What does SPC Control Chart Formula Sheet · LSS.WIKI explain?

A tiered reference sheet for X-bar and R, X-bar and S, I-MR, attribute charts, control-chart constants, EWMA, and CUSUM.

Where should I continue learning?

Continue through the related NEWMETH topic pages or open the corresponding workspace tool.

FH06 · REFERENCE SHEET

SPC 控制图公式总表 SPC Control Chart Formula Sheet

把计量型与计数型控制图、n=2–10 控制图常数、Nelson 规则、EWMA 与 CUSUM 放在同一张速查表中。自上而下重要度递减：先背 T1，再熟练 T2，最后按需查 T3。 One working sheet for variable and attribute charts, n=2–10 constants, Nelson rules, EWMA, and CUSUM. Importance drops as you move down: memorize T1, work fluently with T2, and pull T3 when the monitoring problem calls for it.

覆盖范围 Coverage

T1 → T3

核心 SPC 工具链 core SPC stack

必背 · 先把控制限和过程波动配对 Memorize · Pair Control Limits with the Right Variation Estimate

4 张核心卡 4 core cards

T1-1

控制限通式 General Control-Limit Form

F31 深潜 → F31 deep dive →

UCL/LCL = statistic centerline ± 3·σ̂(statistic)

读作：中心线加减三个标准差——出界就是特殊原因的信号。Read it as: The centerline plus or minus three standard deviations—crossing a limit signals a special cause.

UCL / LCL 上控制限／下控制限 upper and lower control limits
3σ Shewhart 3σ 原则 the Shewhart three-sigma principle

何时用： Use it when: 从统计量的中心线和标准误建立 Shewhart 控制限。 You need Shewhart limits from a statistic's centerline and standard error.

⚠ ✗ 把控制限当规格限判断产品合格 → ✓ 控制限判断过程稳定，规格限判断产品符合要求。 ✗ Use control limits as product specifications → ✓ Use control limits for process stability and specs for conformance.

T1-2

X̄-R 图 X̄-R Chart

F31 深潜 → F31 deep dive →

X̄ chart: x̿ ± A₂R̄
R chart: UCL = D₄R̄, LCL = D₃R̄

读作：均值图的界用 A₂R̄ 撑开；极差图的界用 D₃、D₄ 缩放。Read it as: Open the mean-chart limits with A₂R̄; scale the range-chart limits with D₃ and D₄.

x̿ / R̄ 子组均值总平均／子组极差平均 grand mean of subgroup means and average subgroup range
A₂, D₃, D₄ 由子组大小 n 决定的控制图常数 control-chart constants determined by subgroup size n

何时用： Use it when: 连续数据按小型合理子组采集，需要同时监控均值与组内波动。 Continuous data arrive in small rational subgroups and both mean and within-subgroup variation must be monitored.

⚠ ✗ 只看 X̄ 图而忽略 R 图失控 → ✓ 先确认 R 图稳定，再解释 X̄ 图信号。 ✗ Read the X̄ chart while the R chart is out of control → ✓ Establish range-chart stability before interpreting mean-chart signals.

T1-3

I-MR 图 I-MR Chart

F33 深潜 → F33 deep dive →

I chart: x̄ ± 2.66·M̄R
MR chart: UCL = 3.267·M̄R
2.66 = 3/d₂(n=2)

读作：单值图拿相邻差的平均乘 2.66 当三个 σ。Read it as: Use 2.66 times the average adjacent range as the individuals chart's three-sigma width.

M̄R 相邻单值移动极差的平均值 average moving range between adjacent observations
2.66 / 3.267 移动极差长度为 2 时的控制限常数 control-limit constants for a moving range of length 2

何时用： Use it when: 连续数据只能逐个获得，无法形成合理子组。 Continuous measurements arrive one at a time and rational subgroups are not available.

⚠ ✗ 有天然子组仍拆成单值画 I-MR → ✓ 保留组内信息并改用 X̄-R 或 X̄-S。 ✗ Break natural subgroups into individuals for an I-MR chart → ✓ Preserve within-subgroup information with X̄-R or X̄-S.

T1-4

σ 的控制图估计 Control-Chart Estimate of σ

F30 深潜 → F30 deep dive → PC15 深潜 → PC15 deep dive →

σ̂ = R̄/d₂ or s̄/c₄

读作：极差均值除以 d₂，或标准差均值除以 c₄。Read it as: Divide the average range by d₂, or the average standard deviation by c₄.

R̄ / d₂ 基于平均极差的组内 σ 估计 within-subgroup σ estimate from average range
s̄ / c₄ 基于平均标准差的组内 σ 估计 within-subgroup σ estimate from average standard deviation

何时用： Use it when: 从合理子组估计控制图所需的短期过程波动。 You need short-term process variation estimated from rational subgroups.

⚠ ✗ 不看 n 就套 d₂ 或 c₄ → ✓ 按子组大小查对应常数后再估计 σ̂。 ✗ Drop in d₂ or c₄ without checking n → ✓ Select the constant for the subgroup size before estimating σ̂.

常用 · 常数、属性图与判异规则 Working Set · Constants, Attribute Charts, and Signal Rules

5 张常用卡 5 working cards

T2-5

X̄-S 图 X̄-S Chart

F32 深潜 → F32 deep dive →

X̄ chart: x̿ ± A₃s̄
S chart: UCL = B₄s̄, LCL = B₃s̄

s̄ 子组标准差的平均值 average subgroup standard deviation
A₃, B₃, B₄ 由子组大小 n 决定的控制图常数 control-chart constants determined by subgroup size n

何时用： Use it when: 连续数据按较大子组采集；子组 n≥9 时优于 R 图。 Continuous data arrive in larger subgroups; at n≥9, S uses the data more efficiently than R.

T2-6

控制图常数表 Control-Chart Constants

PC17 常数 → PC17 constants →

n=2–10: d₂ A₂ D₃ D₄ c₄ A₃ B₃ B₄

n	d₂	A₂	D₃	D₄	c₄	A₃	B₃	B₄
2	1.128	1.880	—	3.267	0.7979	2.659	—	3.267
3	1.693	1.023	—	2.574	0.8862	1.954	—	2.568
4	2.059	0.729	—	2.282	0.9213	1.628	—	2.266
5	2.326	0.577	—	2.114	0.9400	1.427	—	2.089
6	2.534	0.483	—	2.004	0.9515	1.287	0.030	1.970
7	2.704	0.419	0.076	1.924	0.9594	1.182	0.118	1.882
8	2.847	0.373	0.136	1.864	0.9650	1.099	0.185	1.815
9	2.970	0.337	0.184	1.816	0.9693	1.032	0.239	1.761
10	3.078	0.308	0.223	1.777	0.9727	0.975	0.284	1.716

何时用： Use it when: 按子组大小 n 查 X̄-R 与 X̄-S 图的精确常数。“—”表示无下限，LCL 取 0。 Look up the exact X̄-R and X̄-S constants by subgroup size n. An em dash means there is no lower limit, so LCL is 0.

T2-7

p 图与 np 图 p and np Charts

F34 深潜 → F34 deep dive →

p chart: p̄ ± 3·√[p̄(1−p̄)/n_i]
np chart: np̄ ± 3√[np̄(1−p̄)]

p / np 不良品率／不良品数 fraction nonconforming and count nonconforming
n_i 第 i 个子组大小；p 图控制限随它伸缩 size of subgroup i; p-chart limits expand and contract with it

何时用： Use it when: 每件只有合格／不合格结果；样本量变化用 p 图，固定样本量可用 np 图。 Each unit is conforming or nonconforming; use p when sample size varies and np when it stays fixed.

T2-8

c 图与 u 图 c and u Charts

F35 深潜 → F35 deep dive →

c chart: c̄ ± 3√c̄
u chart: ū ± 3√(ū/n_i)

c / u 固定机会数的缺陷数／单位机会缺陷数 defect count at fixed opportunity and defects per unit of opportunity
n_i 第 i 个样本的检查单位或机会量 inspection-unit or opportunity size for sample i

何时用： Use it when: 缺陷数近似服从泊松分布；机会量固定用 c 图，变化用 u 图。 Defect counts are approximately Poisson; use c for fixed opportunity and u when opportunity varies.

T2-9

Nelson 8 条判异规则 Nelson's Eight Signal Rules

F37 深潜 → F37 deep dive →

1 beyond 3σ · 9 same side · 6 rising/falling · 14 alternating · 2/3 beyond 2σ · 4/5 beyond 1σ · 15 within 1σ · 8 beyond 1σ

规则 1 · 超界：1 点落在 3σ 之外。 Rule 1 · Out of limits: 1 point beyond 3σ.

规则 2 · 偏移：连续 9 点落在中心线同一侧。 Rule 2 · Shift: 9 in a row on the same side of the centerline.

规则 3 · 趋势：连续 6 点持续递增或递减。 Rule 3 · Trend: 6 in a row steadily increasing or decreasing.

规则 4 · 震荡：连续 14 点上下交替。 Rule 4 · Oscillation: 14 in a row alternating up and down.

规则 5 · 2/3 贴 2σ：连续 3 点中有 2 点落在同侧 2σ 之外。 Rule 5 · 2 of 3 beyond 2σ: 2 of any 3 consecutive points beyond 2σ on the same side.

规则 6 · 4/5 贴 1σ：连续 5 点中有 4 点落在同侧 1σ 之外。 Rule 6 · 4 of 5 beyond 1σ: 4 of any 5 consecutive points beyond 1σ on the same side.

规则 7 · 紧贴中心：连续 15 点都在 ±1σ 内。 Rule 7 · Hugging the center: 15 in a row all within ±1σ.

规则 8 · 远离中心：连续 8 点都在 ±1σ 之外，任意侧。 Rule 8 · Avoiding the center: 8 in a row all beyond ±1σ on either side.

何时用： Use it when: 识别尚未越过 3σ 控制限，但已呈现非随机模式的过程变化。 You need to detect nonrandom process patterns before a point necessarily crosses a three-sigma limit.

进阶 · 小漂移检测与选图决策 Advanced · Small-Shift Detection and Chart Selection

3 张进阶卡 3 advanced cards

T3-10

EWMA EWMA

F36 深潜 → F36 deep dive →

z_t = λx_t + (1−λ)z_t−1
limits: μ₀ ± L·σ·√[λ/(2−λ)·(1−(1−λ)^2t)]
λ=0.2, L≈2.86

λ/(2−λ)平滑系数决定的稳态宽度Steady-state width set by the smoothing coefficient 1−(1−λ)^2t前几点的收敛因子Startup convergence factor for the first few points

λ 新观测的平滑权重，常用 λ=0.2 smoothing weight on the new observation, commonly λ=0.2
L 控制限宽度倍数，常用 L≈2.86 control-limit width multiplier, commonly L≈2.86

何时用： Use it when: 需要通过指数加权历史信息，提高对持续小漂移的灵敏度。 You need exponentially weighted history to improve sensitivity to persistent small shifts.

T3-11

CUSUM CUSUM

F36 深潜 → F36 deep dive →

C⁺_t = max[0, x_t − (μ₀+K) + C⁺_t−1]
C⁻_t = max[0, (μ₀−K) − x_t + C⁻_t−1]
K=δσ/2, H≈5σ

K=δσ/2 针对 δσ 漂移的参考值 reference value targeting a shift of δσ
H≈5σ 常用决策区间 commonly used decision interval

何时用： Use it when: 需要累积连续小偏差，并分别监控向上与向下漂移。 You need small deviations accumulated over time with separate upward and downward signals.

T3-12

选图决策 Chart Selection

F31 深潜 → F31 deep dive → F32 深潜 → F32 deep dive → F33 深潜 → F33 deep dive → F34 深潜 → F34 deep dive → F35 深潜 → F35 deep dive →

continuous + subgroup → X̄-R(small n) / X̄-S(large n)
continuous + individual → I-MR
fraction nonconforming → p/np · defect count → c/u

variable data 连续计量值；按采样结构选 X̄-R、X̄-S 或 I-MR continuous measurements; choose X̄-R, X̄-S, or I-MR by sampling structure
attribute data 不良品率用 p/np，缺陷数用 c/u use p/np for nonconforming units and c/u for defect counts

何时用： Use it when: 从数据类型、子组结构与机会量是否固定三步确定控制图。 You need to select a chart from data type, subgroup structure, and whether opportunity size is fixed.