起源故事 · Biometrika 三巨匠
Origin Story · The Three Giants of Biometrika
1924 年 L.H.C. Tippett 在《Biometrika》先推 E[R/σ],给出 n=2 到 n=1000 的极差期望。
1932 年 E.S. Pearson 跟进推 E[s/σ] 的精确积分形式,引入 c4。
1935 年 H.O. Hartley 完成 Var[R/σ] 推导,给出 d3,至此 X̄-R / X̄-S 图所需常数全齐。
1956 年 Western Electric《Statistical Quality Control Handbook》把它们做成附录表,
1992 年 AIAG SPC 手册附录 A 继承同一份表,n=5 时 d2=2.326 成为车间统计员的"圆周率"。
In 1924 L.H.C. Tippett opened the file in Biometrika, deriving E[R/σ] and tabulating expected range from n = 2 all the way to n = 1000.
In 1932 E.S. Pearson followed with the exact integral form of E[s/σ] and introduced c4.
In 1935 H.O. Hartley closed out Var[R/σ] and delivered d3 — and at that point every constant the X̄-R and X̄-S charts need was on the books.
In 1956 Western Electric's Statistical Quality Control Handbook folded the values into a clean appendix table,
and in 1992 the AIAG SPC manual inherited the same table verbatim. d2 = 2.326 at n = 5 became the shop-floor statistician's value of π.
1 d2 / c4 / d3 随 n 的变化 d2 / c4 / d3 vs Subgroup Size n
n=52 常用 n 值对照表(拖左侧 n 高亮对应行) Lookup Table for Common n (drag the slider to highlight the row)
这就是 AIAG SPC 手册附录 A 的核心表。背下 n=5 那一行:d2=2.326,c4=0.9400,d3=0.864 —— 走遍工厂都够用。 This is the heart of AIAG SPC Appendix A. Memorize the n = 5 row: d2 = 2.326, c4 = 0.9400, d3 = 0.864 — that alone carries you through any plant.
3 现实里的 d2 / c4 / d3 d2 / c4 / d3 in the Real World
X̄-R 图 n=5:车间最常见,d2=2.326,A2/D3/D4 都基于这个 d2 算。
X̄-R chart, n = 5: the most common subgroup on the floor. d2 = 2.326, and the A2 / D3 / D4 control-limit factors are all derived from it.
X̄-S 图 n>10:用 s 法更稳,c4(n=20)=0.9869,几乎不用修正。
X̄-S chart, n > 10: the sample-s route is far steadier. c4(20) = 0.9869 — the bias correction is practically negligible.
GR&R n=2:测同一件两次,c4(2)=0.7979,AIAG MSA 手册必备。
GR&R, n = 2: measure the same part twice. c4(2) = 0.7979 — required reading in the AIAG MSA manual.
Minitab/JMP:都内置完整 d2/c4/d3 表,n=200 以内调用零误差。
Minitab / JMP: both ship the full d2 / c4 / d3 tables built-in. Zero round-off error for any n up to 200.
一句话In One Line
d2、c4、d3 不是魔法常数,是小样本统计量期望值的精确积分解 —— 对一个正态总体,
E[R/σ] 是 n 的确定函数,没有任何拟合或近似。
它们的存在让 R̄/d2 和 s̄/c4 成为 σ 的无偏估计,
才让车间统计员可以"抓 5 个零件量一下"就估出过程波动。
AIAG SPC 手册附录 A 一定要背 n=5 时 d2=2.326,这就是车间统计的"圆周率"。
d2, c4 and d3 are not magic constants — they're exact integral solutions for the expected values of small-sample statistics.
For a normal population, E[R/σ] is a closed function of n alone, with no fitting and no approximation involved.
That's what lets R̄/d2 and s̄/c4 serve as unbiased estimators of σ,
and what lets a shop-floor statistician estimate true process spread from "five parts off the line, measure them once".
The one number to commit to memory from AIAG SPC Appendix A: d2 = 2.326 at n = 5 — the π of shop-floor statistics.
常见误用Common Mistakes
把 d2 写成 d2 = n−1 或 (n+1)/2。d2 是积分常数:n=2 → 1.128;n=5 → 2.326;n=10 → 3.078。没有简单闭式。
Approximating d2 with d2 = n − 1 or (n + 1)/2. d2 is an integral constant: 1.128 at n = 2, 2.326 at n = 5, 3.078 at n = 10. No clean closed form exists.
大子组(n > 10)还硬用 d2 法。大样本下极差损失太多信息,应改 c4 法(s̄/c4),方差估计精度高得多。
Forcing the d2 route on a large subgroup (n > 10). Range throws away too much information as n grows. Switch to the c4 route (s̄/c4) — variance estimates are far more precise.
不同教材的 d2 表略有出入就抄一份了事。国标 GB/T 4091 与 AIAG SPC 附录 A 一致,老教材偶有微差,务必查权威源。
Picking up whatever d2 table happens to be in the nearest textbook. China's GB/T 4091 matches AIAG SPC Appendix A; older textbooks occasionally drift in the last decimal. Always source from an authoritative table.