起源故事 · Clements & Pyzdek 把诚实放回 Cpk
Origin Story · Clements & Pyzdek Put Honesty Back Into Cpk
1990 年代质量界发现 Cp/Cpk 公式假设的『正态』在很多实际场景根本不成立 —— 表面光洁度、平面度、不圆度这些『几何特征』,
物理上有下限 0 但无上限,天生右偏(对数正态 / 威布尔)。1995 年 Clements 和 Pyzdek 推出 Percentile 法:
不预设分布形态,直接找数据的 0.135% 和 99.865% 分位数(对应正态 ±3σ 的相同尾部面积)。
Minitab 把这条路叫『Box-Cox 法』或『分布拟合法』,今天几乎所有 Six Sigma 软件都把它做成『Non-Normal Capability』菜单。
In the 1990s the quality community realized the normality assumption baked into Cp/Cpk simply does not hold for many real characteristics —
surface finish, flatness, roundness and other geometric features are bounded by zero on the left but open on the right,
and they are inherently right-skewed (lognormal / Weibull). In 1995 Clements and Pyzdek proposed the Percentile method:
forget the shape, just locate the 0.135% and 99.865% quantiles of the data (the same tail areas a normal ±3σ would carve out).
Minitab packages this path as "Box-Cox" or "Distribution Fit", and almost every Six Sigma platform today exposes it under a "Non-Normal Capability" menu.
1 右偏分布:P0.135 与 P99.865 在哪里? A Right-Skewed Distribution: Where Are P0.135 and P99.865?
2 三法对比:到底信谁? Three Methods, One Dataset — Whom Do You Trust?
同一份数据 → 三种结果:蓝 = 强行正态(虚高,骗客户)· 紫 = Percentile 法(诚实,慢但真)· 绿 = Box-Cox 变换后(把数据扭成正态再算)。差值越大 → 数据越偏 → 越不能强套正态。 Same data, three answers: blue = forced normal (inflated, misleads the customer) · purple = Percentile method (honest, slower but truthful) · green = Box-Cox transform (warps the data to normal, then runs classic Cpk). The wider the gap, the more skewed the data, and the less you can trust the normal assumption.
3 现实里的非正态过程 Non-normal Processes in the Real World
表面粗糙度 Ra:永远 ≥ 0 的几何量,强套正态 Cpk 常虚高 30%,客户量产首批就翻车。
Surface roughness Ra: a strictly non-negative geometric measure. Force-fitting normal inflates Cpk by about 30%, and the first mass-production lot blows up at the customer.
客户投诉等待时间:天生右偏(指数 / 对数正态族),必须 Percentile,不然 95% 客户被『平均』掩盖。
Customer-service wait time: inherently right-skewed (exponential / lognormal family). Use percentiles, or 95% of customers will hide behind a comfortable-sounding average.
化学含量近上限:物理硬上限 → 数据被『压』向左 → 左偏,同理不能正态。
Chemical assay near upper limit: a hard physical ceiling presses the data leftward — left-skewed by construction, and equally unsuited to normal-Cpk.
Minitab / JMP:『Non-Normal Capability』菜单背后就是 Percentile 法或 Box-Cox 变换,选项就在那里。
Minitab / JMP: behind the "Non-Normal Capability" menu sit exactly the Percentile method and Box-Cox transform — the option is already there.
一句话In One Line
非正态数据强套 Cpk 等于撒谎 —— 分布右偏 + Cpk 假装居中,客户经理被骗着签单,量产第一批就被退货。
Percentile 法虽然慢(需要大样本 N ≥ 125),但是诚实:它直接问数据『你的 0.135% 与 99.865% 在哪』,
不假设任何形状。第二选项是 Box-Cox 变换,把偏态数据『扭』成正态后再走经典 Cpk 公式 —— 速度快、可解释,
但变换后的 σ 与原数据 σ 已经不在同一尺度,不能拿来直接对比。先正态性检验(AD 或 Shapiro),再决定走哪条路。
Forcing Cpk on non-normal data is lying with numbers — the distribution is skewed, Cpk pretends it is centered,
the buyer signs the PO, and the very first lot ships back as rejects.
The Percentile method is slow (it needs N ≥ 125), but it is honest: it asks the data directly
"where is your 0.135% and where is your 99.865%?" with no shape assumption at all.
The other option is the Box-Cox transform, which warps skewed data into a near-normal shape
and then applies the classic Cpk formula — fast and explainable, but the transformed σ
is not on the same scale as the raw σ, so you cannot compare them directly.
Run a normality test first (Anderson-Darling or Shapiro-Wilk), then pick the path.
常见误用Common Mistakes
偏度 +2 还硬用 (USL − μ) / (3σ)。|Skewness| > 1 必须先做正态性检验(AD test),再决定 Box-Cox 还是 Percentile。
Using (USL − μ) / (3σ) when skewness is +2. Once |skewness| > 1, run a normality test (Anderson-Darling) first, then decide between Box-Cox and Percentile.
N = 30 跑 Percentile 法。Percentile 至少 N ≥ 125 数据;否则 P99.865 估计误差太大,反而比正态假设更不可靠。
Running the Percentile method on N = 30. Percentile estimates need N ≥ 125; otherwise the standard error on P99.865 swamps the result and you end up worse than the normal assumption.
Box-Cox 之后还报『σ』给客户。变换后的 σ 与原数据 σ 不可比;只能换算 Cpk / Ppk,不能换算 σ 本身。
Reporting the post-Box-Cox σ back to the customer. The transformed σ is not comparable to the raw σ; you can only translate Cpk / Ppk back, never the σ itself.