起源故事 · 高斯与天文测量误差
Origin Story · Gauss and Astronomical Error
1809 年,德国数学家 Carl Friedrich Gauss 在研究天文测量误差时发现:
一组观测无论怎么重复,误差总是对称地围着真实值,大部分集中在中间,极端值少见 ——
这就是"正态分布"的雏形,而 σ 标准差是它的胖瘦尺。
从此一个钟形分布只需要两个数(μ 中心、σ 胖瘦)就锁定一切,这把尺子后来变成了过程能力公式分母里的 6σ。
In 1809, the German mathematician Carl Friedrich Gauss was puzzling over astronomical measurement error.
No matter how many times he repeated an observation, the errors clustered symmetrically around the true value —
most near the middle, extremes rare. That was the seed of the normal distribution, and σ became its spread meter.
From then on, two numbers (μ for center, σ for spread) were enough to describe an entire bell — and that ruler later became the 6σ in every capability denominator.
1 钟形的胖瘦:±1σ / ±2σ / ±3σ The Bell's Spread: ±1σ / ±2σ / ±3σ
σ = 2.02 把 σ 的计算走一遍 Compute σ Step by Step
n = 303 现实里的 σ σ in the Real World
钻孔深度:σ = 0.02mm,精密加工。±3σ 才 0.06mm,客户图纸 ±0.1mm 绝对够用。
Drilled hole depth: σ = 0.02 mm, precision machining. ±3σ = 0.06 mm, well inside the customer's ±0.1 mm spec.
班次产量:σ = 50 件/天,日均 500。波动小,产能稳定,适合签固定交付合同。
Shift output: σ = 50 pcs/day, mean 500. Low jitter, capacity steady — safe to sign fixed delivery contracts.
客服响应:σ = 30s,均值 90s。±2σ 内 [30s, 150s],服务体验可预期。
Customer-service response: σ = 30 s, mean 90 s. ±2σ keeps replies in [30 s, 150 s] — predictable service.
烤箱均温:同一炉 σ = 2°C,均值 180°C。±3σ 仅 6°C 跨度,烘焙工艺可重复。
Oven uniformity: σ = 2 °C across one batch, mean 180 °C. ±3σ spans just 6 °C — baking is reproducible.
一句话In One Line
σ 量的是日常的诚实波动,不是最大值最小值。
Cp = (USL − LSL) / 6σ 这个公式假设过程是正态的 —— σ 越小,过程门越窄,Cp 越高。
但 σ 不能脱离规格独立看:σ 大不一定不合格,σ 小也不一定 Cp 高,必须和 USL − LSL 一起比。
下一页讲 ±3σ 为什么是 99.73% —— 那个 6σ 的分母从哪儿来。
σ measures honest day-to-day jitter, not max-minus-min.
Cp = (USL − LSL) / 6σ assumes a normal process — smaller σ → narrower bell → higher Cp.
But σ never lives alone: big σ isn't always bad, small σ isn't always Cp-friendly — always compare it against USL − LSL.
Next page: why ±3σ covers 99.73% — the origin of that 6σ denominator.
常见误用Common Mistakes
用极差(Max − Min)代替 σ。极差只看两端,σ 看全员;5 个数和 5000 个数极差差很大,σ 才稳定可比。
Using range (Max − Min) in place of σ. Range sees only the extremes; σ accounts for every point. Range explodes from n = 5 to n = 5000 — σ stays comparable.
样本方差用整体 n 求。样本要用 n − 1(贝塞尔修正),否则系统性低估真实波动。
Dividing sample variance by n instead of n−1. Always use n − 1 (Bessel's correction) for samples — otherwise you systematically underestimate the true spread.
σ 大就一定不合格。σ 大但公差也大照样 Cp 高;σ 必须和 USL − LSL 比,绝对值毫无意义。
Assuming "big σ = no good". A big σ can still hit a high Cp if the tolerance is wide. Sigma is meaningless without USL − LSL alongside it.