该用哪张控制图：选图决策路径与控制图常数表

07 — 选图 · 图 10-5 · § 10.3.2

该用哪张控制图

按手册给出的决策路径。特别留意数据类型和合理子组大小:几乎整个选图结果都由这两点决定。

Are the data attribute data?

手工控制图参考卡

限的计算公式见第 10 章。

x̄ / s — mean and standard deviation§ 10.3.3.2 · pp. 81–82

Standard chart for continuous characteristics with bilateral tolerance, normal behavior, and no trends. Location chart limits are calculated from subgroup size: means narrow the process distribution compared to individual values (a ±2.58 σ limit for individual values narrows to ±1.15 σ with n = 5).

Location: UCL = μ̂ + u₁₋α⁄₂ · σ̂/√n LCL = μ̂ − u₁₋α⁄₂ · σ̂/√n Variation: UCL = √(χ²_{f;1−α/2} / f) · σ̂ LCL = √(χ²_{f;α/2} / f) · σ̂ with f = n − 1 Estimators: μ̂ = x̄̄ σ̂ = √(mean s²)

x̄ / R — mean and range§ 10.3.3.3 · pp. 83–84

Identical to x̄/s but monitoring range. R chart limits can be calculated exactly using standardized range w quantiles, or approximately using normal quantiles and the d₂ and d₃ constants.

Exact: UCL_R = w_{n;1−α/2} · σ̂ LCL_R = w_{n;α/2} · σ̂ Approximate: UCL_R = (1 + u₁₋α⁄₂ · d₃/d₂) · R̄ LCL_R = (1 − u₁₋α⁄₂ · d₃/d₂) · R̄ (if ≤ 0 → LCL_R = 0) Estimator: σ̂ = R̄/d₂

x̃ / R — median and range§ 10.3.3.4 · pp. 85–86

Alternative to means chart when reducing the influence of extreme values within samples is desirable: automated measurements with high variability, tensile tests. Reacts more slowly than the x̄ chart. ISO 7870-2 prefers the mean of sample medians.

UCL = μ̂ + u₁₋α⁄₂ · (cₙ/√n) · σ̂ LCL = μ̂ − u₁₋α⁄₂ · (cₙ/√n) · σ̂ cₙ = σ_x̃ / σ_x̄ (constants table, section 07)

Acceptance chart — tolerance-related§ 10.3.4 · p. 89

The only tolerance-related chart included in the manual. For processes with systematic and accepted location shifts (tool wear, stamping, step drilling). Requirement: instantaneous within-subgroup variation sufficiently small relative to tolerance, typically σ̂ ≤ T/10. Not compatible with zero-defect strategy: admits a defined nonconforming fraction.

x̄ chart: UCL = U − k_A·σ̂ LCL = L + k_A·σ̂ k_A = u₁₋p + u_PA/√n x̃ chart: k_C = u₁₋p + cₙ·u_PA/√n x chart : k_E = u₁₋p + u(ⁿ√P_A) p = accepted out-of-tolerance proportion P_A = action probability

Pearson chart — skewed distributions§ 10.3.5.2 · pp. 91–92

Process-related chart for skewed distributions (time-dependent model A2). Calculated like Shewhart, but standard normal percentiles are replaced by quantiles of an appropriate asymmetric distribution. With individual value charts, or n < 9, always preferred to Shewhart. Application: processes near natural boundaries (runout, flatness, straightness).

CL = x̿ UCL = X99.865% LCL = X0.135% of the fitted distribution Clements approximation (ISO 7870-5, ISO 22514-4): UCL = x̿ + ŝ · P99.865%(γ̂₁, β̂₂) LCL = x̿ − ŝ · P0.135%(γ̂₁, β̂₂)

Shewhart with extended limits§ 10.3.5.3 · pp. 93–94

When the process has inherent and expected mean changes, controlling with classic Shewhart limits proves uneconomical. Limits are widened by incorporating mean fluctuation as an additional term. Applicable to time-dependent models C1, C2, C3, B, and D. Application: processes with trends (wear, environmental influences) or varying process levels (different equipment).

With μ̂max and μ̂min: UCL = μ̂max + u_in · σ̂_in/√n LCL = μ̂min − u_in · σ̂_in/√n With σ_out via ANOVA: UCL = μ̂ + u_in·σ̂_in/√n + u_out·σ̂_out LCL = μ̂ − u_in·σ̂_in/√n − u_out·σ̂_out u_out = 1.5 is a standard empirical value

CUSUM — cumulative sum§ 10.3.5.4 · pp. 95–96

Represents cumulative sum of deviations between sample values and target value. Highly sensitive to small mean shifts: upward slope indicates rising mean, downward falling mean. Key requirement: process variation must be stable. Typical application: chemical processes where slight concentration fluctuations have major consequences. Graphical alternative: V-mask (ISO 7870-4).

CL = 0 UCL = +h·σ/√m LCL = −h·σ/√m CU_i = min{0 , CU_{i−1} + x̄_i − (T − k·σ/√m)} Illustrative ARL for mean shift (in units of σₑ): 0.0 → 370.4 1.0 → 9.9 2.0 → 3.9 See complete values in § 10.3.5.4 of the manual.

EWMA — exponentially weighted moving average§ 10.3.5.5 · p. 96

Weights samples in exponentially decreasing order: most recent samples carry greatest weight. Weighting is set by parameter λ. Detects small mean shifts well, but reacts slower to large shifts: recommended alongside a Shewhart chart to cover both. Application: products deliberately manufactured near specification limit, where detecting very small shifts is critical. Formula details in ISO 7870-6.

可交互生成一条序列,比较检出速度

实验:哪张图先报警

同一条带刀具磨损的序列(自第 9 个样本起线性漂移),由三张图同时监控。放到车间实务里:Shewhart 对大阶跃直观好用;CUSUM 和 EWMA 累积小偏移的记忆,能早得多地发现渐进式漂移。在本模块的参数下(CUSUM k = 0.5、h = 5;EWMA λ = 0.2、L = 3),记忆型图维持着很高的 ARL₀(约 460–550,而 Shewhart 约 370)。

每样本漂移量(以统计量的 σ 计) 40 个样本的序列(漂移自第 9 个样本起)。条件:Shewhart 只用基本的 ±3σ 规则;μ 与 σ 已知;CUSUM 取 k = 0.5、h = 5;EWMA 取 λ = 0.2、L = 3。

Shewhart x̄

样本 21(偏移发生后第 13 个)

CUSUM

样本 17(偏移发生后第 9 个)

EWMA

样本 16(偏移发生后第 8 个)

每个样本的 EWMA 精确动态控制限(±3σ·√(λ/(2−λ)·(1−(1−λ)²ⁱ)))。

Comparison of 1,000 simulated processes (drift of 0.2 σ/sample starting at sample 9, 40-sample horizon)

Chart	% detected (s. 9–40)	Median delay (from shift)	% prior false alarm (s. 1–8)
Shewhart x̄	98.0 %	11 11 samples	2.0 %
CUSUM	99.6 %	9 9 samples	0.4 %
EWMA	98.4 %	9 9 samples	1.6 %

SQA Conclusion: With a drift of 0.2 σ/sample, CUSUM (9 9 samples) and EWMA (9 9 samples) detect the shift before Shewhart (11 11 samples). Memory charts accumulate consecutive deviations, while Shewhart depends on an individual mean crossing ±3σ limits (post-drift detection: 99.6% CUSUM, 98.4% EWMA vs 98.0% Shewhart).

Results breakdown: The three percentages (% detected post-drift, % prior false alarm, and % without alarm) classify all processes into disjoint categories (summing to 100%). Median delay is strictly computed over series with detection after shift onset (samples 9–40).

标准化极差分布的常数

标准化极差的 d₂(期望值)与 d₃(标准差),以及用于精确计算 R 图控制限的 w 分位数。§ 10.3.3.3 · 第 83–84 页。

d₂, d₃ and w quantiles of the standardized range (§ 10.3.3.3). c₄ calculated by the gamma function. c_n = σ_x̃/σ_x̄ for median charts (§ 10.3.3.4).
n	d₂	d₃	c₄	w 99% lower	w 99% upper	w 99.73% lower	w 99.73% upper	c_n median
2	1.128	0.8525	0.7979	0.009	3.970	0.002	4.533	1.000
3	1.693	0.8884	0.8862	0.135	4.424	0.070	4.950	1.160
4	2.059	0.8798	0.9213	0.343	4.694	0.221	5.200	1.092
5	2.326	0.8641	0.9400	0.555	4.886	0.397	5.378	1.198
6	2.534	0.8480	0.9515	0.749	5.033	0.569	5.515	1.136
7	2.704	0.8332	0.9594	0.922	5.154	0.729	5.627	1.214
8	2.847	0.8198	0.9650	1.075	5.255	0.874	5.722	1.159
9	2.970	0.8078	0.9693	1.212	5.341	1.006	5.803	1.223
10	3.078	0.7971	0.9727	1.335	5.418	1.126	5.875	1.175