起源故事 · 一把尺子统天下
Origin Story · One Ruler to Govern Them All
很多过程没有连续测量值,只有『合格 / 不合格』两态 —— 装配扣合、漏液、外观、软件 bug rate。
1980 年代摩托罗拉推 Six Sigma 时遇到同一难题:管理层既要看注塑尺寸的 Cpk,又要看包装漏液的 ppm,两套语言无法对比。
于是把所有缺陷归一到 ppm/DPMO,再反查标准正态分布的 Z_bench,最后除以 3 得『等价 Cpk』。
这让 Motorola、GE、Allied Signal 的高管可以拿同一张排行榜横扫所有过程 —— Six Sigma 的灵魂。
Many processes never produce a continuous measurement, only pass / fail — clip engagement, leak, cosmetic inspection, software bug rate.
When Motorola launched Six Sigma in the 1980s they hit a single wall: management wanted to see molded-part Cpk
alongside packaging-leak ppm, but the two languages did not compare.
Their fix was to push every defect into ppm/DPMO, invert the standard normal to get Z_bench, and divide by 3 for an equivalent Cpk.
That move let Motorola, GE, and Allied Signal execs rank every process on a single leaderboard — the very soul of Six Sigma.
1 从二分饼到正态尾巴:一次翻译 From Pie Slice to Normal Tail: One Translation
2 速查表:常用不良率对应等价 Cpk Quick Lookup: Common Defect Rates Mapped to Equivalent Cpk
| P_defective | DPMO | Z_bench | 等价 Cpkequivalent Cpk | Sigma Level |
|---|
高亮行 = 当前滑块附近的档位。3.4 ppm = Z=4.5 = 等价 Cpk≈1.5 = Sigma Level 6 —— Six Sigma 名字的由来。 The highlighted row is the band closest to the slider. 3.4 ppm = Z = 4.5 = equivalent Cpk ≈ 1.5 = Sigma Level 6 — exactly where the name "Six Sigma" comes from.
3 现实里的离散过程 Discrete Processes in the Real World
装配扣合:抽 1000 件,8 件扣不上 → P=0.8% → Z_bench≈2.41 → 等价 Cpk ≈ 0.80。
Clip engagement: 8 fails out of 1000 → P = 0.8% → Z_bench ≈ 2.41 → equivalent Cpk ≈ 0.80.
包装漏液:100 ppm 漏液 → Z_bench≈3.72 → 等价 Cpk ≈ 1.24。
Packaging leak: 100 ppm leak rate → Z_bench ≈ 3.72 → equivalent Cpk ≈ 1.24.
视觉检验外观:5% 不良 → Z_bench≈1.64 → 等价 Cpk ≈ 0.55 · 这是『大问题』档位。
Visual cosmetic inspection: 5% reject → Z_bench ≈ 1.64 → equivalent Cpk ≈ 0.55 — squarely in the "big problem" band.
软件 bug rate:0.1% 错误率 → Z_bench≈3.09 → 等价 Cpk ≈ 1.03 · 接近 Cpk=1 临界。
Software bug rate: 0.1% error rate → Z_bench ≈ 3.09 → equivalent Cpk ≈ 1.03 — right on the Cpk = 1 borderline.
一句话In One Line
离散数据也能换算成 Cpk,前提是分布近似正态尾巴。等价 Cpk 让管理层不必『连续看 Cpk、离散看 ppm』两套语言,
统一用 Cpk 排序所有过程。这就是Six Sigma 的统一计量学。
但要明确标注『等价』—— 它不是真 Cpk(没有 σ 这个量),只是同尺度的指标。客户审核报告时,
诚实地写『discrete equivalent Cpk = 1.24 (P = 100 ppm)』比写裸 Cpk 更有说服力。
Discrete data can still be translated into Cpk as long as the tail behaves roughly normal. The equivalent Cpk frees executives
from juggling "Cpk for continuous, ppm for discrete" — every process is ranked on the same Cpk scale.
That is the unified metrology of Six Sigma.
But always label it as "equivalent" — it is not a true Cpk (there is no σ here), just a metric on the same scale.
In customer audits, "discrete equivalent Cpk = 1.24 (P = 100 ppm)" is far more credible than a bare Cpk number.
常见误用Common Mistakes
把等价 Cpk 当真 Cpk 报客户。必须标注『discrete equivalent』或『Z_bench 等价』,客户审核才不会反驳『你的 σ 在哪』。
Reporting equivalent Cpk to the customer as if it were a real Cpk. Always tag it "discrete equivalent" or "Z_bench-equivalent", or the audit team will rightly ask "so where is your σ?"
小样本(N < 300)算等价 Cpk。离散数据 N 至少 300 才能稳定估 P;否则 95% 置信区间会比 P 自身还大。
Computing equivalent Cpk on a small sample (N < 300). Binomial P needs N ≥ 300 for a stable estimate; otherwise the 95% confidence interval is wider than P itself.
等价 Cpk = ln(P) / 3。必须 Φ⁻¹ 反查标准正态分位数,不是对数;ln 法是错的,会算出离谱小的 Cpk。
Equivalent Cpk = ln(P) / 3. The right operator is the inverse standard normal Φ⁻¹, not the natural log; using ln gives wildly understated Cpk values.