Tolerance stack-up
A Socratic walk-through of tolerance stack-up — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why can a machine assembled entirely from parts that passed inspection still refuse to fit together?
Every part in the crate has a stamped inspection ticket. Each was measured, each fell inside its drawing limits, none was borderline enough to argue about. And the assembly will not close: the last shaft is a fraction of a millimetre from its bore.
Nobody made a mistake, and yet the outcome is wrong. What exactly did the inspection certify — and what did it never look at?
Reasoning it through
REASONING #Take the simplest version. Six spacers stack inside a housing, and what matters is the gap left at the end. Each is drawn at 20 mm with a tolerance of plus or minus 0.05 mm. Inspection asks one question of each: is this part within its limits? A part at 20.04 mm answers yes. So does the next, and the next.
But the gap at the end is not any part's dimension. It is the housing's length minus the sum of six lengths, and nobody measured that. The inspection certified six independent facts, and the assembly depends on a seventh that was never a fact about any single part. That is the whole shape of the problem, and everything else is arithmetic.
So do the arithmetic. If every spacer came out at its maximum the stack would be 0.30 mm longer than nominal; all at minimum, 0.30 mm shorter. That is worst-case stacking — add the tolerances directly and design the gap to swallow the full plus or minus 0.30. It is completely safe. But is it true? Machining errors are not adversarial; they scatter around a target, and if the parts are independent the chance of all six sitting at their upper limit at once is the product of six small probabilities. Far more often one runs high, another low, and they cancel. Worst-case stacking is not wrong; it is pessimistic, pricing in an event that will not occur in the life of the product.
What is the honest sum? When independent random quantities are added, it is their variances that add, not their spreads. Take the square root at the end and the assembly's spread is the square root of the sum of the squares of the individual spreads. That is statistical, or root-sum-square, stacking — adding in quadrature.
Watch what it does to our six spacers. Worst case gives 6 times 0.05, or 0.30 mm. Root-sum-square gives 0.05 times the square root of 6, about 0.1225 mm. Note what the multiplier is. For n equal tolerances the statistical result is not n times one tolerance but the square root of n times one: six parts multiply the tolerance not sixfold but by about 2.45. Add a seventh and the worst case grows by a full 0.05 while the statistical figure grows by under 0.01. That gap is the payoff of the whole subject, worth a factor of two and a half in how tightly each spacer must be machined — and since cost rises steeply as tolerance tightens, a great deal of money.
But it only works if the assumptions hold. The dimensions must be independent, and often they are not: six spacers cut on the same machine with the same wearing tool drift the same way together, so their errors correlate and the cancelling never happens. The processes must also be centred — quadrature describes scatter about a target, and a process running consistently 0.03 mm high contributes a shift no square-rooting removes. And the stated tolerance must match the actual process spread; the arithmetic is usually set up by treating it as three standard deviations, which is a claim about the shop, not the drawing. Where those assumptions are shaky, inflate the statistical result rather than abandon it.
The analogy
THE ANALOGY #Think of six people asked to walk fifty paces to a mark and stop. Each will be a pace or two off; that is the tolerance. Now line them up head to toe. The pessimist's estimate is that all six overshoot by their full margin, and nothing forbids it. But watch the actual line — one long, one short, one long — and the errors partly cancel, so the end of the chain lands far closer to three hundred paces than the pessimist allowed. Ask them to walk on ground that slopes, though, and every one overshoots together, and the cancelling stops.
the walkers' errors are visible as they occur, so you could correct part-way down the line, whereas a machined chain is measured only after everything is cut — which is why the statistical estimate must be made in advance, on assumptions, rather than observed.
Clarifying the model
THE MODEL #The most common misreading is that statistical stacking is a looser, less rigorous version of worst case. It is not looser about anything except the outcome it promises. Worst case promises that every conforming assembly will fit; statistical stacking promises a stated, large proportion will, and quietly accepts that a small fraction will not. That acceptance is the real decision. If a non-fitting unit means a shim or five minutes of rework, it is a bargain. If it means a failure in the field, worst case is correct and the extra machining cost is the price of certainty.
Both methods also take the tolerance chain as given — but the chain length is itself a design variable, and the more powerful lever. Every part between the two surfaces whose relationship matters adds a term to the sum, so referencing both critical features from a single datum removes terms rather than tightening them. Where the accumulation cannot be designed out it can be absorbed: a shim, an adjustable joint, or selective assembly pairing a high part with a low one.
Finally, a stamped ticket per part is not evidence that the assembly requirement was checked, because that requirement was never a property of a part. Stack-up analysis is the step that turns an assembly requirement into part tolerances; skip it, and the drawing limits are a guess dressed as a specification.
A picture of it
THE PICTURE #How to readThe horizontal axis is how many parts sit in the chain; the vertical axis is the resulting tolerance on the assembly. The bars are worst-case stacking — a straight arithmetic sum, so each added part costs another full 0.05 mm. The line is the statistical sum in quadrature, rising as the square root of the count, so it flattens: the seventh and eighth parts barely move it. At one part the two agree exactly, which is the check that both describe the same thing. By six parts they differ by the square root of six, about two and a half times, and that vertical distance is what worst-case stacking spends to buy certainty.
What became clearer
WHAT CLEARED #Inspection certifies parts; assemblies fail on a quantity no part owns. Once that quantity is written down as a chain, the only question left is how to add its terms — and the two answers differ enormously. Adding directly is safe and expensive, insuring against an alignment of errors independent processes will not produce. Adding in quadrature is the honest expectation, and the arithmetic that matters is that n equal tolerances combine to the square root of n times one, not n times one. The real choice is between a guarantee that every unit fits and a much cheaper tolerance that lets a few not.
Where to go next
ONWARD #- How geometric dimensioning and tolerancing states the assembly requirement directly, instead of hoping part limits imply it.
- Why Monte Carlo simulation of a tolerance chain is now routine, and what it shows that neither formula can.
Key terms
TERMS #| Term | What it means |
|---|---|
| Tolerance chain | the sequence of dimensions between the two features whose relationship the assembly requires. |
| Worst-case stacking | adding tolerances arithmetically, guaranteeing that any set of conforming parts assembles. |
| Root-sum-square stacking | combining tolerances in quadrature, valid for independent, centred processes, giving the square root of n rather than n for equal terms. |
| Selective assembly | sorting parts into size bins and pairing high with low so the accumulated error cancels by construction. |
Every term the collection defines is gathered in the glossary.