起源故事 · Kotz & Lovelace 把"偏心"做成公式
Origin Story · Kotz & Lovelace Turn "Off-Center" into a Formula
1980 年代 Cp/Cpk 大流行,但工程师们说不清"我的 Cp = 1.6 看着不错,客户为啥还在抱怨"—— 根本原因是均值偏了。
1991 年 Samuel Kotz 与 Cynthia Lovelace 在《Process Capability Indices in Theory and Practice》中
把"居中偏移度" k 系统化:k = |μ − M| / (T/2),
一个 0–1 之间的"偏心率"。从此 Cp 和 Cpk 的差距有了精确的名字:Cpk = Cp × (1 − k)。
这个公式让"调中心(便宜,改一个偏置参数)和降 σ(贵,要换设备/改工艺)"之间的取舍,第一次能算清楚。
Cp / Cpk went mainstream in the 1980s, but engineers couldn't quite explain why "my Cp = 1.6 looks great, yet the customer is still complaining" — the real cause was almost always an off-center mean.
In 1991 Samuel Kotz and Cynthia Lovelace, in Process Capability Indices in Theory and Practice, systematized the
off-center fraction k: k = |μ − M| / (T/2) — an off-center ratio between 0 and 1.
From that point on the Cp-versus-Cpk gap had a precise name: Cpk = Cp × (1 − k).
That single equation finally made the tradeoff between re-centering (cheap — flip a setpoint) and shrinking σ (expensive — new machine, new process) a calculation, not a debate.
1 钟形 + 规格门 + 居中条 Bell Curve + Spec Gates + Centering Bar
完美居中perfectly centered2 Cp 与 Cpk 的差距 = Cp × k The Cp − Cpk Gap = Cp × k
k = 0.2
Cpk = 0.80 × Cp
Cpk = 0.80 × Cp
k = 0.5
Cpk = 0.50 × Cp
Cpk = 0.50 × Cp
k = 0.8
Cpk = 0.20 × Cp
Cpk = 0.20 × Cp
3 现实里的 k k in the Real World
CNC 装刀偏 0.05mm:规格 T/2 = 0.1mm,k = 0.5 → Cpk 立刻砍半。重新对刀比换机床便宜 100 倍。
CNC tool offset 0.05 mm: with T/2 = 0.1 mm, k = 0.5 → Cpk is cut in half on the spot. Re-touching the tool is 100× cheaper than buying a new machine.
注塑温度偏低 3°C:规格 ±10°C,k = 0.3 → Cpk × 0.7。改 PID 设定点 5 分钟搞定。
Injection-mold temp 3 °C low: spec ±10 °C, so k = 0.3 → Cpk × 0.7. Bump the PID setpoint — five minutes of work.
自动化产线门槛:k ≤ 0.1 是基本要求,否则 Cpk 损失超过 10%,客户审核过不去。
Bar for automated lines: k ≤ 0.1 is the baseline. Anything more and Cpk loses over 10% — customer audits will not sign off.
Minitab Capability Sixpack:报告里那个 "k value" 字段,就是这条公式的输出。
Minitab Capability Sixpack: that "k value" field on the report is exactly this formula's output.
一句话In One Line
k 真正回答的是一个钱的问题:调中心能救回多少 Cpk?
假设现在 Cp = 1.6, k = 0.5, Cpk = 0.8 —— 把均值校到 M(k→0),Cpk 立刻冲到 1.6,翻倍。
这种"免费午餐"在精益里叫先消偏置再降变异:偏置可能就是 PID 设定点改一行、对刀仪重对一次、夹具垫片加一片;
而降 σ 往往要换设备、改工艺、培训操作工 —— 成本两个数量级。所以见到 Cp 高 Cpk 低,永远先问 k 是多少。
What k really answers is a money question: how much Cpk can re-centering recover?
Say you have Cp = 1.6, k = 0.5, Cpk = 0.8 — pull the mean back to M (k → 0) and Cpk jumps to 1.6 instantly, a full 2× lift.
In lean we call this kill the bias before you fight the variation — and it really is a free lunch:
the bias might be a single line in a PID setpoint, one tool-touchoff, or a shim in the fixture,
while shrinking σ usually means new equipment, new process windows, and operator re-training — two orders of magnitude more expensive.
So whenever you see high Cp and low Cpk, the first question is always: what is k?
常见误用Common Mistakes
k = (μ − M) / USL。分母是规格半宽 T/2 = (USL − LSL)/2,不是 USL 本身。否则 LSL ≠ 0 时数值毫无意义。
Writing k = (μ − M) / USL. The denominator is the spec half-width T/2 = (USL − LSL)/2, not USL itself. The moment LSL ≠ 0 the wrong formula is meaningless.
k > 1 没问题,还能继续生产。k > 1 意味着均值已超规格边界,Cpk < 0(公式给出负数),平均每件都是不良品,必须停线。
Assuming k > 1 is fine and the line can keep running. k > 1 means the mean is already outside the spec limit, Cpk goes negative, and on average every part is a defect. Stop the line.
把 k 当作偏心方向。k 取绝对值,只反映"偏多远"(程度),不反映"偏哪边"(方向)。方向看 μ − M 的符号,做 SPC 调整时要分开看。
Treating k as a signed direction. k is an absolute value — it tells you how far off, not which side. Read the sign of μ − M for direction; SPC adjustments need both.