Tolerance stack-up analysis: worst case and statistical

What a stack-up actually answers

A tolerance stack-up is a calculation of how individual tolerances accumulate across an assembly. It answers a specific question: when every contributing dimension sits at a permitted limit, does the gap, interference or alignment still meet the requirement?

The answer is a range, not a single number. A stack might show that a gap can vary from 0.05 to 1.55 mm. If the design needs at least 0.30 mm for a seal or at most 1.20 mm for a cover, the stack has already failed before a part is made.

The calculation is only as good as the chain. Every dimension between the two features that define the requirement belongs in the stack, including the ones that are awkward to measure. A dimension that is left out does not stop contributing. It just stops being counted.

A stack-up is also a communication tool. It shows which dimensions carry the variation and which are almost irrelevant. That tells the designer where a tighter tolerance will help and where it will only add cost.

Build the chain before doing any arithmetic

Start with the requirement. That might be a clearance between two surfaces, the position of a hole relative to a mating pin, or the compression of a gasket. Draw the assembly in the closed condition and mark the gap or the interference you need to control.

Then trace the path from one side of the requirement to the other. Each dimension on the path is a contributor. Give each one a direction: positive if it opens the gap, negative if it closes it. A dimension that does not lie on the path does not belong in the stack, even if it is on the same drawing.

Decide the units and the sign convention before you start. Mixing millimetres and inches, or adding a tolerance with the wrong sign, is the most common source of a wrong result. A simple table with a column for direction catches most of these errors.

Finally, decide whether the stack is one-dimensional. Most assemblies can be reduced to a single direction for a first check. A two-dimensional stack, such as a hole pattern on a diagonal, needs a vector or a coordinate method. Do not force a two-dimensional problem into a one-dimensional sum.

The worst-case method

Worst case adds the absolute value of every contributor at its full tolerance. If five dimensions each carry plus or minus 0.15 mm and all of them act in the same direction, the total variation is 0.75 mm. The gap can close by 0.75 mm or open by 0.75 mm, depending on which limits the parts land on.

The method assumes that every dimension can sit at its least favourable limit at the same time. That is unlikely in production, but it is possible. The result is a guarantee: if the worst case passes, every permitted combination passes.

Worst case is the right first calculation for a safety feature, a seal, a high-value assembly or a low-volume programme where there is no data to justify a statistical assumption. It is also the right method when the cost of a failure is high and the cost of a tighter tolerance is low.

The weakness is conservatism. A stack with many contributors can fail worst case even when the process is capable and the real defect rate is negligible. That can push a designer into unnecessary tight tolerances on every dimension.

The statistical method

The statistical method, often called root sum square or RSS, combines independent tolerances by squaring each one, adding the squares and taking the square root. For five contributors of plus or minus 0.15 mm, the result is the square root of five times 0.15 squared, which is about 0.34 mm.

The result is smaller than the worst case because it assumes that all contributors are unlikely to sit at their extreme limits at the same time. It predicts a distribution, not a guaranteed bound. The usual assumption is that each dimension is centred on its nominal and behaves like a normal distribution, which is not always true.

RSS is useful for a mature, high-volume process with stable capability data. It can show that a stack which fails worst case is still acceptable in production, or that a particular contributor dominates the variation and should be tightened.

RSS is not a licence to ignore the worst case. If the process is skewed, if the supplier runs near one limit, or if the consequences of a failure are severe, the statistical result can understate the risk. Use it with data, not as a default.

Which method to use

Use worst case when the requirement is safety-related, when the assembly is low volume, when the supplier data is thin or when the cost of a failure is much higher than the cost of a tighter tolerance. It is also the correct method for a first pass on any new design.

Use the statistical method when the process is stable, the volume is high, the contributors are independent and the consequence of an occasional out-of-limit assembly is manageable. A mature automotive or appliance programme may have the data to support it. A new custom part usually does not.

A practical compromise is to calculate both. If the worst case passes, the stack has a margin and the statistical result is reassurance. If the worst case fails but the statistical result passes, the design depends on the distribution. That is a decision for the engineer, not for a spreadsheet.

The method should be stated in the design record. A stack that says only pass or fail hides the assumption. A stack that names the contributors, the limits and the method can be reviewed.

Where the tolerances come from

Every contributor needs a tolerance source. It may be a direct tolerance on the drawing, a general tolerance such as ISO 2768-m, a geometrical control, a supplier process capability or a standard part limit. A dimension without a stated source is a guess.

General tolerances are easy to overlook because they are not written next to the dimension. A 50 mm dimension under ISO 2768-m contributes plus or minus 0.3 mm, whether or not the drawing repeats the value. The stack should use the general class for every untoleranced contributor.

Geometrical controls contribute differently. A position tolerance on a hole pattern moves the hole within a zone, and the effect on the gap depends on the direction of the stack. A flatness call changes the contact height. A profile call can affect both form and location. Do not treat a geometrical tolerance as a simple plus or minus value without checking what it controls.

Purchased parts bring their own limits. A bearing, a fastener or a standard spacer has a tolerance from its catalogue or standard. Use the published limits, not the nominal size, and record the source. If the supplier changes the standard part, the stack may need to be repeated.

Fixing a stack that fails

The first response to a failing stack should not be to tighten every tolerance. Find the contributor with the largest effect. A single dimension that carries half the variation is a better target than five dimensions that each carry a little.

There are four ways to fix a stack. Reduce a contributor, remove a contributor, change the assembly so the requirement has more room, or accept the risk with a documented decision. Tightening a tolerance is only the first of these, and it is often the most expensive.

Sometimes the best fix is a design change. A slot instead of a round hole, a floating fastener, a shim or an adjustable bracket can absorb variation without tightening the process. A stack that is solved by geometry is usually cheaper than one solved by inspection.

If a tolerance is tightened, check that the process can hold it before the drawing is released. A stack that passes on paper but cannot be manufactured is not a solution. The supplier should see the stack and the reason for the critical call.

• Identify the largest contributor before changing anything.
• Prefer a design change that absorbs variation.
• Tighten only the dimension that the function needs.
• Check the process capability behind the new tolerance.
• Record the decision and the revision.

Documenting the assumptions

A stack-up is a snapshot of a design at a revision. Record the drawing revision, the dimensions used, the tolerance source, the direction convention and the method. If the assembly changes, the stack should be reviewed, not quietly reused.

For a critical assembly, keep the calculation with the first article report. The inspector can then see why a dimension was called critical and which limit matters. The supplier can see the same reasoning at RFQ stage.

A short stack is better than a long one that nobody can follow. Show the contributors, the arithmetic and the conclusion on one page. If the calculation needs a spreadsheet, keep the spreadsheet with the design record and name the version.

The final check is physical. Measure the first article and compare the actual gap with the predicted range. If the part sits outside it, the stack is missing a contributor or using a wrong assumption.