Nominal 0.500 mm. Worst case 0.300 to 0.700 mm. RSS 0.378 to 0.622 mm.
- Nominal result
- 0.500 mm
- Worst-case maximum
- 0.700 mm
- Worst-case minimum
- 0.300 mm
- Worst-case band
- 0.400 mm
- RSS mean (mid-limit result)
- 0.500 mm
- RSS ± (3σ)
- 0.122 mmFull value ≈ 0.1224744871391589 mm
- RSS maximum
- 0.622 mmFull value ≈ 0.6224744871391589 mm
- RSS minimum
- 0.378 mmFull value ≈ 0.3775255128608411 mm
With fewer than 4 dimensions, the averaging that RSS relies on barely applies. Use the worst case.
| Dimension | Direction | Share of worst-case band |
|---|---|---|
| Housing inside length | + | 50.0 % |
| Part A length | − | 25.0 % |
| Part B length | − | 25.0 % |
RSS assumes every dimension varies independently, is normally distributed, is centred on the middle of its tolerance band, and has half its band equal to 3σ (±3σ).
What it computes
A stack-up follows a chain of dimensions from one feature to another along a single direction, and asks how much the closing dimension can vary. A typical question is the end gap between parts in a housing. Each dimension in the chain either adds to the gap (+) or subtracts from it (−).
Formulas
For each dimension i with nominal nᵢ, upper deviation uᵢ, lower deviation lᵢ and direction dᵢ of +1 or −1:
- Nominal result N = Σ dᵢ·nᵢ.
- Worst-case maximum = Σ (nᵢ + uᵢ) over the + rows, minus Σ (nᵢ + lᵢ) over the − rows.
- Worst-case minimum = Σ (nᵢ + lᵢ) over the + rows, minus Σ (nᵢ + uᵢ) over the − rows.
- Worst-case band = Σ (uᵢ − lᵢ).
- RSS mean = Σ dᵢ·(nᵢ + (uᵢ + lᵢ)/2).
- RSS half-band T = √(Σ ((uᵢ − lᵢ)/2)²). The RSS limits are the mean ± T.
The tool also shows each dimension's share of the worst-case band. The largest share points to the dimension worth tightening first. Every sum is exact decimal arithmetic in the stack's one unit; only the square root is computed in double precision.
RSS assumptions
RSS is a statistical estimate. It holds only when all four of these are true:
- Independent. No two dimensions come from the same cut, fixture or setup in a way that makes them vary together.
- Normally distributed. The variation of each dimension follows a normal (Gaussian) distribution.
- Centred. Each process is aimed at the middle of its tolerance band. An asymmetric tolerance is first re-centred on its mid-limit, which is why the RSS mean can differ from the nominal result.
- 3σ = half the band. Each half-band equals three standard deviations of its process, so about 0.27 % of parts fall outside each dimension's limits.
Under those assumptions the RSS half-band is a 3σ value for the result, and about 0.27 % of assemblies fall outside the RSS limits. With only a few dimensions the averaging that RSS relies on barely applies. The calculator flags any stack of fewer than four.
When the worst case is required
Use the worst case when every assembly must work, not nearly every one. That covers safety-critical parts, low volumes where statistics do not apply, parts from processes with no capability data, and stacks with few dimensions. Use it also when dimensions come from the same setup and are not independent.
ASME Y14.5 provides a statistical tolerancing symbol. A tolerance that a stack-up treats statistically should be marked that way on the drawing, together with the process control it relies on.
Worked example
01 Write down the chain
A housing has an inside length of 50 ±0.1 mm. Two parts of 20 ±0.05 mm and 29.5 ±0.05 mm sit inside it. The gap is the housing minus part A minus part B.
02 Nominal
50 − 20 − 29.5 = 0.5 mm.
03 Worst case
0.1 + 0.05 + 0.05 = ±0.2 mm, so the gap is 0.3 to 0.7 mm.
04 RSS
√(0.1² + 0.05² + 0.05²) = √0.015 = ±0.1225 mm, so the gap is about 0.378 to 0.622 mm.
05 Read the result
The housing carries half the worst-case band. If a 0.3 mm gap is too small, tightening the housing does the most good. With only three dimensions, the worst case is the safer answer.
Limitations
- One direction only. It does not handle angles, 2D or 3D stacks, or geometric tolerances such as position, which need their own conversion into the chain.
- It does not model thermal growth, deflection or wear.
- It does not model process capability, mean shifts or non-normal distributions. A process aimed off centre moves the RSS result; the worst case still holds.
- At most 30 dimensions, all in one unit, millimetres or inches.
When to involve a qualified engineer
A stack-up is only as good as its chain. A missing dimension, or a nominal used where the feature is really located from another datum, gives a confident wrong answer. Have a qualified engineer check the chain, the directions and the choice between worst case and RSS before the result sets a tolerance on a drawing.
What the product does not check
In the Technical Preview, TruePart does not check tolerance bands.
Questions
What is the difference between worst case and RSS?
Worst case assumes every dimension sits at its least favourable limit at once, so it is a guaranteed bound. RSS assumes the variations are random and independent, so they partly cancel, and it estimates a narrower range that nearly all assemblies fall in.
When can I use RSS?
When the dimensions are independent, normally distributed, centred in their bands and made by capable processes, and when there are enough of them, usually four or more. Otherwise use the worst case.
How do I enter an asymmetric tolerance?
Choose upper and lower deviation and enter both, signed. The worst case uses the limits directly. RSS first moves the dimension to its mid-limit and uses half the band as its ± value.
Why does the RSS mean differ from the nominal result?
An asymmetric tolerance moves the middle of its band away from the nominal. RSS works about those mid-limits, so the shifts add into the mean.
Sources
- ASME Y14.5-2018, Dimensioning and Tolerancing · ASMEPlus and minus tolerancing and the statistical tolerancing symbol.
- ISO 286-1:2010, Geometrical product specifications (GPS), ISO code system for tolerances on linear sizes, Part 1: Basis of tolerances, deviations and fits · ISOUpper and lower limit deviations and the tolerance band.
What TruePart does today
- LimitIn the Technical Preview, TruePart does not check tolerance bands.