故事 · 为什么用极差除以一个常数
Origin Story · Why Divide the Range by a Constant
1924 年贝尔实验室的 Walter Shewhart 发明控制图时,车间没有计算器 —— 让工人算标准差(要平方、开方)几乎不可能。但算极差(最大减最小)人人会做。
统计学家于是证明:对正态数据,子组极差的期望 = d2 × σ,这里 d2 是只跟子组大小 n 有关的常数(n=2 时 1.128、n=5 时 2.326)。
于是 σ̂ = R̄ / d2 成了车间最实用的估计法。后来计算变方便了,又有了用平均标准差的版本 σ̂ = s̄ / c4,
其中 c4 修正了"样本标准差天生偏小"的偏差。两个常数都是为了让简单的车间统计量无偏地还原出 σ。
When Walter Shewhart invented the control chart at Bell Labs in 1924, the shop floor had no calculators — asking an operator to square, sum, and square-root a column of numbers was a non-starter. But anyone could compute the range (max − min).
So statisticians proved a small miracle: for normal data, the expected subgroup range equals d2 × σ, where d2 is a constant that depends only on the subgroup size n (1.128 for n = 2, 2.326 for n = 5).
That made σ̂ = R̄ / d2 the most practical σ estimator on the production floor. Once computation got cheap, the standard-deviation version followed: σ̂ = s̄ / c4,
where c4 corrects the well-known fact that the sample standard deviation is biased downward. Both constants exist for one reason — to let a simple shop-floor statistic recover σ without bias.
1 每组取极差,平均后除以 d2 还原 σ Take Each Subgroup's Range, Average, Divide by d2 → σ
n = 52 子组越大,d2 / c4 越大;短期 vs 长期 σ Larger Subgroups → Larger d2 / c4; Short-term vs Long-term σ
左:d2 随子组 n 增大而上升(黄点是当前 n)。右:短期 σ(组内)与 长期 σ(整体)两根柱子 —— 漂移越大,长期柱越高,缺口就是过程的不稳定。 Left: d2 rises monotonically with subgroup size n (yellow dot marks the current n). Right: two bars — short-term σ (within-subgroup) vs long-term σ (overall). The more drift, the taller the long-term bar — and the gap is your instability.
3 现实里的 σ 估计 σ Estimation in the Real World
控制限:X̄-R 图的 ±3σ 控制限就是用 R̄ / d2 估的短期 σ 算出来的,所以 A2、D3、D4 常数都内含 d2。
Control limits: the ±3σ limits on an X̄-R chart are built from the short-term σ estimated via R̄ / d2 — that's why the A2, D3, D4 constants all carry d2 inside them.
d2 / c4 常数表:n≤8 多用极差法(R̄ / d2),n 较大时标准差法(s̄ / c4)更精确。
d2 / c4 lookup tables: use the range method (R̄ / d2) for n ≤ 8 and the standard-deviation method (s̄ / c4) for larger subgroups, where it's more efficient.
短期 vs 长期:Cp / Cpk 用短期 σ(潜力),Pp / Ppk 用长期 σ(实绩),二者比值反映稳定性。
Short-term vs long-term: Cp / Cpk use the short-term σ (potential capability), Pp / Ppk use the long-term σ (actual performance) — the ratio is a stability index.
1.5σ 漂移:六西格玛常假设长期比短期漂移 1.5σ,本质就是长期 σ > 短期 σ 的经验折算。
The 1.5σ shift: Six Sigma's classic 1.5σ long-term shift is, at heart, an empirical adjustment for the fact that long-term σ tends to exceed short-term σ.
一句话In One Line
关键在"在哪个尺度上量波动"。组内只看同一时刻、同一批的几个点 —— 机器、材料、操作几乎没变,量到的是过程的固有波动(短期 σ)。
整体把几天、几批全搅在一起 —— 还混进了刀具磨损、批次差异、班次切换的漂移,量到的是长期 σ。
d2 和 c4 只是把"极差""样本标准差"这些简单量翻译回真正的 σ 的换算系数(与 n 有关)。
所以一句话:短期 σ 量过程能做到多稳,长期 σ 量它实际有多稳,差距就是改进的空间。
The whole question is at what scale you measure variation. Within-subgroup looks at a handful of points from the same instant, same batch — machine, material, operator barely changed — so you capture the process's inherent jitter (short-term σ).
Overall pools everything across shifts and batches — now you also pick up tool wear, batch-to-batch difference, shift changeovers — and that's long-term σ.
d2 and c4 are just conversion factors (depending on n) that translate simple shop-floor statistics — range and sample standard deviation — back into σ.
So in one line: short-term σ measures how steady the process can be, long-term σ measures how steady it actually is, and the gap is your room for improvement.
常见误用Common Mistakes
把全部数据合起来算 σ 当控制限。控制限要用组内短期 σ(R̄ / d2),否则会把组间漂移塞进控制限、漏掉真异常。
Pooling all data into one σ for the control limits. Control limits must use the within-subgroup short-term σ (R̄ / d2); otherwise drift gets baked into the limits and real special causes slip through.
用错子组大小对应的 d2 / c4。d2、c4 是 n 的函数,查表必须对准当前子组大小,否则 σ 估计系统性偏差。
Pulling d2 / c4 from the wrong row of the table. d2 and c4 are functions of n — look up the row that matches your actual subgroup size, or σ̂ inherits a systematic bias.
子组划分不合理(跨班次 / 跨批)。子组要在"同一条件"内取样,混入漂移会让短期 σ 失去"固有波动"的含义。
Subgroups that straddle shifts or batches. A subgroup must be sampled under the "same conditions" — let drift leak in and the short-term σ stops representing inherent variation.