Tolerance Stack-Up Calculator

The method, and what it rests on

Rebasing, which is where most stack-ups go wrong

Both methods need every contributor expressed as a mean and an equal half tolerance. For each dimension the mean is the midpoint of its limits and the half tolerance is half the span. A dimension of 50.000 +0.200 / -0.000 has a mean of 50.100 and a half tolerance of 0.100, not a mean of 50.000. Skipping this is the most common error in a hand stack-up, and it shifts the answer rather than widening it.

Worst case

Mean gap is the signed sum of the means. The worst case half width is the sum of the absolute sensitivities times the half tolerances. It is a genuine bound. It assumes only that every part stays inside its limits and that your loop is complete and correctly signed. It makes no assumption about distributions and none about independence.

What worst case does not tell you is how likely the extreme is. In a five part stack of centred independent parts the chance of reaching the worst case is vanishingly small. The method never claims otherwise, so it should never be quoted with a percentage.

Statistical, root sum square

The RSS half width is the square root of the sum of the squared, sensitivity-weighted half tolerances. Adding variances in quadrature is exact for independent random variables and needs no distribution assumption. Going from variances to tolerances is where the assumption enters: it holds only if every tolerance is the same multiple of its own standard deviation, so that the common multiplier cancels. If the sigma levels differ between contributors, this formula is wrong.

RSS is not a bound. Parts fall outside it. The calculator reports the sigma of the gap and each contributor's share of the variance so you can see whether one part is carrying the stack.

On standards, plainly

There is no standard that gives a general tolerance stack-up procedure. ISO 286-1 computes fit clearances arithmetically from the limits, which is a worst case stack on a two-part loop, and that is the only standardised piece of this arithmetic we can point at. ASME Y14.5 defines what the tolerances mean and gives no stack-up formulas. The methods here come from engineering practice and from textbooks, notably Fischer's Mechanical Tolerance Stackup and Analysis and Drake's Dimensioning and Tolerancing Handbook. We do not print chapter or page numbers, because we have not verified them.

Where stack-ups go wrong

• The loop is not closed. A stack runs from one face of the gap, through every part that touches, and back to the other face. Every contributor appears exactly once.
• A sign is wrong. This calculator checks every stack twice, once from the rebased means and once straight from the printed limits. If the two disagree it tells you, because that means a sign or a conversion is wrong.
• Unequal or unilateral tolerances are not rebased. 50.000 +0.200 / -0.000 is centred on 50.100.
• Repeated parts are lumped together. Five spacers at plus or minus 0.050 are five variance terms, not one term of 0.250. Enter them separately.
• Independence is assumed when there is a shared cause. Two features in one setup, parts from one mould cavity, or parts from one bar of stock are not independent.
• Centring is assumed. A supplier who drills to the low limit to protect against scrap turns a statistical stack into a worst case one. A known mean offset adds arithmetically, never in quadrature.
• RSS is used on too few contributors, or when one dominates. Below about four terms there is no central limit argument. If one squared term is more than half the sum of squares, the assembly inherits that part's shape. The calculator warns on both.

Common questions

What is the difference between a worst case and an RSS tolerance stack-up?

Worst case adds the tolerances arithmetically. It is the widest the gap can be if every part sits at its worst limit at the same time, so it is a genuine bound and it needs no assumption about distributions or independence. RSS adds them in quadrature, which gives a narrower band that describes a population of assemblies rather than a bound. RSS rests on assumptions that worst case does not need, and parts can and do fall outside an RSS band.

Which method should I use?

Use worst case by default, and always when a single non-conforming assembly is unacceptable. That covers safety-critical fits, sealing, prototypes, single builds, low volume, and spares that must interchange. Use RSS for volume production where a small, quantified and detectable fallout is acceptable and you have capability evidence on every contributor. Never use RSS silently: if the design depends on it, the drawing has to say so, because a supplier shipping every part at one limit is still fully conforming.

Why can I not just root sum square the tolerances?

You can, but only under one specific condition. Adding variances in quadrature is exact for independent random variables. Going from variances to tolerances requires that every tolerance is the same multiple of its own standard deviation, so that the common multiplier cancels. Plus or minus 0.1 at three sigma and plus or minus 0.1 at six sigma are different claims about the same number. If the sigma levels differ between contributors, the plain RSS formula is wrong.

What does RSS assume, and when does it break?

Independence, centred processes, a common sigma level, roughly normal contributors, and no single dominant term. Independence fails for features cut in one setup, parts from one mould cavity, or parts from one bar of stock. Centring fails when a supplier biases toward the safe material condition, and a known mean offset adds arithmetically rather than in quadrature. Normality fails for sorted, fully inspected, or multi-cavity parts. If the largest squared term is more than half the sum of squares, the assembly inherits that one part and the statistical argument is weak.

How wrong can RSS be if the assumptions fail?

Take two contributors of equal tolerance whose values are spread uniformly across their limits, which is what a loosely controlled or sorted process can look like. Their sum is triangular. Classic RSS puts the three sigma half width at 1.414 times the tolerance, but the true central 99.73 percent half width is 1.896 times it, and the fraction falling outside the RSS band is 8.58 percent rather than 0.27 percent. The arithmetic did not fail. The distribution assumption did.

Is there a standard for tolerance stack-up?

Not for the general procedure, and anyone who tells you otherwise should be asked for the clause. ISO 286-1 computes fit clearances arithmetically from the limits, which is a worst case stack on a two-part loop. ASME Y14.5 defines what the tolerances mean but gives no stack-up formulas. The methods on this page come from engineering practice and textbooks, not from a standard, so we name them and show the arithmetic instead of citing a clause we cannot point at.

How is this different from a measurement uncertainty budget?

The arithmetic looks the same and the question is not. A tolerance stack-up combines the permitted variation of parts to predict how an assembly will vary. An uncertainty budget combines sources of doubt about a measurement to state how well you know a value, and it expands the result by a coverage factor to a stated confidence. A tolerance is a permission granted to a process. An uncertainty is a statement about knowledge. Do not carry a number from one into the other without saying which you mean.