What is the difference between Worst-Case (WC) and Root Sum Square (RSS) tolerance analysis?

Worst-Case analysis arithmetically sums the maximum extreme tolerances of every component in the loop, assuming all parts simultaneously hit their most unfavorable size. Root Sum Square (RSS) analysis is a statistical method assuming parts follow a normal distribution; it takes the square root of the sum of squared tolerances, yielding a more realistic and cost-effective assembly tolerance band.

When should I use Worst-Case tolerance stack-up versus RSS?

Use Worst-Case for low-volume production, mission-critical/safety-critical assemblies, or short tolerance chains (fewer than 4 components) where any out-of-spec condition causes catastrophic failure. Use RSS for high-volume manufacturing (Six Sigma, automotive, consumer electronics) with 4+ components where statistical process control is enforced.

How is assembly defect rate (DPMO) calculated from RSS?

By assuming standard ±3σ capability corresponds to the RSS half-band tolerance (T_rss), the assembly gap standard deviation is σ = T_rss / 3. Standard normal CDF Z-scores are computed against the target minimum and maximum gaps, giving the statistical assembly yield percentage and defects per million opportunities (DPMO).

Quality & Tolerancing

ASME Y14.5 / ISO 1101 / Six Sigma

PRO Suite

Tolerance Stack-Up Analysis PRO (Worst Case & RSS Statistical)

Evaluate 1D tolerance loops, calculate arithmetic Worst Case (WC) and Root Sum Square (RSS) statistical boundaries, process capability shifts (Cpk / k-factor), assembly yield percentages, and defect rates (DPMO) per ASME Y14.5-2018 and ISO 1101.

1D Closed-Loop Tolerance Stack-Up Synthesis

Input Parameters

Chain Elements	3links
Unit System	METRICmm
Bender Factor (k)	1.00x
Design Target	[0.050, 0.450]mm

Calculated Results

Mean Centroid	-2.85mm KEY
RSS Band (±3σ)	0.19±mm
Worst Case (WC)	0.3±mm
Assembly Yield	0%

Safety Notes

Estimated assembly reject rate is 100.00% (1,000,000 DPMO). Consider tightening high-variance contributor tolerances.

1D Tolerance Loop Chain Elements

Define vector links in the closed loop path (+Add vs -Sub)

Nominal (L)

+ Upper Tol

- Lower Tol

Cpk Capability

Nominal (L)

+ Upper Tol

- Lower Tol

Cpk Capability

Nominal (L)

+ Upper Tol

- Lower Tol

Cpk Capability

Calculators.ToleranceStackup.results.auditTitle

Calculators.ToleranceStackup.results.nominalCentroid-2.8500mm

Calculators.ToleranceStackup.results.rssTolerance±0.1871mm

Calculators.ToleranceStackup.results.wcTolerance±0.3000mm

Calculators.ToleranceStackup.results.estimatedYield0.00%1,000,000 DPMO

Calculators.ToleranceStackup.results.statisticalRSSLimits

Max Absolute-2.6629 mm

Min Absolute-3.0371 mm

Calculators.ToleranceStackup.results.worstCaseLimits

Max Absolute-2.5500 mm

Min Absolute-3.1500 mm

Calculators.ToleranceStackup.results.contributionTitle

Calculators.ToleranceStackup.results.contributionDesc

Bolt Length Under Head±0.15 (- Sub)

64.3%

Cover Plate Thickness±0.1 (+ Add)

28.6%

Washer Thickness±0.05 (+ Add)

7.1%

Calculators.ToleranceStackup.results.distributionTitle

Calculators.ToleranceStackup.results.distributionDesc

PDF

Nominal Mean

RSS Limits (±3σ)

Target / Reject

Verification Calculation Log

1Convert All Tolerances to Centered Bidirectional Equivalent
- •Element 1: Bolt Length Under Head
- •Direction : [-]
- •Centroid (C): 14.8500 mm
- •Half-Band(t): ±0.1500 mm
- •Element 2: Cover Plate Thickness
- •Direction : [+]
- •Centroid (C): 10.0000 mm
- •Half-Band(t): ±0.1000 mm
- •Element 3: Washer Thickness
- •Direction : [+]
- •Centroid (C): 2.0000 mm
- •Half-Band(t): ±0.0500 mm
2Calculate Assembly Nominal Centroid
- •Summing (Direction × Centroid) for all elements:
- •C_stack = -2.8500 mm
3Worst Case (Arithmetic) Stack-Up Analysis
- •Assuming all parts simultaneously reach absolute maximum bounds:
- •T_wc (Sum of t_i) = ±0.3000 mm
- •Absolute Minimum Assembly = -3.1500 mm
- •Absolute Maximum Assembly = -2.5500 mm
4Statistical / Root Sum Square (RSS) Stack-Up
- •Base RSS Tolerance Band (Sum of t_i^2)^0.5:
- •T_rss = ±0.1871 mm
- •RSS Minimum Assembly Gap = -3.0371 mm
- •RSS Maximum Assembly Gap = -2.6629 mm
5Modified RSS Analysis (Process Capability Shift)
- •Accounting for Cpk variation and Shift Factor (k=1):
- •T_rss_mod = ±0.1407 mm
- •Modified RSS Min Gap = -2.9907 mm
- •Modified RSS Max Gap = -2.7093 mm
6Yield Rate & Defect Probability (assuming ±3σ = T_rss)
- •Target Minimum Gap = 0.0500 mm
- •Target Maximum Gap = 0.4500 mm
- •Z-Score Lower = -46.503
- •Z-Score Upper = 52.918
- •Estimated Yield Rate = 0.0000%
- •DPMO (Defects / Million) = 1000000.0

Mathematical Models & Equations

Centered Mean Dimension (Centroid)

C_i = L_i + \frac{T_{i, \text{upper}} + T_{i, \text{lower}}}{2}

Centered Symmetrical Half-Tolerance Band

t_i = \frac{T_{i, \text{upper}} - T_{i, \text{lower}}}{2}

Assembly Mean Gap Centroid

C_{\text{stack}} = \sum_{i=1}^N \left(D_i \cdot C_i\right)

Worst-Case Arithmetic Stack Tolerance

T_{\text{wc}} = \sum_{i=1}^N t_i, \quad L_{\text{wc}} = C_{\text{stack}} \pm T_{\text{wc}}

Root Sum Square (RSS) Statistical Tolerance (±3σ)

T_{\text{rss}} = \sqrt{\sum_{i=1}^N (t_i)^2}, \quad L_{\text{rss}} = C_{\text{stack}} \pm T_{\text{rss}}

Modified RSS with Process Capability & Inflation Factor

T_{\text{rss, mod}} = k \cdot \sqrt{\sum_{i=1}^N \left(\frac{t_i}{C_{pk, i}}\right)^2}

Assembly Yield & Normal CDF Defect Probability

Z = \frac{\text{Limit} - C_{\text{stack}}}{\sigma_{\text{assy}}}, \quad \text{Yield} = \Phi(Z_{\text{upper}}) - \Phi(-Z_{\text{lower}})

Engineering DisclaimerEngineering calculations are provided for preliminary design and educational/reference purposes only. Users must verify all results according to applicable engineering standards, supplier data, manufacturing requirements, and professional engineering judgment.