| # | Dimension | Dir. | Nominal | + tol | − tol |
|---|
Every assembly has dimensions that nobody draws directly: the gap between a shaft shoulder and a housing face, the overall length of three parts bolted together, the clearance left for a snap ring. Each of those is the result of several toleranced dimensions added and subtracted along a chain, and its variation is the accumulated variation of everything in the chain. A stack-up is the arithmetic that predicts that result before parts are made, so the drawing tolerances can be set to guarantee the assembly works.[1]
The chain is written as a loop: start at one face of the gap, walk through the parts to the other face, and record every dimension crossed with a sign. Dimensions that walk toward the far side of the gap are positive; those that walk back are negative. The resultant is the signed sum of the nominals, and the two analysis methods differ only in how they add the tolerances.
| Worst-case | Statistical (RSS) | |
|---|---|---|
| Tolerance of the result | Sum of every tolerance: T = Σ ti | Root of the sum of squares: T = √(Σ ti²) |
| Assumes | Every part is at its limit in the worst direction at the same time | Each dimension varies independently and is normally distributed, centred, with the tolerance at ±3σ |
| Result | Guaranteed: 100% of assemblies fit if parts are in tolerance | About 99.73% of assemblies fit (3σ); a few in a thousand can stack outside |
| Use it for | Safety-critical fits, low volumes, hand-fitted assemblies, anything where one interference is unacceptable | High-volume production from capable, independent processes where a tiny reject rate is cheaper than tighter tolerances |
| Effect on drawing tolerances | Tighter, more expensive parts | Looser part tolerances for the same assembly tolerance, by a factor of about √n for n equal contributors |
The RSS gain is real but conditional. It only holds when the dimensions come from different, independent processes; two features cut in the same setup with the same tool move together and should be treated as one contributor. It also assumes centred distributions, which a shop running to the top of the tolerance to protect a fit will violate. When in doubt, quote the worst-case number and use RSS as the argument for relaxing a tolerance that worst-case says is too tight.[2]
A dimension toleranced +0.005 / −0.001 is really a dimension whose mean sits 0.002 above the nominal with an equal-bilateral tolerance of ±0.003. The calculator converts every entry that way before adding: the resultant nominal uses the drawn nominals, the worst-case limits use the actual upper and lower limits in each direction, and the RSS band is centred on the mean-shifted resultant with half-ranges of (plus + minus) / 2. That is why an unequal tolerance can move the RSS band without changing the nominal.
A shaft assembly has to leave end play between a shoulder and a housing face. Walking the loop from the housing face: the housing bore depth is 2.000 ±0.005 (positive), then back through two washers at 0.062 ±0.003 each (negative) and the shaft shoulder length of 1.850 ±0.004 (negative).
| Dimension | Direction | Nominal (in) | Tolerance (in) |
|---|---|---|---|
| Housing bore depth | + | 2.000 | ±0.005 |
| Washer 1 | − | 0.062 | ±0.003 |
| Washer 2 | − | 0.062 | ±0.003 |
| Shaft shoulder | − | 1.850 | ±0.004 |
| End play (resultant) | 0.026 | Worst-case ±0.015 (0.011 to 0.041) · RSS ±0.0077 (0.018 to 0.034) |
If the design needs at least 0.015 of end play, worst-case says it can fail (minimum 0.011) while RSS says it passes with margin (minimum 0.018). That disagreement is the whole conversation: either tighten the housing depth, which is the largest contributor at a third of the worst-case tolerance, accept a statistical risk of a few assemblies in a thousand, or add a selective-fit washer. The calculator's contribution bars show which dimension to attack first.
This calculator handles one-dimensional chains of plus/minus dimensions, which covers most shaft, spacer, and housing problems. Position tolerances, datum shift, and features at MMC add bonus tolerance and geometric contributions that need a proper GD&T stack per ASME Y14.5, where each geometric tolerance enters the chain as a ± half-value at the feature's location. The arithmetic is the same; the bookkeeping of what enters the loop is what changes.[3]
C&W reviews stack-critical dimensions at quote time and will tell you when a tolerance is tighter than the assembly needs, or looser than it can afford.