Tolerance stack-up: worst-case and RSS calculation
Published October 17, 2026 — build a tolerance stack-up, calculate the min and max gap with worst-case and root sum square (RSS) methods, a worked example, how to choose between them and a calculator.
In short: a tolerance stack-up links the two surfaces of a gap J through the dimensions of the parts in between. J = Σ positive dimensions − Σ negative dimensions. In worst case, the tolerance of J is the sum of the tolerances: every assembly is guaranteed. In RSS (root sum square), it is the square root of the sum of squares: more realistic in series production, if dimensions are independent and centred.
1. Tolerance stack-up calculator
One line per dimension in the loop: its sign (+ if it increases the gap, − if it reduces it), its nominal value and its upper and lower deviations, in mm. The preloaded example is the one in section 4. Enter the requirement to check the result.
2. Building the tolerance loop
- Define the requirement: the gap, overhang or interference to guarantee, with its limits (min J, max J).
- Walk the loop from one surface of the requirement to the other, through the contact surfaces between parts.
- One dimension per part: each part crossed contributes the dimension between its two contact surfaces. A shorter loop accumulates fewer deviations.
- Sign the dimensions: positive if it increases the requirement as it grows, negative otherwise.
- Calculate nominal J, then its variation, by worst case or RSS, and compare with the requirement's limits.
3. The formulas
Ti is the tolerance range of dimension i (max − min). For a dimension with unequal deviations, use its mean value and the centred half-tolerance: 20 +0.1/0 becomes 20.05 ± 0.05.
4. Worked example
The housing and two parts from the drawing in section 2. Dimensions in mm.
If the requirement calls for a gap of at least 0.35 mm, worst case does not guarantee it (0.3 mm); RSS meets it for 99.73% of assemblies. It is your call whether that risk is acceptable, or to tighten dimension a, the largest contributor: 67% of the variance.
5. Worst case or RSS?
The more dimensions in the loop, the wider the gap between the two methods. With n dimensions of equal tolerance ±0.1 mm:
- Worst case: a few parts, one-off manufacturing, a safety function, or poorly known processes. It is the only method that guarantees every assembly.
- RSS: series production, many dimensions, centred and controlled processes. It assumes independent dimensions, normally distributed and centred on their mean value.
- Monte Carlo simulation: when distributions are not normal or the loop is not linear (angles, 3D dimensions), thousands of random assemblies are drawn to obtain the distribution of J.
- General tolerances also apply to untoleranced dimensions: see ISO 2768, and for holes and shafts, ISO 286 fits.
Printable version: the formulas and example as a PDF (A4, no sign-up), for a design file or a drawing review.
6. FAQ
What is a tolerance stack-up?
It is the sequence of dimensions, from one part to the next, that links the two surfaces of a functional requirement (a gap, an overhang, an interference). Adding these dimensions with their sign gives the requirement's value; adding their tolerances gives its variation.
What is the difference between worst case and RSS?
Worst case adds the tolerances: it guarantees every assembly but forces tight tolerances. RSS (root sum square) adds the squares and takes the root: it assumes independent, centred dimensions and gives a smaller variation. With tolerances at ±3σ and normal distributions, 99.73% of assemblies fall within the RSS range.
When should the statistical method be used?
For series production, with several dimensions in the stack and centred, controlled processes. For a few parts, one-off manufacturing or a safety function, stay with worst case.
How do you choose the sign of a dimension?
A dimension that increases the gap when it grows is positive; one that reduces it is negative. Walking the loop from one surface of the gap to the other, dimensions in the gap's direction are positive, those in the opposite direction negative.
What if the calculated gap does not meet the requirement?
First shorten the loop (fewer parts, one dimension per part), then tighten the tolerances that weigh most: in RSS, a dimension's contribution is its T² divided by the sum of T². The calculator shows it for each line.