When Small Variations Add Up
Every machined feature is made to a tolerance, a small allowable band around its target dimension. Tolerance stacking, also called tolerance stack-up, describes how those individual variations combine when parts are assembled in a chain. The concept explains why a series of in-spec parts can still add up to an out-of-spec assembly.
The Dimensional Chain
When several dimensions line up end to end, the final gap or fit depends on the sum of all of them. Each link in this chain carries its own tolerance, and those tolerances accumulate along the chain. The more parts and dimensions involved, the more room there is for the accumulated variation to grow.
Worst-Case Analysis
The simplest way to bound a stack is worst-case analysis, which adds all the tolerances assuming every part lands at its most extreme allowable size in the same unfavorable direction. This guarantees the assembly will fit even in the rarest bad combination. It is conservative, so it can force tighter, costlier tolerances than are usually necessary.
Statistical Stacking
In practice, parts vary randomly, and it is very unlikely that every one lands at its extreme in the same direction at once. Statistical methods such as root-sum-square (RSS) combine tolerances by summing their squares and taking the square root, yielding a realistic spread smaller than the worst case. This lets designers use sensible tolerances without over-tightening every part.
Datums and Controlling the Stack
Measuring every feature from a common datum, rather than part-to-part down the chain, prevents tolerances from compounding needlessly. Good dimensioning practice deliberately arranges datums and dimensions to keep the critical stack short. This is a major reason standards define coordinated ranges across many related dimensions rather than isolated numbers.
Worked Example
Suppose four spacers are stacked, each with a target length of 10.00 mm and a tolerance of plus or minus 0.05 mm. In the worst case, all four could be 0.05 mm long, so the stack could reach 40.20 mm instead of 40.00, a 0.20 mm accumulation. Using root-sum-square, the realistic variation is the square root of four times 0.05 squared, about 0.10 mm, half the worst case, showing why statistical stacking gives a truer picture.
A Common Misconception
A frequent mistake is assuming that if every individual part passes inspection, the assembly must fit. Parts can each be perfectly in tolerance yet stack to an out-of-specification total when their small errors align. This is exactly why engineers analyze the whole chain, not just each part in isolation.