故事 · 为什么一个指数不够用
Origin Story · Why One Index Was Never Enough
早期工程师只用 Cp 衡量过程：把规格宽度 (USL − LSL) 除以过程的 6σ 散布，比值越大说明波动相对规格越小。可很快有人发现漏洞 ——
Cp 完全不看分布站在哪。一个 Cp = 2.0 的过程若整体偏向 USL，依旧会有大批产品越界。于是引入 Cpk：它取均值到最近那条规格限的距离再除以 3σ，
把偏心的惩罚写进了公式。后来又有 Pp / Ppk —— 形式一模一样，但用的是涵盖全部变异源的整体（长期）σ，
于是 Cpk 与 Ppk 的差距，就成了判断"过程到底稳不稳"的天然信号。
Early process engineers tracked capability with Cp alone: divide the spec width (USL − LSL) by the 6σ spread, and the bigger the ratio the tighter the process relative to the spec. The hole in that story showed up fast —
Cp says nothing about where the distribution actually sits. A process with Cp = 2.0 that has drifted toward USL will still pump out scrap. So Cpk was added: take the distance from the mean to the nearer spec limit and divide by 3σ,
writing the off-center penalty straight into the formula. Later came Pp / Ppk — identical in shape, but using the overall (long-term) σ that swallows every source of variation,
so the gap between Cpk and Ppk turned into a natural signal for "is this process actually stable?"
1 把分布塞进规格里：偏一点，废品就出来 Fitting the Distribution Inside the Spec: A Small Drift, A Real Scrap Stream
Cp 0.00 / Cpk 0.002 Cp 与 Cpk 的差距，就是偏心的代价 The Gap Between Cp and Cpk Is the Cost of Being Off-Center
两根柱子并排：Cp（潜力）恒定不变，Cpk（现实）随偏心下沉。居中时两者等高，越偏心缺口越大 —— 那道缺口 = 你白白浪费掉的过程潜力。虚线 1.33 是常用合格门槛。 Two bars side by side: Cp (potential) stays put while Cpk (reality) sinks as the mean drifts. They are equal when centered; the further off-center, the wider the gap — and that gap is the process potential you are throwing away. The dashed line at 1.33 is the common pass threshold.
3 现实里的过程能力 Process Capability in the Real World
Cp 看潜力：它只问"波动相对规格够不够窄"，是过程能做到的最好，与对中无关。
Cp is potential: it only asks "is the spread narrow enough relative to the spec?" — the best the process can do, regardless of centering.
Cpk≥1.33：多数行业把 Cpk ≥ 1.33（约 4σ）当作合格线，汽车 / 航空常要求 1.67 甚至更高。
Cpk ≥ 1.33: most industries treat Cpk ≥ 1.33 (roughly 4σ) as the pass bar; automotive and aerospace usually require 1.67 or higher.
居中 vs 偏心：Cp = Cpk 说明已对准中心；Cpk < Cp 说明先调中心比降波动更划算。
Centered vs off-center: Cp = Cpk means the process is already on target. Cpk < Cp means recentering is cheaper than shrinking variation.
短期 vs 长期：Cp/Cpk 用组内 σ（短期潜力），Pp/Ppk 用整体 σ（长期实绩），缺口反映稳定性。
Short term vs long term: Cp/Cpk use within-subgroup σ (short-term potential), Pp/Ppk use overall σ (long-term performance), and the gap between them reflects stability.
一句话In One Line
能力指数本质是在数同一件事：规格留给我的空间，是过程散布的几倍。
Cp 把规格宽度除以 6σ，问的是"理论上我有多宽裕"—— 这是潜力。
Cpk 改成"到最近那条规格线还剩几个 3σ"，于是偏心会被狠狠扣分 —— 这是现实。
至于 Pp / Ppk，公式一样，只是把短期组内 σ 换成长期整体 σ：当过程会漂移、批次有差异，整体 σ 更大，Ppk 就低于 Cpk。
所以三句话记住：Cp 看潜力，Cpk 看现实，Cpk 与 Ppk 的差看稳定。
Every capability index is counting the same thing: how many process spreads fit inside the room the spec gives me?
Cp divides the spec width by 6σ, asking "in theory, how much headroom do I have?" — that is potential.
Cpk rephrases it as "how many 3σ steps remain to the nearer spec limit?", so any off-center drift gets penalized hard — that is reality.
Pp / Ppk share the same algebra but swap short-term within-subgroup σ for long-term overall σ — when the process drifts or batches differ, σ_overall is larger and Ppk falls below Cpk.
Three lines to remember: Cp is potential, Cpk is reality, and the Cpk vs Ppk gap is stability.
常见误用Common Mistakes
只报 Cp 就说过程合格。Cp 高而偏心时照样出废品，必须看 Cpk；只有居中才有 Cp = Cpk。
Calling a process capable from Cp alone. A high Cp with an off-center mean still produces scrap — you must report Cpk too. Only a perfectly centered process gives Cp = Cpk.
把 Cpk 和 Ppk 混着用。Cpk 用组内（短期）σ，Ppk 用整体（长期）σ，混用会高估或低估真实能力。
Mixing Cpk and Ppk interchangeably. Cpk uses within-subgroup (short-term) σ; Ppk uses overall (long-term) σ. Swapping them silently over- or under-states true capability.
数据非正态也硬套公式。偏态 / 多峰数据须先变换或用非正态法，否则废品率估计严重失真。
Applying the standard formulas to non-normal data. Skewed or multi-modal data must be transformed or modeled with a non-normal method first — otherwise the scrap-rate estimate is badly distorted.