Worst case and the square root

There are two rules for adding tolerances up, they disagree by about a factor of two, and both are correct. Which one to use is not a matter of taste; it is a question about how the parts are made, and the answer is usually not written down anywhere.

The two rules #

Worst case adds the magnitudes: if the four contributions to the output error are c₁ … c₄, the band is |c₁| + |c₂| + |c₃| + |c₄|. It assumes nothing about how the errors are distributed and it is a genuine bound. No part that passes inspection can be outside it.

Root-sum-square takes √(c₁² + c₂² + c₃² + c₄²). It treats each error as a random variable with the tolerance as roughly a standard deviation, and it is the standard deviation of the sum when the four are independent.

Neither is an approximation of the other. They answer different questions. Worst case answers could any acceptable part be this far out? RSS answers how far out is a typical part? — and a designer needs both, at different moments, for different reasons.

The ceiling, which is arithmetic and not statistics #

The ratio between them has an exact upper bound and it is worth deriving because it is so often quoted as though it were a rule of thumb.

By Cauchy–Schwarz, for k contributions,

$$\sum_{i=1}^{k} |c_i| \le \sqrt{k}\,\sqrt{\sum_{i=1}^{k} c_i^2}$$

with equality exactly when all the |cᵢ| are equal. So the worst-case sum can exceed the root-sum-square by at most √k, and it reaches that only when every contribution is the same size.

For a four-bar, k = 4 and the ceiling is exactly 2. The “RSS halves the stack-up” folklore is therefore not folklore at all in one respect — 2 is the right number — and completely wrong in another: 2 is the best possible case, achieved only by a mechanism whose four lengths contribute equally, and this linkage’s contributions are 38, 26, 25 and 11 per cent.

Measured through the turn, the ratio runs from 1.42 to 1.96. It approaches 2 where the four contributions happen to be nearest to equal and falls towards 1.4 where one dominates. In the limit where a single length is responsible for everything, the ratio goes to 1: a single error has nothing to be independent of, and the statistical rule has nothing to offer.

That is the first thing this figure is for. The saving is not a constant, it is not 2, and it is largest exactly where a tolerance allocation has already done its work — because a well-allocated design is one whose contributions are nearly equal, which is also the design where RSS helps most. The two techniques compound rather than competing.

What the saving is bought with #

Everything above is arithmetic. The engineering is in one word: independent.

RSS is the standard deviation of a sum of independent random variables. If the four errors are correlated, the formula is wrong, and it is wrong in the unsafe direction — it under-states the spread.

Correlation is not exotic. It is the normal case:

• Two features made in one setup. If the two fixed pivot holes are bored without moving the workpiece, their positions share the machine’s error; the distance between them may be better than either position, or worse, and either way it is not independent of them.
• A shared fixture. Two links located from the same datum inherit that datum’s error identically.
• Thermal effects. Every dimension on an aluminium part grows together when the shop is warm. A tolerance stack across parts of one material at one temperature has a common-mode component that no amount of statistics removes.
• One worn tool. A drill that has drifted produces a batch whose errors are all in the same direction. The population is not centred where the drawing says.

That last one is the killer, and it is why RSS is properly a process capability statement rather than a tolerance statement. It assumes each dimension is centred on its nominal and spread symmetrically. A shifted process satisfies the drawing and violates the assumption, and the resulting band is worst-case-like without anybody having decided to accept a worst-case band.

The half of the assumption nobody states #

There is a second assumption inside RSS which is separate from independence and is almost never mentioned: that the tolerance means a standard deviation.

A drawing that says 3.5 ± 0.01 has specified limits. It has said nothing about the distribution between them. Treating ±0.01 as one standard deviation implies that about a third of parts are outside the limits, which is absurd; treating it as three standard deviations implies a process far better than the drawing demands, and gives a band three times smaller than the one-sigma reading.

Common practice is to take the tolerance as three standard deviations and then quote a three-sigma output band, which cancels the factor and gets back to something sensible. But the cancellation only works if every one of the contributing dimensions is made by a process centred and capable in the same way — and a stack that mixes a ground shaft with a cast boss mixes two processes with completely different relationships between their tolerance and their spread.

The practical upshot is that an RSS band is a number about a factory wearing the clothes of a number about a mechanism. This site can compute the sensitivities exactly and it cannot compute the distributions at all, which is why the figures here report both rules and neither is labelled correct.

The two rules meet in the middle of the cycle #

There is a small pleasing fact visible in the hero figure that is worth pointing out, because it is a check as well as a curiosity.

The gap between the two curves is not widest where the band is widest. It is widest where the four contributions are nearest to equal, which is a different place. Around 120° of crank rotation the band is at its narrowest and the two rules are furthest apart in ratio; around 30° the band is at its widest and the ratio is nearer 1.4, because there one length has come to dominate.

So the statistical rule saves most exactly where there is least to save, and least where there is most. That is not a designed-in irony; it follows directly from Cauchy–Schwarz, and it means the absolute saving is far flatter around the cycle than either curve is. A designer looking at the worst position — which is the one that sets the specification — gets the smallest proportional benefit from switching rules.

Where the contributions came from #

One step is being taken for granted and it is the step this essay depends on.

The four contributions cᵢ are not measured; they are computed, as ∂ψ/∂ℓᵢ times δᵢ, from the derivative route. Both combination rules are functions of those four numbers and neither can be more reliable than they are.

That is why the agreement between the two routes to a sensitivity matters here rather than only there. A worst-case sum built on a wrong derivative is a wrong bound, and — because it is a bound rather than an estimate — it will be wrong in whichever direction the derivative was wrong, with nothing to indicate which. A bound that might be an under-estimate is not a bound.

The check that the four contributions add up to the measured corner band is therefore not a formality. It is the only thing standing between a stack-up sheet and a sheet of plausible numbers, and it costs sixteen solves at one crank position. Every figure in this field runs it.

Which to use #

The honest answer is both, for different decisions.

Use worst case when the consequence of being outside the band is unacceptable rather than merely unwelcome — an interference, a jam, a mechanism that cannot be assembled. There is no comfort in “only one in ten thousand will bind” if binding destroys the machine, and the calculation for whether a mechanism assembles at all is a worst-case calculation by its nature.

Use root-sum-square when the consequence is a distribution of performance — a batch of pumps with slightly different strokes, a set of mechanisms whose timing varies. Then the question really is about the typical part, and worst case will lead to tolerances several times tighter than the application needs.

And when using RSS, say what it assumed. A stack-up sheet that reports a number with no note about independence is reporting a number whose validity nobody can check later.

Where both rules fail together #

Both rules are built on the same foundation: that the output error is a sum of contributions, each linear in its own length error. Where that fails, both fail, and they fail in the same direction and by the same enormous factor.

At a change point every contribution diverges, so worst case and root-sum-square both go to infinity and their ratio stays a perfectly reasonable-looking number somewhere between 1 and 2. A stack-up sheet near a singularity therefore looks entirely normal — the two rules agree with each other to the usual factor, both are enormous, and nothing in the comparison between them says why.

The check that catches it is not between the two rules. It is between either of them and the corner enumeration, which does not assume linearity because it does not assume anything: it builds sixteen mechanisms and solves them. That comparison costs sixteen solves at one position, and it is the only thing in this field that can tell a large band from a meaningless one.

A worked example of the independence failing #

The abstract statement — errors that share a cause do not combine as independent ones — is easy to nod at. Here is what it costs.

Suppose the two fixed pivot holes are bored in one casting, on one machine, in one setup. The distance between them is the ground length g, and it is one of the four contributors at 26%.

If the machine’s positioning error is δ, both holes move by δ in the same direction and the distance between them does not change at all. The ground length is far better than the position of either hole, and its contribution to the output band is smaller than a naive stack-up says — perhaps much smaller.

Now suppose instead that the two pivots are on separate brackets bolted to a frame. Each bracket’s position carries the frame’s error plus its own, and the two errors add. The ground length is now worse than either bracket’s position.

Same drawing, same nominal tolerances, same mechanism. Two manufacturing routes, and the contribution of the largest-but-one term to the output band differs between them by a factor that is not small. Neither worst case nor root-sum-square, applied to the numbers on the drawing, sees any of this — both are functions of the tolerances alone, and the tolerances are identical.

What sees it is a stack-up done on the features rather than on the dimensions: the two hole positions as the independent variables, with the ground length derived from them. That is more work and it is the only version that is right, and it is why tolerance analysis in serious practice is done on a datum scheme rather than on a list of lengths.

This site cannot do that, and says so. It computes the sensitivities of the output to the mechanism’s dimensions exactly. Which of those dimensions are independent is a fact about a factory.

Neither rule says anything about assembly #

A limitation both rules share, and it is the one that catches people out on mechanisms rather than on stacks of washers.

Both rules compute a band on an output given that the mechanism assembles and reaches the position. Neither of them asks whether it does.

For a stack of parts along a line that distinction does not arise: the parts fit together whatever their lengths, and the only question is where the far end lands. A closed kinematic chain is different. Four bars of the wrong lengths do not make a four-bar that is slightly wrong; they make a four-bar that cannot be assembled at some crank angles, or at all.

The check for that is not a band. It is the count of corner linkages that failed to solve, and this site’s generators report it: the tolerance-band figure asserts that all sixteen corners assemble at essentially every position, and it would fail the build rather than draw a band if they did not.

That check has no analogue in either rule. A worst-case sum of sensitivities is a perfectly well-defined number for a mechanism that cannot be put together, and it will be computed and reported without complaint. The first question about a toleranced closed chain is therefore not how accurate it is; it is whether every part that passes inspection is a mechanism at all.

What a batch actually looks like #

It is worth describing the population the two rules are arguing about, because neither of them draws it and the picture settles which is which.

Take four independent errors, each uniform across its tolerance. Their weighted sum — the output error — is not uniform. It is the convolution of four rectangles, which is a bell-shaped curve with hard edges: it reaches zero exactly at the worst-case limits and it is small there.

That shape is the whole story in one object. The support of the distribution is the worst-case band, and it is genuinely reachable — a part at all four extremes exists, and if enough parts are made one will occur. The standard deviation is the root-sum-square figure, and the bulk of the population sits within a fraction of the worst-case band.

The fraction is not small enough to ignore in either direction, which is why both numbers survive in practice. For four equal uniform contributions the standard deviation is a shade over a quarter of the half-width of the support, so a three-sigma spread covers about three quarters of the worst case and the remaining quarter is where the rare parts live.

None of that arithmetic is in this site’s figures, because it is a statement about a distribution of manufactured parts rather than about a mechanism. What the site computes is the four cᵢ, exactly; what a factory supplies is the distribution each error is drawn from; and the combination rule is the join between them. Getting the join wrong is much easier than getting either side wrong, and it does not show up in any single number.

The number to remember #

If one thing survives from this essay it should be the ceiling, because it is exact and it kills a whole class of wishful arithmetic. The most that a statistical combination can ever save over a worst-case one is the square root of the number of contributions, and it saves that much only when the contributions are all the same size.

Four lengths: at most 2. Seven, for a six-bar: at most 2.65. Twenty parameters in a spatial mechanism: at most 4.47.

And every one of those is a ceiling on a saving that is paid for with an assumption about how the parts were made — an assumption which is a fact about a factory, not about a mechanism, and which this site cannot compute for anybody.

The saving being bought entirely with an assumption of independence is the finding, and there is a practical form of it that makes the choice between the two rules decidable rather than a matter of taste. The assumption is about the process, not the mechanism, so it can be checked without touching the geometry at all: were these four dimensions produced by four separate operations, or by one setting of one machine? A frame whose four holes were bored in a single setup has correlated errors and the root-sum-square is optimistic by up to the full factor of two; four parts from four suppliers are as independent as anything gets. So the question a tolerance analyst should ask first is not which rule but what made these parts, and the answer decides. That is a more useful instruction than the usual advice to use worst case for safety-critical work and statistics elsewhere, because it is a question with a factual answer — and, as the repeated-unit case shows, the factual answer is sometimes the uncomfortable one.