BIZOMAT MAKEOS · Quality
Statistical (RSS) Tolerance Stack
Will this gap stay within its limits, worst case and statistically?
Worst-case and root-sum-square (RSS) results for a linear stack of up to five dimensions, checked against the required gap limits.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Dimension 1 | ||
| Direction | choice | Whether dimension 1 increases or decreases the gap. |
| Nominal | mm | Nominal size from the drawing. |
| Tolerance ± | mm | Symmetrical ± tolerance; convert unequal tolerances to a mean nominal with equal ±. |
| Dimension 2 | ||
| Direction | choice | Direction of dimension 2. |
| Nominal | mm | Nominal of dimension 2. |
| Tolerance ± | mm | Tolerance of dimension 2. |
| Dimension 3 | ||
| Direction | choice | Direction of dimension 3. |
| Nominal | mm | Nominal of dimension 3. |
| Tolerance ± | mm | Tolerance of dimension 3. |
| Dimension 4 | ||
| Direction | choice | Direction of dimension 4. |
| Nominal | mm | Nominal of dimension 4. |
| Tolerance ± | mm | Tolerance of dimension 4. |
| Dimension 5 | ||
| Direction | choice | Direction of dimension 5, or not used. |
| Nominal | mm | Nominal of dimension 5. |
| Tolerance ± | mm | Tolerance of dimension 5. |
| Requirement and statistics | ||
| Minimum allowed gap | mm | Smallest gap the function allows, from the design requirement. |
| Maximum allowed gap | mm | Largest gap the function allows. |
| RSS inflation factor | × | 1.0 for plain RSS; a larger factor (your organisation’s rule, e.g. Bender’s 1.5) allows for shifted or non-normal processes. |
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.
Dimension 1
What this means: Direction
- What it is
- Whether this dimension in the stack makes the gap bigger (adds) or smaller (subtracts) as it grows, or is not part 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 round the assembly from one side of the gap to the other; dimensions pointing the same way as the gap add.
What this means: Nominal
- What it is
- The basic size of this dimension in the stack, as shown on the drawing, before tolerance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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
- If the tolerance is unequal, use the mid-limit size as the nominal and half the band as the ± tolerance.
What this means: Tolerance ±
- What it is
- The equal plus-or-minus tolerance on this dimension: half its tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 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.
- Effect on the result
- Each tolerance adds directly to the worst case but only as its square to the RSS result, so the largest ones dominate.
- Get it right
- Enter the ± value, not the full band. Convert an unequal tolerance to a mid-limit nominal with an equal ± first.
Dimension 2
What this means: Direction
- What it is
- Whether this dimension in the stack makes the gap bigger (adds) or smaller (subtracts) as it grows, or is not part 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 “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Walk the loop round the assembly from one side of the gap to the other; dimensions pointing the same way as the gap add.
What this means: Nominal
- What it is
- The basic size of this dimension in the stack, as shown on the drawing, before tolerance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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
- If the tolerance is unequal, use the mid-limit size as the nominal and half the band as the ± tolerance.
What this means: Tolerance ±
- What it is
- The equal plus-or-minus tolerance on this dimension: half its tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 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.
- Get it right
- Enter the ± value, not the full band. Convert an unequal tolerance to a mid-limit nominal with an equal ± first.
Dimension 3
What this means: Direction
- What it is
- Whether this dimension in the stack makes the gap bigger (adds) or smaller (subtracts) as it grows, or is not part 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 “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Walk the loop round the assembly from one side of the gap to the other; dimensions pointing the same way as the gap add.
What this means: Nominal
- What it is
- The basic size of this dimension in the stack, as shown on the drawing, before tolerance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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
- If the tolerance is unequal, use the mid-limit size as the nominal and half the band as the ± tolerance.
What this means: Tolerance ±
- What it is
- The equal plus-or-minus tolerance on this dimension: half its tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 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.
- Get it right
- Enter the ± value, not the full band. Convert an unequal tolerance to a mid-limit nominal with an equal ± first.
Dimension 4
What this means: Direction
- What it is
- Whether this dimension in the stack makes the gap bigger (adds) or smaller (subtracts) as it grows, or is not part 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 “Subtracts from the gap” selected as a worked example. Choose the option that matches your case.
- Get it right
- Walk the loop round the assembly from one side of the gap to the other; dimensions pointing the same way as the gap add.
What this means: Nominal
- What it is
- The basic size of this dimension in the stack, as shown on the drawing, before tolerance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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
- If the tolerance is unequal, use the mid-limit size as the nominal and half the band as the ± tolerance.
What this means: Tolerance ±
- What it is
- The equal plus-or-minus tolerance on this dimension: half its tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 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.
- Get it right
- Enter the ± value, not the full band. Convert an unequal tolerance to a mid-limit nominal with an equal ± first.
Dimension 5
What this means: Direction
- What it is
- Whether this dimension in the stack makes the gap bigger (adds) or smaller (subtracts) as it grows, or is not part 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 “Not used” selected as a worked example. Choose the option that matches your case.
- Get it right
- Choose not used when the stack has fewer dimensions; its nominal and tolerance are then ignored.
What this means: Nominal
- What it is
- The basic size of this dimension in the stack, as shown on the drawing, before tolerance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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 the direction is set to not used.
What this means: Tolerance ±
- What it is
- The equal plus-or-minus tolerance on this dimension: half its tolerance band.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 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
- Enter the ± value, not the full band. Ignored when the direction is set to not used.
Requirement and statistics
What this means: Minimum allowed gap
- What it is
- The smallest gap the assembly can accept and still work, from the design requirement. A negative value means a minimum interference.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 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
- Take it from the functional requirement, not from the nominal gap the stack happens to give.
What this means: Maximum allowed gap
- What it is
- The largest gap the assembly can accept and still work, from the design requirement.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 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
- It must not be smaller than the minimum allowed gap.
What this means: RSS inflation factor
- What it is
- A multiplier applied to the plain root-sum-square result to allow for processes that are off-centre or not normally distributed. A value of 1 gives plain RSS.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts 1 × to 3 ×. 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 proportion to this factor; the worst case is unchanged.
- Get it right
- Use the factor your organisation’s tolerancing rule specifies; do not choose it to make a stack pass.
Method and formulas
- Nominal gap = Σ(+ nominals) − Σ(− nominals).
- Worst case = nominal ± Σ tolerances.
- RSS = nominal ± k√(Σ tolerance²), with k the inflation factor.
- Each dimension’s share of variance = tolerance² ÷ Σ tolerance².
- Example: the default stack has a nominal gap of 0.5 mm, worst case ±0.28 mm and RSS ±0.14629 mm.
Limitations
- RSS assumes independent, centred, roughly normal dimensions with tolerances at the same number of standard deviations.
- Linear one-dimensional stack only; angular and geometric tolerances need their own analysis.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check an assembly gap or clearance
- Decide whether statistical tolerancing is justified
- Find the dimension that drives the variation
Who it is for
- Design engineers
- Quality engineers
What you get
- Nominal gap
- Worst-case limits
- RSS limits
- Largest contributor
Interpreting the result
- If only the RSS result fits, the design depends on process control: confirm capability of the main contributors.
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.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Set fits from measured coupons
In Printed Mechanisms That Run
What it covers: Print calibration coupons, measure how holes and shafts really come out, and turn the results into named CAD offsets for press, slip and running fits.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.