故事 · 当一件可以有很多个毛病
Origin Story · When One Unit Can Carry Many Flaws
p 图把世界看成黑白:每件非合格即不合格。可这套二分法装不下现实 —— 一块印制板上同时有焊桥、虚焊、错件,
一匹布上分布着零星的污渍和断纱。这些"缺陷个数"不是"是 / 否",而是"在一段范围里,稀有事件发生了几次" ——
这正是泊松分布的舞台。于是 Shewhart 体系再添两员:c 图盯每个固定单位(一块板、一米布)的缺陷个数,
u 图盯单位缺陷率(当检验单位大小不一时,把缺陷数除以单位量再比较)。
泊松的招牌特征是均值 = 方差 = λ,标准差就是 √λ,所以控制限简洁得惊人:c̄ ± 3√c̄。
The p-chart paints the world black and white: every unit is either good or defective. Reality refuses that dichotomy — one PCB can have a solder bridge, a cold joint and a wrong component all at once,
and a bolt of fabric carries scattered stains and broken threads. Those "defect counts" aren't yes/no events; they are "how many times did a rare event happen in this region of opportunity?" —
which is exactly the stage the Poisson distribution was built for. So Shewhart's framework added two more charts: the c-chart tracks the count of defects on each fixed-size unit (one PCB, one meter of fabric),
and the u-chart tracks defects per unit (when units vary in size, you divide defect count by the area of opportunity before comparing).
The signature of the Poisson is mean = variance = λ, so the standard deviation is simply √λ — and the control limit shrinks to something startlingly clean: c̄ ± 3√c̄.
1
受控In control2 c / u(泊松)vs p / np(二项):到底哪个 c / u (Poisson) vs p / np (Binomial): Which One Fits?
一句话区分:问的是"一件上有几个缺陷"→ c / u 图(泊松);问的是"一批里有几件不合格"→ p / np 图(二项)。 同一块板有 3 个焊点不良,c 图记"3 个缺陷",p 图只记"1 件不良"。 The litmus test in one breath: asking "how many defects on this unit?" → c / u chart (Poisson); asking "how many defective units in this batch?" → p / np chart (binomial). The same PCB with three bad solder joints is logged by the c-chart as "3 defects" and by the p-chart as just "1 defective unit".
3 现实里的 c / u 图 c / u Charts in the Real World
缺陷计数:每块板的焊点缺陷、每台设备的报警次数、每页文档的错误数,都是 c / u 图的对象。
Defect counting: solder-joint defects per PCB, alarms per machine per shift, errors per page of a document — natural c / u territory.
泊松控制图:缺陷服从泊松、均值=方差=λ,控制限 c̄±3√c̄ 由这一性质直接推出。
Poisson-based charts: defects follow a Poisson with mean = variance = λ, and that single property hands you the limit c̄ ± 3√c̄ for free.
c vs p:数"几个缺陷"用 c/u,数"几件不良"用 p/np —— 选错图,控制限就算错。
c vs p: count "how many defects" with c / u; count "how many defective items" with p / np. Pick the wrong family and the control limits are wrong from the start.
表面瑕疵监控:布匹污点、涂层针孔、玻璃气泡这类按面积/长度计的瑕疵,单位不齐时用 u 图。
Surface-defect monitoring: stains on fabric, pinholes in coatings, bubbles in glass — flaws counted per area or per length. When inspection units differ in size, the u-chart is the right call.
一句话In One Line
属性控制图分两大家族,全看你数的是什么。数"不良品个数"(一件只能不合格一次)→ 二项 → p / np 图。
数"缺陷个数"(一件可以有好几个毛病)→ 泊松 → c / u 图。泊松的精髓是均值 = 方差 = λ,
所以标准差就是 √λ,控制限简洁到 c̄ ± 3√c̄ —— 不需要像二项那样带 (1−p)。c 图与 u 图的区别只在检验单位齐不齐:
单位固定(每块都是标准板)用 c 图盯个数;单位可变(布一卷长一卷短)用 u 图盯每单位的缺陷率,控制限随单位大小 n 起伏。
Attribute control charts split into two families, and the deciding question is simply what are you counting? Counting "defective items" (each unit can fail at most once) → binomial → p / np chart.
Counting "defects" (one unit can carry several) → Poisson → c / u chart. The genius of the Poisson is mean = variance = λ,
so the standard deviation is √λ and the limit shrinks to c̄ ± 3√c̄ — no (1−p) factor like the binomial. The c-chart and the u-chart differ on one thing only: are the inspection units the same size?
Same size (every board is the standard board) → c-chart, count defects directly. Different sizes (fabric rolls vary in length) → u-chart, track defects per unit, with limits that breathe with each unit size n.
常见误用Common Mistakes
把缺陷数当不良品数画 p 图。一件可有多个缺陷就该用 c / u 图(泊松);p 图会把多缺陷的板只记一次,丢信息。
Treating defect counts as defective counts on a p-chart. If one unit can carry several defects, use a c / u chart (Poisson). The p-chart logs a multi-defect board as a single defective and throws away information.
单位大小不一还用 c 图。检验面积/长度不齐时必须用 u 图(除以单位量),否则大单位天然更多缺陷会误判。
Using a c-chart when inspection units differ in size. When the area or length under inspection varies, switch to a u-chart (divide defects by the unit measure). Otherwise larger units will naturally show more defects and be falsely flagged.
缺陷会扎堆还硬套泊松。缺陷聚集(同一批料同一处针孔)导致过度离散时,控制限假性收窄,要改用负二项或分层。
Forcing a Poisson model when defects cluster. Clustered defects (the same lot of material with pinholes in the same spot) overdisperse the data, the Poisson limits collapse too tightly — switch to a negative binomial or stratify before charting.