故事 · 从厚厚的 Z 表到一行计算
Story · From a Thick Z Table to a One-Line Calc
上世纪的质量工程师,桌上都压着一张标准正态分布表(Z 表):要算"超过 10.5 mm 的比例",
先把 x 减 μ 除以 σ 换成 Z,再翻表查面积。表是死的,但思路一点没变 —— 概率,就是曲线下的面积。
今天计算器、Excel 的
NORM.DIST 一行就出结果,但你必须先想清楚要的是哪块面积:
左尾(小于某值的比例)、右尾(超规格的不良率),还是中间区间(落在公差内的合格率)。
想清楚要哪块面积,剩下的只是让曲线替你积分。这页就是那张 Z 表的会动的现代版。
Last century's quality engineers all kept a standard normal table (Z table) under the desk glass: to find "the share above 10.5 mm,"
you'd subtract μ, divide by σ to convert x into Z, then look up the area in the table. The table was static, but the idea hasn't budged — probability is the area under the curve.
Today a calculator or Excel's NORM.DIST gives the same answer in one line, but you still have to be clear about which area you want:
the left tail (share below some value), the right tail (out-of-spec defect rate), or the central interval (yield inside tolerance).
Once you've picked the area, the curve does the integration for you. This page is the live, moving version of that Z table.
1 拖一刀,曲线替你把面积算出来 Drag a Cutoff and the Curve Integrates for You
左尾left tail2 百分比刻度:合格 vs 超规格 Percentage Bar: In-Spec vs Out-of-Spec
把同一个概率画成一条 100% 的横条:高亮段是你关注的那部分(合格率或不良率),剩下的是它的补集。面积 概率 比例,三者是同一件事的三种说法。 The same probability drawn as a 100% horizontal bar: the highlighted segment is the piece you care about (yield or defect rate), the rest is its complement. Area probability share — three names for the same thing.
3 现实里的概率计算 Probability Calcs in the Real World
超规格比例:尺寸超出 USL 的右尾面积,就是这批零件的预估不良率,直接换算成 PPM。
Out-of-spec share: the right-tail area beyond the USL is the expected defect rate for the batch — convert it straight into PPM.
分位数:给 P 算 x —— 设计公差时反问"覆盖 99% 的产品需要多宽的范围",得到上下限。
Quantiles: P → x — when designing tolerances, ask "how wide a window covers 99% of parts?" and the curve hands you the upper and lower limits.
Z 表的现代版:当年翻表查的累积概率,如今拖一下边界就有,原理还是"化成 Z、查面积"。
The modern Z table: the cumulative probabilities people once looked up in a table now appear the moment you drag the cutoff. Same idea — convert to Z, read the area.
不良率估算:双侧超规格 = 左尾 + 右尾,两块面积相加,是 Cpk 与 DPMO 的计算基础。
Defect-rate estimate: two-sided out-of-spec = left tail + right tail, summed. This is the foundation underneath Cpk and DPMO.
一句话In One Line
一句话记住:概率就是面积。正态曲线下的总面积永远是 1,你要的任何概率,
都是从这块"1"里切出来的一片。给 x 算 P 是把边界画好、量出面积;
给 P 算 x 是反过来 —— 先要一块指定大小的面积,再问它的边界落在哪。
前者回答"有多少件会超差",后者回答"公差该定多宽"。它们是同一条曲线、同一块面积的正反两看,
也是过程能力、不良率 PPM、置信区间共同的底层语言。
One line to remember: probability is area. The total area under the normal curve is always 1, and every probability you ever want
is a slice carved out of that 1. x → P draws the boundary first, then measures the area;
P → x reverses it — start with an area of a given size, then ask where its boundary lands.
The first answers "how many parts will fall out of spec?"; the second answers "how wide should the tolerance be?" Same curve, same area, two directions —
and the shared underlying language of capability, defect-rate PPM, and confidence intervals.
常见误用Common Mistakes
左尾右尾搞反,把合格率当成不良率。先想清楚要的是"小于"还是"超过",再读对应那块面积。
Flipping left and right tails and reading yield as defect rate. Be clear whether you want "less than" or "greater than" before you read the area off the curve.
数据非正态还硬套正态概率。先做正态性检验(上一页),偏态数据的尾部面积会算偏。
Applying normal probabilities to non-normal data. Run the normality test from the previous page first — skewed data throws the tail area off.
把 Z 分数直接当成概率。Z 是"几个 σ",要经过曲线积分才变成面积/概率。
Treating the Z-score itself as a probability. Z is "how many σ" — only after integrating the curve does it become area, and therefore probability.