Tolerance Stack Calculator
Calculate worst-case and statistical tolerance stack-up for assembled parts. Enter values for instant results with step-by-step formulas.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
RSS = sqrt(t1^2 + t2^2 + ... + tn^2)
Where RSS = Root Sum Square tolerance, t1 through tn are individual part tolerances. Worst-case simply sums all tolerances. The RSS method assumes normal distributions and independent dimensions, producing a statistically likely total tolerance that is smaller than worst-case.
Worked Examples
Example 1: Four-Part Linear Stack-Up
Problem:Four parts with nominals 10, 15, 20, 12 mm and tolerances +/-0.05, 0.08, 0.10, 0.06 mm are assembled. Calculate worst-case and RSS total tolerance.
Solution:Nominal total = 10 + 15 + 20 + 12 = 57 mm\nWorst-case = 0.05 + 0.08 + 0.10 + 0.06 = 0.29 mm\nRSS = sqrt(0.05^2 + 0.08^2 + 0.10^2 + 0.06^2) = sqrt(0.0025 + 0.0064 + 0.0100 + 0.0036) = sqrt(0.0225) = 0.15 mm\nRSS reduction = (1 - 0.15/0.29) x 100 = 48.3%
Result:Assembly: 57 +/- 0.29mm (worst-case) or 57 +/- 0.15mm (RSS, 48.3% tighter)
Example 2: Six-Sigma Stack-Up for Precision Assembly
Problem:Three precision parts with tolerances +/-0.02, 0.03, 0.025 mm need 6-sigma quality. What is the statistical tolerance?
Solution:RSS = sqrt(0.02^2 + 0.03^2 + 0.025^2) = sqrt(0.0004 + 0.0009 + 0.000625) = sqrt(0.001925) = 0.04389 mm\n6-sigma tolerance = 0.04389 x (6/3) = 0.04389 x 2 = 0.08778 mm\nWorst-case = 0.075 mm\nNote: 6-sigma is larger than worst-case, showing extra margin for process variation.
Result:3-sigma: +/-0.044mm | 6-sigma: +/-0.088mm | Worst-case: +/-0.075mm
Frequently Asked Questions
What is tolerance stack-up analysis and why is it important?
Tolerance stack-up analysis is the process of calculating the cumulative effect of individual part tolerances on the overall assembly dimension. When multiple parts are assembled together, each part contributes its own dimensional variation, and these variations can add up to create a total assembly variation that is much larger than any single part tolerance. This analysis is critical for ensuring that assembled products function correctly, parts fit together properly, and quality standards are met. Without stack-up analysis engineers risk creating designs that cannot be reliably manufactured or assembled.
What is the difference between worst-case and RSS tolerance analysis?
Worst-case analysis assumes all parts are simultaneously at their extreme tolerance limits, giving the maximum possible variation in the assembly. This is the most conservative approach but is statistically unlikely for assemblies with more than a few parts. RSS (Root Sum Square) analysis takes a statistical approach, calculating the square root of the sum of squared tolerances. RSS assumes tolerances follow a normal distribution and that it is statistically improbable for all parts to be at their extremes simultaneously. RSS typically produces a total tolerance 30 to 60 percent smaller than worst-case, allowing tighter assembly specifications without tightening individual part tolerances.
When should I use worst-case versus statistical tolerance analysis?
Use worst-case analysis when the assembly is safety-critical, when the number of parts in the stack is small (fewer than four), when 100 percent conformance is required with no rejects, or when production volumes are low. Use statistical (RSS) analysis when the assembly has many contributing dimensions, when a small percentage of out-of-tolerance assemblies is acceptable, when production volumes are high enough for statistical behavior, and when tightening individual tolerances would be too costly. Many engineers use both methods: worst-case to understand the absolute extremes and RSS to set realistic manufacturing specifications.
What does the sigma level mean in tolerance analysis?
The sigma level represents the number of standard deviations used in the statistical tolerance calculation and directly determines the expected yield or conformance rate. At 3-sigma (the most common default), approximately 99.73 percent of assemblies will be within specification. At 4-sigma the conformance rises to 99.9937 percent, and at 6-sigma it reaches 99.99966 percent or about 3.4 defects per million opportunities. Higher sigma levels result in larger calculated tolerance zones, meaning tighter individual part tolerances are needed to achieve the desired assembly quality level. The appropriate sigma level depends on production volume and acceptable defect rates.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy