A tolerance analysis answers one question: will these parts assemble and function across the full range of production variation? The arithmetic is easy — the discipline is defining the loop correctly and picking the right method for what's at stake.
1. Define the loop
Trace the chain of dimensions from one side of the gap you care about to the other. Each dimension gets a sign: + if it opens the gap, − if it closes it; the gap itself is the closing dimension. Miss a contributor — a bearing shoulder, a snap-ring groove, a coating thickness — and the whole analysis is wrong, so build the loop from the actual mating features, not just the drawing's obvious dims.
2. Gather real tolerance data
Use tolerances that will actually be produced, not aspirational ones: pull them from the drawings, the supplier's demonstrated capability, or standard tables like ISO 2768 for unspecified dimensions. A GD&T position callout converts to an equivalent ± of half its diametral zone in each axis — see GD&T explained.
3. Choose worst-case, RSS or Monte Carlo
Worst-case sums the extremes: if it assembles worst-case it always assembles, but it's expensive and usually over-tight for long chains. RSS (root-sum-square) assumes independent, roughly normal contributors whose extremes rarely line up, giving a realistic spread for higher-volume production. Monte Carlo samples the real distributions and handles shifted means, non-normal shapes and non-linear stacks that RSS can't. Run all three in the Tolerance Stackup Analyzer.
4. Judge it against capability
A predicted spread only means something next to the limits. Convert it to a Cpk: below ~1.33 you'll see rejects; above ~1.67 you may be paying for tolerance you don't need. If the stack fails, don't reflexively tighten everything — RSS makes the culprits obvious (the biggest tol² dominates), so fix the one or two contributors that drive the variation, or redesign the loop to remove one entirely.
Common traps
Dimensions sharing a datum aren't independent, so RSS under-predicts. Temperature changes the loop for mixed materials (see thermal expansion). And a stack that looks fine in 1D can fail once angular and positional variation are added — when in doubt, model it explicitly rather than trusting a single linear chain.