The tolerance that cost a week

Every part of an assembly passed inspection. The assembly did not fit, and nobody had made a mistake.

A drawing specifies a nominal dimension and a tolerance: the amount by which a real part may differ and still be acceptable. Each of the seven parts in this assembly was within its stated tolerance. Stacked in series, their errors added, and the sum exceeded the clearance at the end. The phenomenon is called tolerance stack-up, and it is a property of the design rather than of the manufacturing.

There are two ways to think about the sum, and choosing between them is the whole decision. The worst case assumes every part sits at the unfavourable extreme simultaneously, which is safe and expensive, because designing for it demands tighter tolerances on every part. The statistical approach treats each part's error as a draw from a distribution and adds variances rather than limits, which gives a far smaller expected total and accepts that a small proportion of assemblies will not fit. Which is correct depends on the cost of a failure and on whether the parts are independent — and they often are not, because parts made on the same machine in the same hour share whatever that machine was doing.

The practical remedies are mostly about structure. Reduce the number of parts in series. Put the loose tolerance at the end, where one adjustable component can absorb the sum. Measure the stack in the first batch rather than the hundredth.

The habit that generalises beyond machining is to look for places where small permitted errors are added in sequence. A seven-step approval process with a two-day target at each step is a tolerance stack, and it fails in exactly the same way.