起源故事 · 当数据只能"数"不能"量" Origin Story · When Data Can Only Be Counted, Not Measured
1924 年 Shewhart 的控制图最初服务于连续测量(X̄-R 图)。可现实里大量质量数据是属性的:合格还是不合格、有没有划痕、几个气泡。 这些数据没法取平均,只能计数。于是衍生出四张计数图。分水岭在两件事:① 数不良品(一件要么合格要么不合格,服从二项分布,用 p/np) 还是 数缺陷(一件上可以有多个缺陷,服从泊松分布,用 c/u);② 样本量恒不恒定(恒定用 np/c,可变用 p/u)。选对图,控制限才算得准。 Shewhart's 1924 control chart was born for continuous measurements (the X̄-R chart). But a huge chunk of real-world quality data is attribute in nature: pass or fail, scratched or not, three bubbles or none. You can't average that — you can only count it. Four attribute charts emerged. Two forks decide everything: ① Are you counting defective items (each item is pass or fail — binomial distribution, use p/np) or defects (one item can carry several — Poisson distribution, use c/u)? ② Is the sample size constant (np/c) or varying (p/u)? Pick the right chart and your control limits actually mean something.

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2 四种图的控制限公式,凭什么不同 Why the Four Charts Have Different Formulas

高亮行是当前选中的图。p/np 用二项方差(含 1−p̄ 项),c/u 用泊松方差(方差 = 均值,无 1−p 项)。样本量 n 越大,控制限越窄 —— 抽得越多,越能分辨真异常。 The highlighted row is your current chart. p/np use binomial variance (carries the 1−p̄ term); c/u use Poisson variance (variance = mean, no 1−p term). The larger n gets, the tighter the limits — more sampling means sharper resolution of real anomalies.

3 现实里的计数型控制图 Attribute Charts in the Real World

出货不良率(p 图):每批抽检件数不一,监控不合格品占比,是质检最常用的图。 Outgoing defective rate (p chart): batches sampled at different sizes; track the proportion non-conforming. The everyday workhorse of incoming/outgoing QC.
每批不良数(np 图):固定每批检 200 件,直接盯不良"个数",现场最直观。 Defectives per batch (np chart): inspect a fixed 200 pcs every time and track the raw count — the most shop-floor-friendly view.
单板缺陷数(c 图):每块电路板大小一样,数它上面的焊点缺陷总数。 Defects per PCB (c chart): every board is identical in size, so just count the solder defects on each — apples to apples.
每米缺陷率(u 图):布匹、钢卷长度不等,折算成"每单位长度的缺陷数"再比。 Defects per meter (u chart): fabric rolls and steel coils have unequal lengths, so normalize to "defects per unit length" before comparing.
一句话In One Line
连续型控制图(X̄-R)盯的是"尺寸偏了多少",计数型控制图盯的是"出了多少个不合格"。后者的关键不是公式难,而是先分清你在数什么: 不良品(整件判定,二项分布,p/np)还是缺陷(一件可多处,泊松分布,c/u)。 再看样本量恒不恒定 —— 恒定就能用"个数"(np/c),可变就必须折算成"率"(p/u),否则点之间没法比。 选对图,控制限才对;选错图,整张图白做。 Variables charts (X̄-R) ask "how far off the dimension is"; attribute charts ask "how many came out non-conforming". The hard part isn't the formula — it's naming what you're counting: defective items (whole-item pass/fail, binomial, p/np) or defects (a single item can carry several, Poisson, c/u)? Then check if sample size is fixed — if it is, you can plot raw counts (np/c); if it varies, you must convert to a rate (p/u), otherwise the points aren't comparable. Right chart = right limits. Wrong chart = the whole SPC effort is wasted.
常见误用Common Mistakes
把缺陷当不良品,套 p 图。一件可有多个缺陷时用泊松的 c/u 图,别用二项的 p/np。 Treating defects as defectives and forcing a p chart. When one item can carry multiple defects, use Poisson-based c/u — not binomial p/np.
样本量变了还用 np/c。样本量可变必须折算成率(p/u),控制限才随 n 自动调整。 Sticking with np/c after sample size starts varying. Variable sample size demands a rate (p/u) so the limits flex with n.
把 LCL 算成负数还画出来。计数下限不能为负,LCL 小于 0 时直接取 0。 Plotting a negative LCL. Counts can't go below zero — when the formula gives a negative LCL, floor it at 0.

计数型控制图