Dimensions in the loop
Walk the loop from one side of the gap to the other. Dimensions pointing in the positive direction are +, those pointing back are −.
| Description | Dir. | Nominal | + Tol | − Tol (a positive entry is read as minus) |
|---|
Worst case
-RSS (statistical)
-Monte Carlo (200,000 runs)
-Loop diagram
Monte Carlo distribution of the gap
What drives the result
Share of RSS variance
Share of worst-case range
Worst case vs RSS: which one to use
Worst case adds every tolerance at its limit. If the worst-case gap meets the requirement, every assembly built from in-tolerance parts will fit, with no statistics needed. The price is tight, expensive tolerances once a stack has more than a few parts.
RSS (root sum square) assumes each dimension varies randomly and independently around its mid-tolerance value, so the extremes rarely line up. The stack variation is √(Σti²) instead of Σti. With tolerances set at ±3σ it predicts about 99.73% of assemblies inside the RSS range. It relies on centred, roughly normal, independent processes: shifted means, tool wear or a single setup making all parts the same will break it.
Monte Carlo builds hundreds of thousands of virtual assemblies from the distributions you choose and counts how many fall outside the spec. It shows the same picture as RSS for normal inputs, and lets you test a pessimistic case with uniform distributions (every value in the tolerance band equally likely).
| Method | Stack range | Use when |
|---|---|---|
| Worst case | Σ ti | Safety-critical fits, few parts, low volume, unknown suppliers |
| RSS | √(Σ ti²) | Many parts, capable and centred processes, a small reject rate is acceptable |
| Monte Carlo | Simulated | Non-normal or skewed processes, checking a reject rate against a spec |
How the calculator handles unequal tolerances
A dimension like 0.500 +0.002/−0.000 is converted to its mean (0.501) and a symmetric half-band (±0.001) before the statistics. Worst case uses the real limits. That is why the RSS and Monte Carlo centre can differ from the nominal gap: the nominal is not where the parts actually land.
Reading the results
- Cpk = min(USL − μ, μ − LSL) / 3σ of the simulated gap. 1.33 or more is a common target in industry; the right value depends on your customer and the consequence of a reject.
- Predicted reject rate is the share of simulated assemblies outside your required gap, in parts per million. Below about 20 ppm (a handful of rejects in 200,000 runs) the simulated rate is not reliable; use the Cpk instead.
- Contribution shows where to spend money: tightening a dimension with a large share of the variance improves the stack the most.
Questions
What is a tolerance stack-up?
A tolerance stack-up adds the dimensions and tolerances along a chain of parts to find the range of a resulting gap or dimension, for example the end play of a shaft between a bearing and a retaining ring.
What is the difference between worst case and RSS tolerance analysis?
Worst case adds all tolerances at their limits, so the stack range is the sum of the tolerances. RSS treats each dimension as an independent random variable, so the stack range is the square root of the sum of the squared tolerances, which is smaller and predicts the range most assemblies fall in rather than the absolute extremes.
How do I choose the sign of each dimension?
Pick a direction across the gap and walk the loop from one face of the gap to the other through the parts. A dimension that points the same way as your direction is positive; one that points back is negative. The nominal gap is the sum of the signed nominals.
Can I use this for GD&T?
Yes for simple cases: convert a position or profile tolerance into an equivalent ± value along the stack direction and add it as a row. Bonus tolerance and datum shift need a more detailed analysis.
Engineering estimate for guidance. Statistical results assume independent dimensions with the distributions you choose; you remain responsible for the design. Nothing you enter leaves your browser except through a share link you copy yourself.