Engineering Reference

Tolerance Stack-Up Calculator

Combine up to six tolerances by the worst-case and root-sum-square methods, and see how much the statistical approach saves.

Page last changed 2026-10-19 · based on n/a — standard engineering relationships, no single governing revision

Quick Answer

Worst case adds the tolerances: ±(t₁ + t₂ + … + tₙ). The statistical (RSS) method combines them as ±√(t₁² + t₂² + … + tₙ²). With ten equal tolerances, RSS gives ±3.16t against a worst case of ±10t — a 68% reduction.

Tolerance Stack-Up

The Formulas Used

Worst case (arithmetic): T = Σ tᵢ
Statistical (root-sum-square): T = √(Σ tᵢ²)
For n equal tolerances: worst case n·t, RSS t·√n

Worst Case vs Statistical — and When Each Is Right

The worst-case method assumes every dimension lands at its limit simultaneously, in the direction that makes the assembly fail. It is guaranteed correct — if the parts pass inspection, the assembly works — but it is expensive, because it demands tight tolerances on every contributor regardless of how many there are.

The RSS method assumes the errors are independent and normally distributed, so they partially cancel. It is statistically valid for a large production run and lets tolerances be much looser. But it carries a real risk: a small percentage of assemblies will be out of tolerance, and the method is only valid if the processes are actually centred and in statistical control.

The choice is a business decision, not a mathematical one. Worst case is right where a single failure is unacceptable — safety-critical assemblies, aerospace, medical. RSS is right for high-volume commercial production where a small reject rate is cheaper than tight tolerances on every part. A common compromise is a shifted RSS using 1.5 times the RSS value, which approximates a process that has drifted off centre.

Frequently Asked Questions

What is the difference between worst-case and RSS tolerance stack-up?
Worst case adds all tolerances arithmetically and assumes they all land at their limits in the same direction. RSS combines them as the square root of the sum of squares, assuming independent normally distributed errors that partly cancel. RSS gives a much smaller total but accepts a small statistical risk.
When should I use worst-case tolerance analysis?
When a single out-of-tolerance assembly is unacceptable — safety-critical parts, aerospace, medical devices, or any assembly where the consequence justifies the cost. Worst case guarantees that any combination of conforming parts will assemble.
How much does RSS save?
For n equal tolerances, worst case is n·t and RSS is t·√n, so the reduction is a factor of √n. Ten tolerances give RSS at 32% of worst case; four tolerances give 50%. The saving grows with the number of contributors, which is why RSS is most valuable on long stacks.
What are the assumptions behind RSS?
That each dimension is independent, normally distributed and centred on nominal, and that the processes are in statistical control. If a process drifts off centre, or if dimensions are correlated — as they are when several features are machined in one setup — RSS understates the real variation.
What is the 1.5 sigma shift?
A convention from Six Sigma practice that assumes a process mean drifts up to 1.5 standard deviations off centre over time. Applying it widens the RSS estimate substantially and is a more realistic model of production than a perfectly centred process.

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