Guide
Tolerance stack up: should you use worst-case or RSS?
A tolerance stack up is the calculation that adds the tolerances of every dimension in a chain to predict the range of one result you care about, such as a gap, an overlap, a wall or an axial float. Worst-case and RSS (root sum square, or statistical) are the two standard ways to do that addition, and they answer different questions: worst-case asks whether any legal combination of parts can fail, RSS asks how often a real production mix will.
By Saola Manufacturing engineering team · Published · Updated
This guide works one 1D stack both ways, lists the assumptions each method quietly makes, and shows how the result should change the tolerances and notes on the drawing. The governing vocabulary is ASME Y14.5, which also defines the statistical tolerance symbol used when a drawing relies on RSS.
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What does a tolerance stack up actually tell you?
It tells you the smallest and largest value a functional result can take when every contributing dimension is somewhere inside its tolerance. If that range falls outside what the assembly needs, at least one tolerance on the drawing is wrong, however reasonable each one looks on its own.
Every stack starts with a requirement, not with the drawing. Write it down as a number range: the minimum clearance that keeps a retaining ring from binding, the maximum float a bearing preload allows, the minimum edge distance for a seal. That range is the target. The stack is the chain of dimensions that connects one side of the result to the other, and each link in the chain is a dimension from one of the parts, a feature of a mating part, or a geometric tolerance that moves a surface.
The answer is only as good as the chain. A link left out (a chamfer that changes where a part seats, a flatness callout, a coating, a datum that is not the one the assembly locates on) makes the result look better than reality. That is why the method choice, worst-case or RSS, matters less than getting the loop right, and why engineers who review stacks look at the loop sketch before they look at the arithmetic.
How do you set up a 1D stack-up loop?
Draw the assembly along one axis, pick a direction as positive, and walk from one side of the gap to the other through every surface that touches. Each dimension you cross becomes a contributor with a sign, a mean value and an equal half-band.
- Name the result and its limits. For example, “axial gap G between the spacer face and the cover, required between 0.30 and 0.70 (values in millimeters).”
- Sketch the loop. Start at one face of the gap, follow contacting surfaces around the assembly, and end at the other face. Only surfaces that actually touch in assembly belong in the loop.
- Assign signs. Pick right (or up) as positive. A dimension crossed in the positive direction adds to the gap; one crossed in the negative direction subtracts.
- Convert every tolerance to mean and equal half-band. A unilateral or unequal tolerance, such as a part width of 15.00 with limits +0.00 and −0.12, becomes a mean of 14.94 with a half-band of 0.06. Working from the nominal instead of the mean is the most common arithmetic error in hand stacks.
- Add geometric contributors. Flatness, perpendicularity, position and runout can all move the contacting surface. Enter each as its own line with a mean of zero and a half-band of half the tolerance zone, unless the feature control frame and datum scheme say otherwise.
- Sum the means for the nominal result, then combine the half-bands by the method you choose, below.
Keep the loop to one axis only when the parts genuinely locate along that axis. If a part can tilt or shift sideways inside a clearance, the 1D model underestimates the variation and a 2D or 3D analysis is the right tool.
Worked example: what do worst-case and RSS give for the same stack?
In the example below, worst-case predicts a gap range that breaks the requirement while RSS predicts one that passes comfortably. Neither number is wrong; they are answers to different questions, and the drawing has to state which question it was designed to.
The assembly is a housing cavity A that holds three stacked parts: a sleeve B, a spacer C and a bushing D. The gap is G = A − (B + C + D). The housing dimension is crossed in the positive direction and the three parts in the negative direction. All values are illustrative and in millimeters, and every tolerance is already symmetric about its mean.
| Contributor | Sign | Mean (mm) | Half-band, each side (mm) | Half-band squared (mm²) |
|---|---|---|---|---|
| A, housing cavity length | + | 50.00 | 0.10 | 0.0100 |
| B, sleeve length | − | 20.00 | 0.05 | 0.0025 |
| C, spacer thickness | − | 14.50 | 0.04 | 0.0016 |
| D, bushing length | − | 15.00 | 0.06 | 0.0036 |
| Gap G, nominal (sum of signed means) | 0.50 | |||
| Worst-case half-band (sum of half-bands) | 0.25 | |||
| RSS half-band (square root of 0.0177) | 0.133 | 0.0177 | ||
| Gap range, worst-case | 0.25 to 0.75 | Fails the 0.30 minimum | ||
| Gap range, RSS | 0.37 to 0.63 | Passes | ||
| Gap range, RSS with 1.5 inflation factor | 0.30 to 0.70 | On both limits, no margin | ||
| Revised: A and D half-bands tightened to 0.05 and 0.04, worst-case | 0.32 to 0.68 | Passes with no statistical assumption | ||
Read the result in three steps. Worst-case says that if every part arrives at its unfavorable limit at the same time, the gap can close to 0.25 against a requirement of 0.30, so the design as drawn does not guarantee function. RSS says that if the four dimensions vary independently and roughly normally, centered on their means, with each tolerance equal to three standard deviations of its process, the gap will sit between 0.37 and 0.63 for nearly every assembly. Setting each tolerance at three standard deviations is the same as asking every contributing process to run centered with a capability index of at least 1; the NIST/SEMATECH e-Handbook defines those indices against a spread of three standard deviations either side of the process mean.
The inflated RSS line is the honest middle ground. Real processes drift off center and are not perfectly normal, so many design groups multiply the RSS half-band by a factor (1.5 is a common choice, as industry guidance rather than a standard value). Here that lands on the limits with almost nothing to spare, which tells you the design has no margin left once ordinary process drift is allowed for. The last row shows the fix that removes the argument: tightening only the two largest contributors makes the stack pass even worst-case.
When is worst-case the right method, and when is RSS safe?
Use worst-case when one failure is unacceptable, when there are few contributors, or when you cannot rely on the suppliers’ process statistics. Use RSS when there are many independent contributors made in volume by capable, centered processes, and accept that a small, known fraction of assemblies may fall outside the predicted range.
The RSS formula is the same root-sum-square rule metrologists use to combine independent standard uncertainties, described in NIST Technical Note 1297. It inherits the same conditions: the contributors must be independent, and the spread used for each one must reflect how the part is actually made. Each condition fails in a recognizable way on real parts.
- Few contributors. With two or three links, RSS saves very little and the statistical argument is weak. The benefit grows with the number of links, because the chance of every part sitting at its bad limit at once shrinks quickly.
- Correlated dimensions. Two lengths cut in one setup, or two features formed by the same die cavity, move together. Their variations add arithmetically, not in quadrature. Group them into a single contributor before you take the square root.
- Off-center processes. A process that sits near one limit (a turned diameter run on the high side to leave stock for wear, a casting that shrinks consistently) violates the centered-mean assumption. Use the actual mean in the stack, or inflate.
- Low volume or mixed sources. Ten parts from a prototype run, or a lot assembled from two suppliers with different setups, is not a normal population.
- Safety-critical results. If the gap closing means a seal leaks, a pin shears or a medical device jams, the cost of the rare failure usually justifies worst-case and the tighter tolerances it brings.
Monte Carlo simulation is the third option. It samples each contributor from a chosen distribution and reports the predicted spread of the result, which handles non-normal distributions, nonlinear loops and mean shifts. It is still only as good as the distributions entered, so the same question applies: where did those numbers come from?
How should the stack results drive the tolerances on the drawing?
Tolerances should be allocated from the stack, not copied from a title-block default: give the most tolerance to the dimensions that are expensive to hold and tighten the ones that are cheap to hold. If the design depends on RSS, the drawing has to say so, because a supplier who meets every tolerance individually has met the drawing even if the assembly fails.
Start by ranking contributors by their share of the result. In the example, the housing length supplies more than half of the squared total, so it is the first place to look; the spacer barely matters. Then ask what each dimension costs to tighten. A turned length on a part already held to a close tolerance elsewhere often tightens almost for free; a cast length that would need a new machining pass does not. The guide to the cost of tight tolerances covers that trade.
Once the allocation is set, put it on the drawing in a form the shop and the inspector can act on:
- Worst-case design. Plain limits or bilateral tolerances on each contributing dimension are enough. Mark the contributing dimensions so they are inspected at first article and watched in production, and make sure each one is dimensioned directly from the surface that locates in the assembly, not chained from another feature.
- Statistical design. Add the statistical tolerance symbol defined in ASME Y14.5 next to each statistically allocated tolerance, together with a note requiring those features to be produced under statistical process control, and state the capability index you expect, for example a minimum Cpk. Without the symbol and the note, the RSS assumption lives only in your spreadsheet.
- Dual tolerancing. Y14.5 also allows a drawing to show both the statistical tolerance and a tighter arithmetic tolerance, with a note that the statistically toleranced features are produced under statistical process control or to the more restrictive arithmetic limits. A supplier without capability data then still has a tolerance that protects the stack, and the relaxed statistical tolerance is earned only with process control.
- General tolerance notes. A title-block note such as an ISO 2768 class is fine for non-functional features, but no dimension in a stack should rely on it. The ISO 2768 and NADCA tolerance guide explains what those notes allow.
Which drawing mistakes break a tolerance stack up?
Most failed stacks fail because of the drawing, not the arithmetic. The usual causes are chain dimensioning, datums that differ from the assembly’s locating surfaces, geometric tolerances and coatings left out of the loop, and nominal values used instead of means.
- Chain dimensioning. When a functional length is built from three dimensions on one part, all three tolerances enter the stack. Dimension the functional length directly and the part contributes one tolerance instead of three.
- Datums that do not match the assembly. If the drawing measures from a face that does not touch anything in the assembly, the inspection report can pass while the assembly fails. The guide to datums on cast parts shows how to pick datums that match how the part locates.
- Geometric tolerances left out. A face that is within its size limits but out of flat or out of square still moves the contact point. Position tolerances at maximum material condition add bonus tolerance as the feature departs from MMC, so the stack must use the largest zone the drawing allows; see the MMC bonus tolerance guide.
- Coatings and finishes. Plating, anodizing and powder coat add material to every coated face, and the coating has its own thickness range. State whether dimensions apply before or after coating; the anodizing dimensional change guide covers how much growth to expect.
- Mixed standards. A part drawn to ISO GPS and its mate drawn to ASME Y14.5 can interpret the same callout differently, especially for size and envelope. The ASME vs ISO GPS guide lists the differences that matter.
- RSS without the note. Allocating tolerances statistically but releasing a drawing with plain tolerances transfers the risk to assembly with no one aware of it.
What should the RFQ say when a dimension comes from a stack?
Tell the supplier which dimensions are stack contributors, which method you used, and what evidence you expect for each one. A supplier can only protect a stack it knows exists.
- Mark the contributors on the drawing with a flag or key characteristic symbol, and list them in the RFQ.
- State the method. For a worst-case design, every part inside tolerance is acceptable. For a statistical design, state the capability index and the sample plan that proves it, and whether the tighter arithmetic limit applies when that proof is missing.
- Ask for the data, not a summary. A first-article report with actual values for every contributor, and lot measurement records for the same dimensions, let you rerun the stack with real means instead of nominals. The first article inspection guide lists what that report should contain.
- Send the mating-part context where you can, even a simple assembly sketch. A supplier who sees the gap can flag a datum or dimensioning scheme that does not match it.
At Saola, drawing review returns a marked-up copy of your own drawing, and the first-article report measures every toleranced dimension on the first piece from new tooling. When a drawing marks a dimension as a process-capability control point, it is treated as one of the values watched through the production run, not only at first article, as set out on the quality page. On machined castings, using one consistent datum scheme on the casting and machining drawings keeps the part to one setup, which also keeps the datum-related links in your stack from multiplying; see CNC milling. ISO 9001 certification is in progress, not yet held. To have a stack-driven drawing reviewed with a price, send it through the quote form, and see the RFQ guide for the rest of the package.
Frequently asked
Is RSS always less conservative than worst-case?
For the same contributors, yes: the square root of a sum of squares can never exceed the plain sum. That is exactly why it needs justification. RSS buys tolerance by accepting that a rare combination of parts may not fit, and the justification is process data showing the contributors are independent, centered and capable.
How many contributors does RSS need to be worthwhile?
There is no fixed number in the standards. With two contributors the saving is small and the statistical argument is thin; the advantage grows as more independent links are added. Many teams default to worst-case for short loops and reserve RSS for long ones built from high-volume parts.
What if my contributors have unequal tolerances, like plus zero, minus something?
Convert each one to its mean and an equal half-band before you start. Using the nominal with an unequal tolerance shifts the predicted result, and the error grows with every unilateral contributor in the loop.
Does a 1D stack work for parts that can tilt?
Not reliably. If clearance lets a part rock or shift sideways, the contact point moves in more than one direction and a 1D sum misses part of the variation. Use a 2D loop with the angular terms, or a 3D variation analysis tool, for those assemblies.
Can the supplier choose the method for me?
No. The method is a design decision about acceptable risk, and it has to be visible on the drawing. A supplier can supply the measurement data that shows whether a statistical assumption holds, but meeting every printed tolerance is all a plain drawing obliges them to do.
Sources
- ASME Y14.5, Dimensioning and Tolerancing, ASME, 2018 (reaffirmed 2024)
- NIST Technical Note 1297, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, B. N. Taylor and C. E. Kuyatt, NIST, 1994
- What is process capability?, NIST/SEMATECH e-Handbook of Statistical Methods, NIST, accessed October 2026
- ISO 2768-1:1989, General tolerances, Part 1: Tolerances for linear and angular dimensions without individual tolerance indications, ISO, 1989