故事 · 「全员同时作弊」的概率有多小
Origin Story · How Unlikely Is "Everyone Cheats at Once"?
想象一根由五段杆件首尾相接拼成的长轴。每段都有 ±0.1mm 的加工容差。
总长的误差最大能有多少?最坏情况法说:万一五段全都偏长 0.1mm,总长就偏 0.5mm —— 按这个定容差,绝对不会装不上。
可工艺工程师会反问:五段同时、同方向偏到极限,这概率有多小?
如果每段误差是独立随机的,它们更可能互相抵消:有的偏长、有的偏短,加起来接近于零。
统计学给出答案:当各误差独立时,总误差的标准差是各标准差的平方根和,而不是直接相加。
于是 RSS 法用 √(0.1²×5) ≈ 0.224mm 取代了 0.5mm —— 同样的装配良率,容差却宽松得多,子件加工成本大降。
Picture a long shaft built from five rod segments joined end to end. Each segment carries a ±0.1 mm machining tolerance.
What's the worst possible total-length error? Worst-case says: if all five happen to run +0.1 mm long, the total drifts +0.5 mm — design to that and it'll never refuse to assemble.
But a process engineer pushes back: five segments drifting to the limit, in the same direction, at the same time — how likely is that, really?
If every segment's error is independent, they're far more likely to cancel out: some long, some short, the sum near zero.
Statistics answers cleanly: when errors are independent, the total standard deviation is the root-sum-square of the per-part standard deviations, not the linear sum.
So RSS replaces 0.5 mm with √(0.1²×5) ≈ 0.224 mm — same assembly yield, dramatically looser per-part tolerances, and machining cost drops accordingly.
1 容差堆叠:最坏情况 vs RSS Tolerance Stack-Up: Worst-Case vs RSS
拖子件容差drag part tolerances2 实际装配误差分布(为什么 RSS 够用) Actual Assembly Error Distribution (Why RSS Is Enough)
独立子件误差相加 → 总误差近似正态分布,绝大多数装配体落在中间窄窄一带, RSS 边界(约 3σ)已覆盖 99.7% 的装配。 最坏情况边界远在外侧,那里几乎没有真实装配体 —— 为极小概率付全额保险,太贵。 Independent part errors sum to a total that is approximately normal, with the bulk of assemblies clustered in a narrow band around zero. The RSS limit (≈ 3σ) already covers 99.7% of the population. The worst-case limit sits far outside that, in a zone almost no real assembly ever reaches — paying full insurance for a vanishingly small probability is just expensive.
3 现实里的容差设计 Tolerance Design in the Real World
容差分配:把装配总容差合理拆给各子件,关键件收紧、非关键件放宽。
Tolerance allocation: split the assembly total across components — tighten the critical ones, loosen the rest.
RSS vs 最坏情况:最坏情况绝对安全但贵,RSS 用统计抵消换更经济的容差。
RSS vs worst-case: worst-case is rock solid but expensive; RSS trades a tiny risk for far cheaper tolerances using statistical cancellation.
装配公差:电机轴系、镜头组、堆叠 PCB 等多件叠加场景,RSS 是行业常用法。
Assembly tolerancing: motor shaft trains, lens groups, stacked PCBs — wherever many parts stack, RSS is the industry default.
成本-良率平衡:容差越紧加工越贵,RSS 在可接受良率下把容差放到最经济。
Cost vs yield: tighter tolerances cost more to machine. RSS opens tolerances as far as the target yield allows — the cheapest point that still ships.
一句话In One Line
容差设计是田口三段论的最后一步:参数设计求稳健(不花钱)在前,容差设计精修(花钱)在后。
而 RSS 的核心洞察是 —— 独立随机误差会互相抵消,不会集体作弊。
最坏情况法为「全员同时偏到极限」这个近乎不可能的事件买全额保险,于是把子件容差压得过死、成本飙升;
RSS 承认统计现实,用 √(Σ容差²) 给出既保良率又经济的总容差,子件越多,平方根和相对线性和省得越多。
但 RSS 有前提:各容差近似独立、近似正态、且分布对称居中。
若子件偏度大、强相关,或工艺中心严重偏移,RSS 会低估总容差 —— 这时就得回到更保守的最坏情况法或做修正。
算法是工具,前提才是关键。
Tolerance design is the last step of Taguchi's three-stage logic: parameter design for free robustness first, tolerance design for paid precision last.
The core insight of RSS — independent random errors cancel each other; they don't conspire to cheat in unison.
Worst-case buys full insurance against the near-impossible event of every part drifting to the limit in the same direction, so it crushes per-part tolerances and inflates cost.
RSS accepts statistical reality and uses √(Σ tolerance²) to deliver a total that protects yield while staying economical — the more components you stack, the bigger the saving relative to the linear sum.
But RSS comes with strings attached: tolerances must be roughly independent, roughly normal, and centered with no big bias.
If components are skewed, strongly correlated, or the process mean has drifted off-center, RSS will underestimate the total — fall back to worst-case or apply a shift-corrected variant.
The algorithm is just a tool; the assumptions are what matter.
常见误用Common Mistakes
不管前提一律用 RSS。RSS 要求各容差独立、近似正态、居中对称,否则会低估总容差。
Reaching for RSS without checking the assumptions. RSS needs independent, roughly normal, centered tolerances — otherwise it underestimates the total.
对安全关键尺寸也用 RSS 省料。失效后果严重的关键链宁可保守,用最坏情况法兜底。
Using RSS to "save material" on safety-critical dimensions. For chains where failure is severe, stay conservative — fall back to worst-case for hard guarantees.
忽略工艺中心偏移,只算容差。中心偏移会吃掉余量,需用偏移修正或动态 RSS 评估。
Ignoring process mean shift and computing tolerances only. Off-center drift eats the margin — use shift-corrected (inflation-factor) or dynamic RSS instead.