Dimensional Chain Tolerance Calculator

Estimate worst-case, root-sum-square, and one-sided probabilistic tolerance for a three-contributor dimensional chain.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Tolerance-stack inputs

One independent normal mode. Each positive tolerance is a centred ±3σ half-width.

Tolerance analysis

Results use millimetres and preserve the staged archived rounding.

One-sided probabilistic tolerance—mm

Combined σA— mm

Reliability factor k—

Worst-case Aworst— mm

Three-sigma ARSS— mm

Probabilistic results depend on independent, centred, normal contributors. Use the worst-case sum when that statistical model is not justified.

Equations used

All tolerance contributions are positive additive magnitudes in millimetres. Each is interpreted as a centred ±3σ half-width.

σᵢ = Tᵢ / 3, i ∈ {B,C,D}

σA = √(σB² + σC² + σD²)

Aworst = round₃(TB + TC + TD)

ARSS = round₃(3σA) = round₃(√(TB² + TC² + TD²))

p = 1 − R(A); q = 0.5 − |p − 0.5|; w = √[−2 ln(q)]

z = sgn(p − 0.5){w − (2.30753 + 0.27061w)/[1 + w(0.99229 + 0.04481w)]}

k = −round₃(z); Aprob = −round₄(σA z)

Symbols: Tᵢ is a component tolerance half-width; σᵢ is its standard deviation; σA is combined chain scatter; R(A) is one-sided reliability; p is lower-tail probability; z approximates Φ⁻¹(p); k is the positive reliability factor; and Aprob is the reported probabilistic tolerance magnitude.

Theory and method

For independent normal variables, variances add. The worst-case result instead adds all magnitudes, while the RSS result reports three combined standard deviations. The rational inverse-normal approximation maps the selected one-sided reliability to a quantile. The model assumes centred independent normal contributors, linear additive geometry, and ±3σ input definitions. It excludes correlation, process mean shifts, asymmetric limits, thermal drift, nonlinear geometry, selective assembly, and safety factors.

References

NIST/SEMATECH: normal distribution.
Abramowitz and Stegun, Handbook of Mathematical Functions, formula 26.2.23.

How to use

Use the single independent-normal mode only for three additive centred deviations represented by ±3σ half-widths.
Enter TB, TC, and TD in mm, then enter one-sided R(A) greater than 0.5 and less than 1.
Select Calculate or edit an input to update all results. Select Reset to restore the worked example.
Read Aprob as a one-sided statistical tolerance magnitude. Compare it with the three-sigma RSS and conservative worst-case results.
Correct any warning before interpretation. Do not use this model for correlated, shifted, asymmetric, non-normal, nonlinear, differently sized, or safety-critical stacks.

Worked example

For TB = 0.1 mm, TC = 0.2 mm, TD = 0.2 mm, and R(A) = 0.99, σA = 0.100000 mm, k = 2.328, Aworst = 0.500 mm, ARSS = 0.300 mm, and Aprob = 0.2328 mm. Under the model, about 99% of the one-sided chain deviation lies below 0.2328 mm.

Interpretation and limits

Aprob is not an assembly-yield guarantee. Verify the signed tolerance loop and distribution assumptions with process data. Use a correlated analytical or Monte Carlo model when contributors interact or their means are off target.

Probabilistic fit and dimensional tolerance analysis Continue in TriboSolver

Use TriboSolver or another validated simulation workflow to refine larger, correlated, or nonlinear tolerance stacks.