Every dimension on a drawing carries a tolerance — the allowable deviation from nominal. A shaft called out as Ø10.00 ±0.05 mm can legitimately measure anywhere from 9.95 to 10.05 mm and still pass inspection. When parts assemble in series, those individual tolerances accumulate. Ignore the math and you'll ship assemblies that bind, rattle, or refuse to mate.

Worst-case (arithmetic) stack: Add nominals; add tolerances. Five blocks each 20.0 ±0.1 mm stacked end-to-end give a total of 100.0 ±0.5 mm. This is the conservative method — it guarantees fit even if every part is at its worst extreme simultaneously. Use it for safety-critical interfaces, low-volume builds, and anywhere a failure means rework.

Statistical (RSS) stack: Real parts cluster near nominal; all five being at the extreme limit at once is vanishingly rare. Root-Sum-Square treats tolerances as standard deviations: T_total = √(T₁² + T₂² + ... + T_n²). For the same five blocks: √(5 × 0.1²) = 0.224 mm. That's less than half the worst-case envelope. RSS is appropriate for high-volume production where process capability (Cpk) is known and a small reject rate is acceptable.

Real-world example: A bearing-shaft assembly. The housing bore is Ø52.00 +0.030/-0.000, the bearing OD is Ø52.00 +0.000/-0.013, and the shoulder depth tolerance is ±0.05 mm. Worst case, the bearing has 0.043 mm of radial clearance instead of the intended light press fit — the inner race spins under load, galls the shaft, and the unit fails in service. Engineers caught this kind of stack-up failure in early jet engine accessory drives; modern GD&T (geometric dimensioning and tolerancing) exists largely to prevent it.

Rules of thumb:

• Tight tolerances cost money exponentially. A ±0.5 mm feature is cheap; ±0.005 mm needs grinding and inspection.
• If a worst-case stack fails, don't tighten every tolerance — find the one dimension contributing most and tighten only that.
• Use datums intelligently. Stacking from a single reference face beats chaining dimensions part-to-part.
• Always check both maximum material condition (MMC) and least material condition (LMC) — interference fails at MMC, clearance fails at LMC.
• For ten or more components in series, RSS is essentially mandatory; worst-case becomes absurdly pessimistic.

Software engineers will recognize the pattern: tolerance stack-up is the mechanical analog of floating-point error accumulation in long computations. Each operation is "correct" within its specified precision, but compose enough of them and the result drifts outside acceptable bounds.