起源故事 · 休哈特与 1924 年备忘录
Origin Story · Shewhart and the 1924 Memo
1924 年 5 月 16 日,贝尔实验室年轻物理学家 Walter Shewhart 给老板递上一张一页纸备忘录,
画的就是后来人称"X̄-R 控制图"。他要从"小子组"快速估 σ,但车间统计员算不出复杂的标准差公式——
于是 ASTM 团队推导出极差 R 的期望与 σ 的简洁关系 E[R] = d2·σ,
反过来 σ̂ = R̄/d2。查表 + 一次除法,文化程度有限的女工也能上手。这一招让 SPC 真正走进车间。
On 16 May 1924, a young Bell Labs physicist named Walter Shewhart handed his boss a one-page memo —
the sketch that would become the X̄-R control chart. He needed a way to estimate σ from small subgroups,
but the shop-floor inspectors couldn't crunch the full standard-deviation formula in their heads.
The ASTM committee then derived a clean relation between the expected range R and σ: E[R] = d2·σ,
inverted as σ̂ = R̄/d2. Look up a constant, do one division — even an operator with limited schooling could run it.
That move is what finally brought SPC into the factory.
1 k 个子组 · 每组 n 个点 · 红条 = 组内极差 k Subgroups · n Points Each · Red Bar = Within-subgroup Range
σ̂ ≈ —2 d2 系数对照表(拖左侧 n 看高亮) d2 Constants (drag n on the left to highlight)
n ≤ 10 时 R 法稳健range method is robust for n ≤ 10| n 子组大小n (subgroup size) | d2 | D3(R 下控制限系数)D3 (lower R-chart limit factor) | D4(R 上控制限系数)D4 (upper R-chart limit factor) |
|---|
3 R̄/d2 在车间的真实用法 R̄/d2 in Real Shop-Floor Practice
抽检节奏:质检员每小时抽 5 件量直径,X̄-R 图就用 σ̂ = R̄/d2;d2(n=5)=2.326 是车间最熟数字。
Sampling cadence: an inspector pulls 5 parts per hour to measure diameter — the X̄-R chart uses σ̂ = R̄/d2. d2(n=5) = 2.326 is the most familiar number on the shop floor.
心算速估:R̄ 心算容易,除 2.326 一秒搞定;比 Σ(x−x̄)² 那一坨简单太多,这就是当年它能流行的关键。
Mental-math estimator: R̄ is easy to average in your head, and dividing by 2.326 takes a second. Far simpler than the full Σ(x − x̄)² sum — which is exactly why it took off.
Minitab 默认:X̄-R 用 R̄/d2;X̄-S(大子组)用 s̄/c4(见 PC17)。两条路最后估的都是 σwithin。
Minitab default: X̄-R uses R̄/d2; X̄-S (for larger subgroups) uses s̄/c4 (see PC17). Both routes end up estimating the same σwithin.
SPC 软件内嵌:现代 SPC 软件这些 d2/D3/D4 表全内嵌,你看不到查表,但底层算的就是这套公式。
Baked into SPC software: modern SPC tools embed the d2/D3/D4 tables — you never see the lookup, but the same formula is running under the hood.
一句话In One Line
σ̂within 是"瞬时波动"——只算每组内部,不算组间漂移。
这就是 Cp/Cpk 该用的那个 σ;而 Pp/Ppk 用的是把所有数合并后的 σoverall(下一页)。
两个 σ 的差距,正好反映过程稳不稳定 —— 如果 σwithin ≈ σoverall,过程稳;
如果 σoverall ≫ σwithin,说明组间在偷漂,Cp 漂亮但 Pp 难看,真实客户感受要看 Pp。
σ̂within captures "instantaneous jitter" — it lives inside each subgroup and ignores between-subgroup drift.
That is the σ Cp/Cpk should use; Pp/Ppk uses σoverall, computed from all data pooled together (next page).
The gap between the two σ values is your stability thermometer — if σwithin ≈ σoverall, the process is stable;
if σoverall ≫ σwithin, between-subgroup drift is sneaking in: Cp looks great while Pp tells the truth — and the truth is what the customer feels.
常见误用Common Mistakes
子组大小 n > 10 还用 R̄/d2。R 法只对 n ≤ 10 稳健;大子组要改用 s̄/c4(信息利用率高得多)。
Still using R̄/d2 when n > 10. The range method is only robust for n ≤ 10. For larger subgroups switch to s̄/c4 — it uses each point's information far more efficiently.
d2 记错(常见 n=5 写成 2.236 而非 2.326)。查表别背;n=2 → 1.128,n=5 → 2.326,n=10 → 3.078。
Mis-remembering d2 (a classic: writing n = 5 as 2.236 instead of 2.326). Look it up — don't memorize: n = 2 → 1.128, n = 5 → 2.326, n = 10 → 3.078.
用 R̄/d2 估所有 σ(包括长期)。R̄/d2 只估短期组内 σ;长期 σoverall 要用 √(Σ(x−x̿)²/(N−1))。
Using R̄/d2 for every σ, including long-term. R̄/d2 estimates short-term within-subgroup σ only. For long-term σoverall use √(Σ(x − x̿)² / (N − 1)).