standard, your drawing, or your customer's requirements.