起源故事 · Shewhart 与 ±3σ
Origin Story · Shewhart and the ±3σ Line
1809 年高斯(Gauss)发表《天体运动论》,写下正态分布积分。一百多年后的 1924 年 5 月 16 日,贝尔实验室的
Walter A. Shewhart 给上司 George Edwards 递了一张备忘录,附了张图——后世称为统计过程控制图。
他选 ±3σ 当上下控制限,不是因为这条线在数学上"对称漂亮",而是因为:
① 千次抽样平均误报 2.7 次,车间统计员可以接受;
② 异常一旦出现,必定显著大于 3σ,能可靠捕获。
这个"99.73%"后来被 Juran、Deming 写进质量管理教科书,成了 Cp = (USL−LSL)/(6σ) 公式分母的来源—— 所谓"六西格玛",本质就是这条 Shewhart 当年画的线。 In 1809 Gauss published Theoria Motus and wrote down the integral of the normal distribution. A century later, on May 16, 1924, Bell Labs' Walter A. Shewhart handed his boss George Edwards a one-page memo with a sketch — what the world now calls the statistical process control chart. He picked ±3σ for the upper and lower control limits, not because the line was "mathematically symmetric and pretty", but because:
① 2.7 false alarms per 1000 samples — a rate the shop-floor statistician could live with;
② any real anomaly would land well past 3σ and be caught reliably.
That "99.73%" was later written into the quality bibles by Juran and Deming, and it became the denominator in Cp = (USL − LSL) / (6σ) — "Six Sigma" itself is, at heart, the line Shewhart drew that day.
① 千次抽样平均误报 2.7 次,车间统计员可以接受;
② 异常一旦出现,必定显著大于 3σ,能可靠捕获。
这个"99.73%"后来被 Juran、Deming 写进质量管理教科书,成了 Cp = (USL−LSL)/(6σ) 公式分母的来源—— 所谓"六西格玛",本质就是这条 Shewhart 当年画的线。 In 1809 Gauss published Theoria Motus and wrote down the integral of the normal distribution. A century later, on May 16, 1924, Bell Labs' Walter A. Shewhart handed his boss George Edwards a one-page memo with a sketch — what the world now calls the statistical process control chart. He picked ±3σ for the upper and lower control limits, not because the line was "mathematically symmetric and pretty", but because:
① 2.7 false alarms per 1000 samples — a rate the shop-floor statistician could live with;
② any real anomaly would land well past 3σ and be caught reliably.
That "99.73%" was later written into the quality bibles by Juran and Deming, and it became the denominator in Cp = (USL − LSL) / (6σ) — "Six Sigma" itself is, at heart, the line Shewhart drew that day.
1 钟形 + ±kσ 阴影:覆盖与尾部 Bell Curve + ±kσ Shading: Coverage and Tails
k=3 · 99.73%2 k = 1~6 对照表 k = 1 to 6 — Reference Table
| k | 覆盖率Coverage | 两侧尾部 ppmTwo-tail ppm | 门宽 = 2kσ 时 CpCp when gate = 2kσ | 车间含义Shop-floor meaning |
|---|
注:±6σ 含 1.5σ 长期漂移修正后 ≈ 3.4 ppm(见 18-shift-1.5-sigma),此处是纯短期正态尾部。 Note: ±6σ with the 1.5σ long-term shift correction lands at ≈ 3.4 ppm (see 18-shift-1.5-sigma). The numbers above are pure short-term normal tails.
3 不同 σ 的钟形塞进同一规格门(LSL=−3, USL=+3) Bells of Different σ Inside the Same Spec Gate (LSL = −3, USL = +3)
门宽 = 6σ 时 Cp = 1gate = 6σ → Cp = 16 个钟形(σ=0.5/1/1.5/2/2.5/3)放在同一规格门里。σ=1 那条恰好钟全宽 = 6σ = 门宽,Cp=1.00;σ 越小钟越瘦,Cp 越大。 Six bells (σ = 0.5 / 1 / 1.5 / 2 / 2.5 / 3) inside the same gate. The σ = 1 curve has full width 6σ exactly equal to gate width — Cp = 1.00. Smaller σ → leaner bell → higher Cp.
4 现实里的 ±3σ 与 6σ ±3σ and 6σ in the Real World
SPC 控制图:UCL/LCL 永远用 μ±3σ,全行业一致标准——换 ±2σ 误报太多,换 ±4σ 又错过真异常。
SPC control chart: UCL/LCL is always μ ± 3σ — the universal standard. ±2σ raises too many false alarms; ±4σ misses real anomalies.
6σ 战略:要求 Cp ≥ 2.0(门宽 12σ),相当于钟塞进门后两侧还各留 3σ 缓冲。
The Six Sigma initiative: requires Cp ≥ 2.0 (gate width 12σ) — the bell fits inside the gate with a 3σ buffer on each side.
USP <905>:美国药典含量均匀度标准内嵌 6σ 思想,限值大致按 ±3σ 设。
USP <905>: the US Pharmacopeia content-uniformity standard bakes 6σ thinking right in — limits are set roughly at ±3σ.
ASQ 黑带:题库永远在考"3σ 外多少 ppm"——2700、63、0.57,这三个数必须背熟。
ASQ Black Belt: exams forever ask "ppm beyond 3σ?" — memorize 2700, 63, 0.57. No shortcuts.
一句话In One Line
6σ 这个分母不是迷信,是 1924 年 Shewhart 留给后人的工程妥协。
99.73% 覆盖率刚好让"日常波动"和"异常"分得开——少一点会误报,多一点会漏报。
所有 Cp/Cpk/Pp/Ppk 公式的分母 6σ,都是这条线的延伸:
· Cp ≥ 1 = 钟形刚塞进规格门;
· Cp ≥ 1.33 = 钟两侧各留 0.33σ 缓冲(汽车一类件最低要求);
· Cp ≥ 2.0 = 真正"六西格玛级",门宽是钟宽的两倍。 The 6σ denominator is not folklore — it's Shewhart's 1924 engineering compromise, handed down. 99.73% coverage is exactly enough to separate "daily jitter" from "real anomaly" — a hair less means false alarms, a hair more means missed signals. Every Cp / Cpk / Pp / Ppk denominator is an extension of that single line:
· Cp ≥ 1 — the bell just fits inside the spec gate;
· Cp ≥ 1.33 — 0.33σ buffer on each side of the bell (automotive critical-part minimum);
· Cp ≥ 2.0 — true "Six Sigma grade", gate is twice as wide as the bell.
· Cp ≥ 1 = 钟形刚塞进规格门;
· Cp ≥ 1.33 = 钟两侧各留 0.33σ 缓冲(汽车一类件最低要求);
· Cp ≥ 2.0 = 真正"六西格玛级",门宽是钟宽的两倍。 The 6σ denominator is not folklore — it's Shewhart's 1924 engineering compromise, handed down. 99.73% coverage is exactly enough to separate "daily jitter" from "real anomaly" — a hair less means false alarms, a hair more means missed signals. Every Cp / Cpk / Pp / Ppk denominator is an extension of that single line:
· Cp ≥ 1 — the bell just fits inside the spec gate;
· Cp ≥ 1.33 — 0.33σ buffer on each side of the bell (automotive critical-part minimum);
· Cp ≥ 2.0 — true "Six Sigma grade", gate is twice as wide as the bell.
常见误用Common Mistakes
改用 ±2σ 当控制限,"早点抓到异常"。±2σ 千次误报 46 次,车间受不了;Shewhart 早算过这笔账。
Switching to ±2σ control limits to "catch anomalies sooner". ±2σ gives 46 false alarms per 1000 — the shop floor won't stand for it. Shewhart already ran the math.
把分母 6σ 改成 4σ,让 Cp 看起来更高。分母一改和全行业基准对不上,客户审核直接打回。
Swapping 6σ for 4σ in the denominator to puff Cp up. Change the denominator and you can't compare against any industry benchmark — the customer auditor sends it back on sight.
认为 ±3σ 必然包 99.73%,套到任何数据上。只在正态前提下成立;偏态/重尾分布另算(见 PC21 非正态能力)。
Assuming ±3σ always covers 99.73%, no matter the data. Only holds under normality. Skewed or heavy-tailed distributions need their own treatment (see PC21 non-normal capability).