起源故事 · Sheppard 的 Z 表与 Abramowitz Handbook Origin Story · Sheppard's Z Table and the Abramowitz Handbook
1809 年高斯写下钟形曲线,但当时没人能快速算出"−2σ 到 +2σ 的面积"。 1894 年英国统计学家 W.F. Sheppard 第一次把 Φ(z) 累积分布函数列成 4 位精度表,发表在《Philosophical Transactions》。 1950 年代美国国家标准局 Abramowitz & Stegun 编《Handbook of Mathematical Functions》, 其中 Φ(z) 表跨 6 位精度铺满 30 页,成为 20 世纪被引用最多的工程参考书之一。 今天 Excel 的 NORM.S.DIST(z, TRUE)、Minitab 的 PPM 报告、ASQ 黑带题库里的"Cpk=1.5 求 ppm", 全都是查这张表的两侧尾巴。 Gauss wrote down the bell curve in 1809, but at the time nobody could quickly compute the area from −2σ to +2σ. In 1894 the British statistician W.F. Sheppard published the first 4-digit table of the Φ(z) cumulative function in Philosophical Transactions. In the 1950s the U.S. National Bureau of Standards' Abramowitz & Stegun edited the Handbook of Mathematical Functions, whose Φ(z) tables run 30 pages at 6-digit precision — one of the most cited engineering references of the 20th century. Today Excel's NORM.S.DIST(z, TRUE), Minitab's PPM report, and every "Cpk = 1.5, find the ppm" question on an ASQ Black Belt exam are all reading the two tails of that same table.

1 双尾面积 = 不良率 Two-Tail Area = Defect Rate

yield 99.99%+

2 Φ(z) 累积函数 · 查表过程 Φ(z) Cumulative Function · How the Table Is Read

把 Z_USL、−Z_LSL 投影到 Φ(z) 曲线,纵坐标就是面积。 上尾 = 1 − Φ(Z_USL),下尾 = Φ(−Z_LSL)。这就是查 Z 表的几何意义。 Project Z_USL and −Z_LSL onto the Φ(z) curve — the y-coordinate is the tail area. Upper tail = 1 − Φ(Z_USL), lower tail = Φ(−Z_LSL). That is the geometric meaning of "looking up the Z table".

3 现实里的 % Defective % Defective in the Real World

Minitab Capability:直接报告 PPM < LSL / PPM > USL / PPM Total,就是这两段尾巴的积分。 Minitab Capability: reports PPM < LSL / PPM > USL / PPM Total directly — those are the integrals of the two tails.
半导体多步良率:每步 Φ 单独算,再连乘 —— 100 步每步 99.99% = 总良率 99.0%,复利效应惊人。 Multi-step semiconductor yield: compute Φ per step, then multiply — 100 steps at 99.99% each = 99.0% overall. The compounding effect bites hard.
ASQ 黑带必考:"Cpk=1.5 求 ppm",答 3.4 ppm(含 1.5σ 漂移)或 6.8 ppm(不含),都靠查 Φ 表。 ASQ Black Belt classic: "Cpk = 1.5, find the ppm." Answer 3.4 ppm (with the 1.5σ shift) or 6.8 ppm (without). Both come straight from the Φ table.
Excel 一键算:=NORM.S.DIST(z, TRUE) 就是 Φ(z),不用查表了。 One-cell Excel: =NORM.S.DIST(z, TRUE) is Φ(z) — no more flipping through paper tables.
一句话In One Line
% Defective 是把 工程语言 σ 翻译成客户语言 ppm/良率的最后一公里。 经典换算表(双侧居中、无 1.5σ 漂移): Cpk 1.00 → 2700 ppm(99.73%)· Cpk 1.33 → 63 ppm(99.9937%)· Cpk 1.67 → 0.6 ppm · Cpk 2.00 → 0.002 ppm。 这串数字,就是和销售经理、品质总监开会时真正能听懂的语言。 % Defective is the last mile that translates σ as engineers speak it into ppm and yield as customers hear it. The classic two-sided lookup (centered process, no 1.5σ shift): Cpk 1.00 → 2,700 ppm (99.73%) · Cpk 1.33 → 63 ppm (99.9937%) · Cpk 1.67 → 0.6 ppm · Cpk 2.00 → 0.002 ppm. That short string of numbers is the only language that actually lands in a meeting with sales and the head of quality.
常见误用Common Mistakes
单侧规格也用双侧公式(比如表面粗糙度只有 USL)。单边规格只算一边尾巴,否则 ppm 会算大一倍。 Using the two-sided formula on a one-sided spec (e.g. surface roughness with USL only). A one-sided spec gets one tail only; otherwise you double-count the ppm.
非正态分布直接套 Φ 公式。偏态/双峰分布要先做 Box-Cox / Johnson 变换,或改用 Percentile 法(见 PC21)。 Slapping the Φ formula onto a non-normal distribution. Skewed or bimodal data needs a Box-Cox or Johnson transform first, or switch to the percentile method (see PC21).
ppm 直接乘良率算成品率。良率 = 1 − ppm / 1,000,000,别乘错。多步串联时是良率连乘,不是 ppm 相加。 Multiplying ppm by yield to get the rolled throughput. Yield = 1 − ppm / 1,000,000 — don't mix the two. Multi-step lines compound yields, they don't sum ppm.

% Defective = Φ(−Z_LSL) + (1 − Φ(Z_USL))