[ T_assembly = \sqrtT_1^2 + T_2^2 + T_3^2 + \dots + T_n^2 ]
Worst-case analysis adds the absolute values of all tolerances. For ten parts, the total possible variation is (10 \times 0.1 = \pm 1.0) mm. To ensure assembly under the worst possible scenario, every part must be manufactured at the extreme edge of its tolerance simultaneously. This is profoundly unlikely. Worst-case forces designers to tighten individual tolerances dramatically—driving up machining, inspection, and rejection costs—to avoid a near-impossible event. It is the engineering equivalent of buying flood insurance for a house on the moon. RSS tolerance analysis replaces the linear sum with a geometric one: rss tolerance analysis
Using the same ten parts with (\pm 0.1) mm, the RSS prediction is: [ T_assembly = \sqrtT_1^2 + T_2^2 + T_3^2
[ T_assembly = \sqrt10 \times (0.1)^2 = \sqrt0.1 \approx \pm 0.316 \text mm ] This is profoundly unlikely
Where (T_i) are the individual tolerances (expressed as standard deviations or half-bands). This formula emerges from a beautiful fact: .