Worst case, statistics, and the twenty per cent we saved by choosing
Picking a stacking method deliberately rather than by default.
Choosing deliberately between worst-case and statistical tolerance stacking, instead of defaulting to whichever method felt safer by habit, saved roughly a fifth of a drawing's manufacturing cost, because worst-case stacking assumes every dimension in a chain lands at its worst extreme at the same moment, an outcome so unlikely across enough independent parts that designing to survive it means paying for a scenario that will essentially never occur.
A drawing came back quoted at three times the expected price, and the cause was two numbers nobody had thought about, the individual part tolerances that had been added up the simple, cautious way instead of combined the way their actual statistics justified.
Two ways to add up a stack
A dimension that depends on several other dimensions stacked in sequence, such as the length of a shaft passing through a series of stacked components, can be analysed in two very different ways. Worst-case stacking adds every individual tolerance together directly, assuming the least favourable case, in which every part in the stack lands at its own extreme limit at the same time and in the same direction.
Statistical stacking treats each part's actual dimension as an independent, randomly distributed value that clusters mostly near its target. It combines the tolerances by a method reflecting how unlikely it is for many independent extremes to line up at once, usually by squaring each tolerance, adding the squares and taking the square root of the total. For the identical stack of parts, that gives a combined tolerance considerably tighter than worst case predicts.
A long line of people walking single file
Picture a long line of people walking single file, each straying slightly left or right of the person ahead by their own small, independent amount. The back of the line ends up displaced sideways by far less than the sum of everyone's maximum sway, since some lean left and some lean right, and over a long enough line those small strays mostly cancel. Worst-case stacking assumes every person in the line leans the same way, as far as they possibly can, all at once. That is technically possible and wildly unlikely. Statistical stacking expects the realistic outcome, a long line of independent little strays partially cancelling, which leaves the worst practical result far milder than the worst theoretical one.
The arithmetic follows the picture. Four parts each held to plus or minus a tenth of a millimetre stack to plus or minus four tenths by worst case, and to plus or minus two tenths statistically, half the spread. With nine such parts the gap widens further, nine tenths against three.
Where the saving comes from
Because worst-case stacking defends against the least forgiving possible combination, it forces every part in the chain to a tighter tolerance than the statistics require, so the design survives even the vanishingly unlikely case where every part sits at its extreme together. Statistical stacking lets individual parts be toleranced more loosely for the same overall assembly requirement. Looser individual tolerances mean cheaper manufacturing processes, higher yields, and fewer parts rejected for being honestly ordinary.
On this drawing, loosening the chain of part tolerances by a modest amount, justified by switching to statistical stacking on a design where the underlying assumptions held, brought the quoted cost down by roughly a fifth. No feature of the part was redesigned. The saving came entirely from matching the analysis to how the parts actually behave.
When the statistics do not hold
Statistical stacking carries its own risk when it is chosen without checking its assumptions. It depends on each part's dimension varying independently and clustering near its target, with no lean toward one edge of the tolerance band. That holds well for a mature, well-controlled production process. It can fail badly for a process still being tuned in, for a small batch where only a few parts will ever exist, or for a supplier whose output habitually skews toward one side of its stated tolerance. In any of those cases the parts behave less like the independent walkers in the line and more like a group that all drift the same way, and the cancelling the method relies on never happens.
Choosing statistical stacking without confirming those conditions trades a calculable saving for a risk that is harder to see, and the saving on this drawing was trustworthy only because that check was done first. The next article in this set follows that more dangerous side of the choice, a design that applied statistical assumptions to a stack that never behaved the way the statistics required.
Making the choice visible on the drawing
The habit that came out of this was treating the choice of stacking method as an engineering decision that needs its own justification on the drawing, and never as a silent default baked into whichever calculation sheet happened to be open. A design review that asks which method was used, and why its assumptions fit the parts in question, catches both the unnecessary cost of an over-cautious worst-case default and the hidden risk of an unjustified statistical one. Finding either mismatch at review costs very little compared with finding it in a quote, or in a batch of parts that will not assemble.
My key error with this
For a long time I stacked tolerances the worst-case way on the grounds that it was the responsible thing to do, since worst case cannot be argued with and nobody has ever been criticised for a part that fits. What that produced, on a perfectly ordinary machined part, was a set of tolerances tight enough to change how the part had to be made and what it cost, in order to protect against a combination of extremes that would essentially never occur together. The cost was real and recurring, and it was invisible on the drawing because a tight tolerance looks like diligence. Learning how process capability is actually measured, and then setting the drawings against a defined control level rather than against the theoretical extreme, cut a meaningful amount off the cost of that part without any change to how well it worked. What replaced the belief is that worst case is not the safe choice, it is the choice that declines to find out how the process actually behaves, and that measuring the process is what lets you tell a tolerance that matters from a tolerance you are paying for out of habit.