故事 · 当"偷懒的极差"不够用了
Story · When the Lazy Range Runs Out of Road
极差法 (X̄-R) 之所以流行,是因为极差好算 —— 当年车间没计算器,最大减最小谁都会。但它有个隐患:极差只看两端两个数,
子组越大,中间被忽略的信息越多,对真实波动的估计就越不稳、效率越低。统计学家算过:n=5 时极差还能拿到约 95% 的"效率",
到 n=10 以上就掉得明显。于是当自动化采样让大子组变得容易,工程师改用标准差 s —— 它吃进组内每个点,是更精确的波动度量。
公式骨架完全不变,只是把 R̄ / d2 那套换成基于 s̄ 的 A3 / B3 / B4 常数(内含 c4 的无偏修正)。
The X̄-R chart became popular because range is easy to calculate — back when shop floors had no calculators, max-minus-min was something anyone could do. But it has a hidden cost: range looks only at the two extremes.
As the subgroup grows, more middle points are ignored and the estimate of true variation becomes noisier and less efficient. Statisticians measured it: at n = 5 range still retains about 95% efficiency, but by n = 10 it drops noticeably.
So once automated sampling made large subgroups easy, engineers switched to standard deviation s — every point in the subgroup contributes, making it a much more precise measure of spread.
The skeleton of the formula is unchanged; the R̄ / d2 machinery is swapped for the s̄-based constants A3 / B3 / B4 (which carry the c4 unbiasing correction inside).
1 X̄ 图盯中心,S 图盯波动,出界即报警 X̄ Watches the Center, S Watches the Spread — Any Breach Is an Alarm
受控in control2 子组越大,极差法效率掉得越多 The Larger the Subgroup, the More Efficiency Range Bleeds
横轴是子组大小 n,纵轴是极差法相对标准差法的相对效率(≈ 用了多少信息)。n 小时两者接近, n > 10 后极差法效率明显下滑 —— 黄点是当前 n,落在虚线右侧就该改用 X̄-S 图。 Horizontal axis: subgroup size n. Vertical axis: the relative efficiency of the range method against the standard-deviation method (roughly how much of the available information it uses). At small n the two are nearly tied, but range efficiency drops sharply once n > 10 — the amber dot marks the current n. If it sits right of the dashed line, switch to the X̄-S chart.
3 现实里的 X̄-S 图 The X̄-S Chart in the Real World
大子组 SPC:自动化采样、计量数据量大时(n≥10),X̄-S 图是首选,波动估计更精确。
Large-subgroup SPC: when automated sampling produces lots of variables data (n ≥ 10), X̄-S is the chart of choice — its spread estimate is much more precise.
A3 / B3 / B4:控制限常数内含 c4 无偏修正,n 大时 B3 才大于 0(S 图才有非零下限)。
A3 / B3 / B4 constants: these limit constants embed the c4 unbiasing correction. Only at sufficiently large n does B3 rise above zero — only then does the S chart get a non-zero lower limit.
vs X̄-R:原理与判读完全一致,只是波动度量从极差换成标准差;小子组仍常用 R 图(更简单)。
vs X̄-R: the logic and interpretation are identical — only the spread metric changes from range to standard deviation. Small subgroups still favor the R chart for its simplicity.
先看 S 图:和 R 图一样,S 图不受控则波动失稳,X̄ 控制限不可信,判读顺序先 S 后 X̄。
Read S first: just like with R, if the S chart is out of control the variation itself is unstable, so the X̄ limits can't be trusted. Always interpret S before X̄.
一句话In One Line
X̄-S 与 X̄-R 是同一套思想的两个版本:都用子组、都靠 ±3σ 控制限分辨常见原因与特殊原因、都是 X̄ 盯中心 + 第二张图盯波动。
唯一的差别是"怎么量组内波动":极差 R 只取两端,简单但浪费信息;标准差 s 吃进每个点,精确但要多算几步。
当子组小,两者几乎等效,极差胜在省事;当子组大,标准差的精度优势压倒一切 —— 这就是"n > 10 用 S"的全部道理。
常数从 A2/D3/D4 换成 A3/B3/B4,只是把 σ 的估计源从 R̄/d2 换成 s̄/c4,骨架分毫未动。
X̄-S and X̄-R are two versions of the same idea: both use subgroups, both lean on ±3σ limits to separate common cause from special cause, and both pair an X̄ chart for the center with a second chart for the spread.
The only real difference is "how within-group variation is measured": range R takes only the extremes — simple but wasteful, while standard deviation s uses every point — precise but a few more arithmetic steps.
For small subgroups the two are essentially equivalent and range wins on convenience. For large subgroups, standard deviation's precision advantage dominates — that's the whole reason behind the "n > 10, use S" rule.
Swapping the constants from A2/D3/D4 to A3/B3/B4 simply changes the σ estimator from R̄/d2 to s̄/c4. The skeleton is untouched.
常见误用Common Mistakes
大子组还硬用极差 R 图。n > 10 时极差只看两端、估计粗糙,应改用 X̄-S 把每个点都用上。
Forcing the R chart onto a large subgroup. Once n > 10, range looks only at the two extremes and the estimate is coarse — switch to X̄-S so every point gets used.
用错 n 对应的 A3 / B3 / B4。这些常数都是子组大小的函数,查表必须对准 n;B3 在 n 较小时为 0。
Using the wrong A3 / B3 / B4 for your n. These constants are functions of subgroup size — always look them up for the actual n. B3 is exactly 0 at small n.
S 图失控也照算 X̄ 控制限。S 图不稳 → s̄ 不可靠 → X̄ 限失真,必须先让 S 图受控再判 X̄。
Computing X̄ limits while the S chart is out of control. Unstable S → unreliable s̄ → distorted X̄ limits. Bring the S chart into control first, then read X̄.