BIZOMAT MAKEOS · Quality
Tolerance Stack Design (Worst Case, RSS and Predicted Rejects)
Will this assembly gap stay within its limits by worst case, by root-sum-square, and what share of assemblies will fall outside at my process capability?
One-dimensional tolerance stack of up to six dimensions with symmetric tolerances: nominal gap, worst-case limits, root-sum-square (RSS) limits with an inflation factor (as the RSS tolerance stack tool), the statistical spread from each dimension’s process capability index Cp, the predicted share of assemblies outside the required gap assuming normal distributions centred on nominal, and the dimension contributing most.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Dimension 1 | choice | Whether dimension 1 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 1 nominal | mm | Nominal size of dimension 1. |
| Dimension 1 tolerance (±) | mm | Plus-or-minus tolerance of dimension 1. |
| Dimension 2 | choice | Whether dimension 2 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 2 nominal | mm | Nominal size of dimension 2. |
| Dimension 2 tolerance (±) | mm | Plus-or-minus tolerance of dimension 2. |
| Dimension 3 | choice | Whether dimension 3 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 3 nominal | mm | Nominal size of dimension 3. |
| Dimension 3 tolerance (±) | mm | Plus-or-minus tolerance of dimension 3. |
| Dimension 4 | choice | Whether dimension 4 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 4 nominal | mm | Nominal size of dimension 4. |
| Dimension 4 tolerance (±) | mm | Plus-or-minus tolerance of dimension 4. |
| Dimension 5 | choice | Whether dimension 5 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 5 nominal | mm | Nominal size of dimension 5. |
| Dimension 5 tolerance (±) | mm | Plus-or-minus tolerance of dimension 5. |
| Dimension 6 | choice | Whether dimension 6 adds to or subtracts from the gap (or is unused), going round the loop. |
| Dimension 6 nominal | mm | Nominal size of dimension 6. |
| Dimension 6 tolerance (±) | mm | Plus-or-minus tolerance of dimension 6. |
| Requirement | ||
| Smallest gap allowed | mm | Smallest gap the function allows. |
| Largest gap allowed | mm | Largest gap the function allows. |
| RSS inflation factor | × | Factor on the RSS result to allow for non-normal or off-centre processes (often 1.0–1.5). |
| Process capability index of each dimension | Capability of the processes making the parts: tolerance = 3 × Cp × standard deviation. | |
| Largest reject share accepted | % | Largest share of assemblies outside the gap you accept. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Dimension 1
- What it is
- How dimension 1 acts on the gap as you go round the dimension loop: making the gap bigger, making it smaller, or left out of the stack.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Adds to the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Walk the loop in one direction from one side of the gap to the other; dimensions you travel in the gap’s direction add.
What this means: Dimension 1 nominal
- What it is
- The nominal (basic) size of dimension 1 in the loop, from the drawing. Whether it adds or subtracts is set by its direction.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it as a positive size; the direction choice gives the sign. For an unequal tolerance, use the mid-limit size.
What this means: Dimension 1 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 1: half the total band between its upper and lower limits.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value only, not the full band: for limits of 10.1 and 9.9 enter 0.1.
What this means: Dimension 2
- What it is
- How dimension 2 acts on the gap round the loop: adding to it, subtracting from it, or not used.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Keep the same direction of travel round the loop for every dimension.
What this means: Dimension 2 nominal
- What it is
- The nominal (basic) size of dimension 2 in the loop, as a positive size from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For an unequal tolerance, use the mid-limit size and the half band.
What this means: Dimension 2 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 2: half its total tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value, not the full band.
What this means: Dimension 3
- What it is
- How dimension 3 acts on the gap round the loop: adding to it, subtracting from it, or not used.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Keep the same direction of travel round the loop for every dimension.
What this means: Dimension 3 nominal
- What it is
- The nominal (basic) size of dimension 3 in the loop, as a positive size from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For an unequal tolerance, use the mid-limit size and the half band.
What this means: Dimension 3 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 3: half its total tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value, not the full band.
What this means: Dimension 4
- What it is
- How dimension 4 acts on the gap round the loop: adding to it, subtracting from it, or not used.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Keep the same direction of travel round the loop for every dimension.
What this means: Dimension 4 nominal
- What it is
- The nominal (basic) size of dimension 4 in the loop, as a positive size from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 29.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For an unequal tolerance, use the mid-limit size and the half band.
What this means: Dimension 4 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 4: half its total tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.08 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value, not the full band.
What this means: Dimension 5
- What it is
- How dimension 5 acts on the gap round the loop: adding to it, subtracting from it, or not used.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Not used” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose Not used when the loop has fewer dimensions; its size and tolerance are then ignored.
What this means: Dimension 5 nominal
- What it is
- The nominal (basic) size of dimension 5 in the loop, as a positive size from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when dimension 5 is set to Not used.
What this means: Dimension 5 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 5: half its total tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value, not the full band. Ignored when the dimension is not used.
What this means: Dimension 6
- What it is
- How dimension 6 acts on the gap round the loop: adding to it, subtracting from it, or not used.
- Units and format
- Choose one of 3 options from the list: Adds to the gap; Subtracts from the gap; Not used.
- Example
- The form opens with “Not used” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose Not used when the loop has fewer dimensions; its size and tolerance are then ignored.
What this means: Dimension 6 nominal
- What it is
- The nominal (basic) size of dimension 6 in the loop, as a positive size from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when dimension 6 is set to Not used.
What this means: Dimension 6 tolerance (±)
- What it is
- The plus-or-minus tolerance on dimension 6: half its total tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger tolerance widens both the worst-case and the RSS range.
- Get it right
- Enter the ± value, not the full band. Ignored when the dimension is not used.
Requirement
What this means: Smallest gap allowed
- What it is
- The smallest gap the assembly can have and still work, for example the least clearance that avoids rubbing.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A negative value means interference is allowed; it must not exceed the largest gap.
What this means: Largest gap allowed
- What it is
- The largest gap the assembly can have and still work, for example the most end float or play allowed.
- Units and format
- Enter a number in mm: millimetres. The form accepts -10000 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.9 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from the functional requirement, not from the stack result.
What this means: RSS inflation factor
- What it is
- A multiplier on the root-sum-square range to allow for processes that are not normal or not centred on nominal.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts 1 × to 2 ×. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 × as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The RSS range widens in direct proportion; worst case and predicted rejects do not change.
- Get it right
- Use 1 for the plain RSS result; a larger factor gives a more cautious statistical range.
What this means: Process capability index of each dimension
- What it is
- The process capability of the processes making each dimension: the tolerance divided by three standard deviations. One value is used for every dimension.
- Units and format
- Enter a plain number with no unit. The form accepts 0.5 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.33 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher capability narrows the gap spread and lowers the predicted rejects.
- Get it right
- Take it from your process capability studies; a value for a process that drifts off centre overstates the yield.
What this means: Largest reject share accepted
- What it is
- The largest share of assemblies you accept with a gap outside the limits, compared with the predicted reject share.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 50%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.27% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type the percentage itself, not a fraction or parts per million.
Method and formulas
- Gap = Σ (sign × nominal); worst-case half-range = Σ tolerance; RSS half-range = inflation × √Σ tolerance² (as the RSS tolerance stack tool).
- Each dimension σ = tolerance ÷ (3 Cp); gap σ = √Σσ².
- Predicted reject share = Φ((min − gap) ÷ σ) + 1 − Φ((max − gap) ÷ σ), with Φ the normal distribution.
- Example: the gap is 0.5 mm, worst case ±0.28 mm, RSS ±0.14629 mm; predicted rejects 0%.
Limitations
- Linear one-dimensional stack with symmetric tolerances; geometric tolerances and non-linear stacks need more.
- Assumes centred normal processes; real processes drift.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check an assembly gap before releasing a drawing
- Decide whether worst-case or statistical tolerancing is justified
- Find the dimension to tighten first
Who it is for
- Design engineers
- Quality engineers
- Drafters
What you get
- Nominal gap, worst-case and RSS limits
- Statistical spread and predicted reject share
- Largest contributor
Interpreting the result
- Tighten the largest contributor first: it gives the biggest RSS reduction for the cost.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 628–634, 128–130. Topic reference; no source tables or recommended settings are reproduced.
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