1. Theory

Control charts: Plot a process statistic against limits set from in-control data, signalling special causes when the statistic leaves the limits or follows non-random patterns (MetricGate; NIST CUSUM page).
Run length: The average number of subgroups before a signal measures both false-alarm rate and sensitivity (NIST CUSUM page; Quality Gurus).
Memory charts: EWMA and CUSUM charts accumulate past data and detect small shifts faster than Shewhart charts (NIST EWMA page; NIST CUSUM page).
Capability: Indices compare process spread and centring with specification limits (NIST capability page).

2. Working Principle

Phase I: Collect rational subgroups, estimate the mean and spread, remove assignable causes and fix the limits (MetricGate; Phase I naming my reading).
Phase II: Plot new data and apply the Western Electric rules (NIST e-Handbook 6.3.2).
Chart choice: Shewhart for large shifts, EWMA or CUSUM for shifts of about 1 sigma or less (NIST CUSUM page).
Capability: Estimate Cp and Cpk with confidence intervals from a stable process (NIST capability page).

3. Applications

Gauge control: Check-standard charts that keep metrology tools in calibration (Gauge R&R page; Calibration and Traceability page).
Yield and reliability: Capability indices feed parametric yield and specification setting (NIST capability page).

Shewhart Charts

Name	Formula	Related Formulae	Variables/Constants
X-bar and R chart limits	$U C L_{\overset{x}{ˉ}} = \overset{ˉ}{\overset{x}{ˉ}} + A_{2} \overset{ˉ}{R}, L C L_{\overset{x}{ˉ}} = \overset{ˉ}{\overset{x}{ˉ}} - A_{2} \overset{ˉ}{R}$	$U C L_{R} = D_{4} \overset{ˉ}{R}, L C L_{R} = D_{3} \overset{ˉ}{R}, A_{2} = \frac{3}{d _{2} n}$	$\overset{ˉ}{\overset{x}{ˉ}} : g r an d m e an o f s u b g r o u p m e an s$ $\overset{ˉ}{R} : m e an s u b g r o u p r an g e$ $n : s u b g r o u p s i z e$ $A_{2}, D_{3}, D_{4}, d_{2} : t ab u l a t e d co n s t an t s; M e t r i c G a t e : l imi t f or m u l a s, co n s t an t s d e p e n d o n s u b g r o u p s i z e, c h ec k t h e R c ha r t f i r s t b ec a u se X - ba r l imi t s a r e u n r e l iab l e i f i t i s o u t o f co n t r o l; F ab r i co : f or n = 5, d_{2} = 2.326, D_{3} = 0, D_{4} = 2.114 an d A_{2} = 0.577$ $(co m p u t e d : A_{2} = 0.577 f or n = 5)$
Shewhart run length	$A R L = \frac{1}{p}$	$p_{0} = 2Φ (- 3) = 0.0027, A R L_{0} = 370$	$p : p r o babi l i t y t ha t o n e p o in t f a l l s o u t s i d e t h e l imi t s$ $Φ : s t an d a r d n or ma l c u m u l a t i v e d i s t r ib u t i o n; N I S T C U S U M p a g e : A R L i s t h e a v er a g e n u mb er o f s u b g r o u p s b e f or e a s i g na l, 3 - s i g ma S h e w ha r t A R L 370.37 in co n t r o l (t ab u l a t e d a s 371) an d 43.96 f or a 1 - s i g ma s hi f t w i t h p = 0.02275; Q u a l i t y G u r u s : A R L_{0} ab o u t 370 an d A R L_{1} ab o u t 44$ $(co m p u t e d : 370.4 in co n t r o l an d 43.9 f or a 1 - s i g ma s hi f t in c l u d in g t h e f a r t ai l)$
Run rules	$A R L_{0, W E} \approx 92$	$\frac{A R L _{0}}{A R L _{0, W E}} \approx 4$	$W E : f u l l W es t er n E l ec t r i c r u l e se t (a p o in t b ey o n d 3 s i g ma, 2 o f 3 b ey o n d 2 s i g ma, 4 o f 5 b ey o n d 1 s i g ma, 8 in a r o w o n o n e s i d e; N I S T e - H an d b oo k 6.3.2 r u l e l i s t); Q u a l i t y G u r u s : f u l l r u l e se t d r o p s A R L_{0} f r o m ab o u t 370 t o ab o u t 92, r o ug h l y f o u r t im es a s man y f a l se a l a r m s; M e t r i c G a t e : r u n - r u l e v i o l a t i o n s f l a g g e d, N e l so n r u l es o f f er e d$ $(a ss u m e d : 1000 c ha r t s e a c h pl o tt in g o n e p o in t p er d a y; co m p u t e d : ab o u t 2.7 f a l se a l a r m s p er d a y w i t h 3 - s i g ma l imi t s a l o n e an d 10.9 w i t h t h e f u l l r u l e se t)$

EWMA and CUSUM Charts

Name	Formula	Related Formulae	Variables/Constants
EWMA statistic	$E W M A_{t} = λ Y_{t} + (1 - λ) E W M A_{t - 1}$	$E W M A_{0} = target, 0 < λ \leq 1$	$Y_{t} : o b ser v a t i o n a t t im e t$ $λ : w e i g h t; N I S T : λ u s u a l l y 0.2 t o 0.3 (H u n t er 1986), λ = 1 r e d u ces t o a S h e w ha r t c ha r t; e x am pl e t a r g e t 50, 20 o b ser v a t i o n s, f ina l E W M A 51.99$ $(co m p u t e d : t h e r ec u r s i o n w i t h λ = 0.3 r e p r o d u ces a l l 20 l i s t e d E W M A v a l u es)$
EWMA control limits	$U C L, L C L = E W M A_{0} \pm k s \frac{λ}{2 - λ}$	$s_{E W M A, t}^{2} = s^{2} \frac{λ}{2 - λ} [1 - (1 - λ)^{2 t}]$	$s : hi s t or i c a l s t an d a r d d e v ia t i o n$ $k : l imi t m u l t i pl i er; N I S T : k = 3 or f r o m Lu c a s an d S a cc u cc i t ab l es, e x am pl e s = 2.0539, λ = 0.3, l imi t s 47.4115 an d 52.5884, a l l p o in t s in s i d e, u pw a r d t r e n d o v er t h e l a s t 5 p er i o d s; M o n t ana S t a t e S T A T 528 n o t es : e x a c t t im e - d e p e n d e n t v a r ian ce an d l imi t s a pp r o a c hin g t h e a sy m pt o t i c v a l u es$ $(co m p u t e d : 3 t im es 0.4201 t im es 2.0539 = 2.588; a t t = 1 t h e e x a c t f a c t or i s 0.3 in s t e a d o f 0.42, so e a r l y l imi t s a r e na r r o w er)$
Tabular CUSUM	$S_{r} = \sum_{i = 1}^{r} (\overset{x}{ˉ}_{i} - k)$	$k = \frac{μ ^ _{0} + μ _{1}}{2}, h_{s} = \frac{h n}{σ}$	$\overset{x}{ˉ}_{i} : s u b g r o u p m e an s$ $k : r e f er e n ce v a l u e$ $μ_{1} : r e j ec t ab l e m e an$ $h : d ec i s i o n in t er v a l; N I S T : s t an d a r d i z e d k = 0.5, s i g na l w h e n t h e r i se ab o v e t h e l o w es t p r e v i o u s p o in t r e a c h es h, A R L f or h_{s} = 4 an d 5 : 336 an d 930 in co n t r o l, 8.38 an d 10.4 a t 1 s i g ma, a g ain s t S h e w ha r t 371 an d 44.0; Q u a l i t y G u r u s : C U S U M k = 0.5, h = 5 g i v es A R L_{0} ab o u t 465 an d A R L_{1} ab o u t 10$ $(co m p u t e d : tw o o n e - s i d e d c ha r t s w i t h A R L 930 e a c h co mbin e t o 465)$

Process Capability

Name	Formula	Related Formulae	Variables/Constants
Capability indices	$C_{p} = \frac{U S L - L S L}{6 σ}, C_{p k} = min [\frac{U S L - μ}{3 σ}, \frac{μ - L S L}{3 σ}]$	$C_{p k} = C_{p} (1 - k), k = \frac{∣ m - μ ∣}{( U S L - L S L ) /2}$	$U S L, L S L : s p ec i f i c a t i o n l imi t s$ $m : s p ec i f i c a t i o n mi d p o in t; N I S T e x am pl e U S L 20, L S L 8, s = 2, m e an 16 : C_{p} = 1.0, C_{p k} = 0.667, C_{p u} = 0.667, C_{pl} = 1.333, C_{p k} a t l e a s t 1.0 d es i r e d; ce n t r e d r e j ec t s 0.27 p er ce n t, 64 pp m, 0.6 pp m an d 2 pp b a t C_{p} = 1.00, 1.33, 1.66 an d 2.00$ $(co m p u t e d : 2700, 66, 0.64 pp m an d 0.002 pp m a t t h e r o u n d e d v a l u es; 63 an d 0.57 pp m a t e x a c tl y 4/3 an d 5/3)$
Taguchi capability	$C_{p m} = \frac{U S L - L S L}{6 σ ^{2} + ( μ - T ) ^{2}}$	$C_{p m} \leq C_{p}$	$T : t a r g e t v a l u e; N I S T : d e f ini t i o n an d s am pl e es t ima t or w i t h \overset{x}{ˉ} an d s$ $(co m p u t e d : f or t h e N I S T e x am pl e w i t h t a r g e t 14, C_{p m} = 12/ (68) = 0.707)$
Capability confidence interval	$C_{p k} = \hat{C}_{p k} \pm z_{1 - α /2} \frac{1}{9 n} + \frac{C ^ _{p k}^{2}}{2 ( n - 1 )}$	$n \geq 25, preferably n \geq 100$	$n : s am pl e s i z e$ $z : n or ma l q u an t i l e; N I S T : co mm o n l y u se d a pp r o x ima t i o n, a t l e a s t 25 s am pl es f or v a l i d i t y an d g e n er a l l y 100 f or c a p abi l i t y s t u d i es, v a r ia t i o n o f t h e in d i ces n o t n e g l i g ib l e$ $(co m p u t e d, 95 p er ce n t : es t ima t e 1.33 g i v es 0.93 t o 1.73 a t n = 25, 1.05 t o 1.61 a t n = 50 an d 1.13 t o 1.53 a t n = 100)$

Important Questions

Ques: How are Shewhart X-bar and R limits set and what does an ARL of 370 mean?

The X-bar chart places limits at $\overset{ˉ}{\overset{x}{ˉ}} \pm A_{2} \overset{ˉ}{R}$ and the R chart at $D_{3} \overset{ˉ}{R}$ and $D_{4} \overset{ˉ}{R}$ , with constants depending on subgroup size (MetricGate). For five wafers per subgroup $A_{2}$ = 0.577 (computed from Fabrico’s $d_{2}$ = 2.326, which also lists 0.577), and the R chart should be judged first, since its instability invalidates the mean limits. Three-sigma limits leave a probability of 0.0027 per point outside, so an in-control process gives a false signal on average every 370 subgroups (NIST CUSUM page; Quality Gurus). NIST quotes 370.37 in its text and 371 in its table, a rounding difference; 370.4 is the computed value. A 1-sigma shift in the mean is caught after about 44 subgroups on average, which is slow (NIST CUSUM page). Reference: MetricGate, ‘Shewhart X-bar / R Control Chart’ calculator page (fetch-tool summary); Fabrico, ‘X-bar R chart’ guide: for subgroup size 5, A2 = 0.577, D3 = 0, D4 = 2.114, d2 = 2.326 (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2.3.1, ‘Cusum Average Run Length’ (fetch-tool summary); Quality Gurus, ‘Understanding Average Run Length in Control Charts’ (fetch-tool summary).

Ques: Why do run rules matter in a fab with many charts?

They raise sensitivity but also false alarms. Quality Gurus reports that the full Western Electric rule set drops the in-control ARL from about 370 to about 92, roughly four times as many false alarms. A fab may run a thousand or more charts (my reading); with one point per chart per day, 3-sigma limits alone would give about 2.7 false alarms a day and the full rule set about 10.9 (computed, assumed 1000 charts). Engineers then learn to ignore signals, which defeats the purpose (my reading). A common compromise is to use only the 3-sigma rule on most parameters, and add an EWMA or CUSUM chart where small drifts matter (my reading; NIST CUSUM page). Their probabilities assume normally distributed data with known parameters (NIST e-Handbook 6.3.2), and correlated points, such as wafers from one lot, would break that assumption (my reading). Reference: Quality Gurus, ‘Understanding Average Run Length in Control Charts’ (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2.3.1, ‘Cusum Average Run Length’ (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2, ‘What are Variables Control Charts?’: Western Electric rules (any point beyond 3 sigma, 2 of the last 3 beyond 2 sigma, 4 of the last 5 beyond 1 sigma, 8 consecutive on one side), probability statements for a normal distribution with known parameters (fetch-tool summary); Calibration and Traceability page of this vault.

Ques: How does the NIST EWMA example work?

Each EWMA value is $λ Y_{t} + (1 - λ) E W M A_{t - 1}$ , starting at the target of 50 (NIST EWMA page). With $λ$ = 0.3 the recursion reproduces all 20 listed values, ending at 51.99 (computed). The limits are $50 \pm 3 s λ / (2 - λ)$ , where s = 2.0539, giving 47.41 and 52.59, and every point lies inside, though NIST notes an upward trend over the last five periods. The handbook uses the asymptotic variance; the exact variance includes a factor of $1 - (1 - λ)^{2 t}$ , so at the first point the limit factor is 0.30 rather than 0.42 and early limits are narrower (Montana State STAT 528 notes; computed). NIST recommends $λ$ of 0.2 to 0.3, and $λ$ = 1 recovers a Shewhart chart. Smaller $λ$ gives more memory and faster detection of small drifts, such as slow chamber wall build-up (my reading). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2.4, ‘EWMA Control Charts’ (fetch-tool summary); Montana State University, STAT 528 course notes, section 9.7 ‘Exponentially Weighted Moving Average Control Charts’: variance sigma^2 lambda/(2 - lambda)[1 - (1 - lambda)^(2i)], time-varying limits approaching asymptotic limits (fetch-tool summary, formula garbled).

Ques: Is the in-control ARL of a CUSUM with h = 5 equal to 930 or 465?

Both figures appear. NIST tabulates an in-control ARL of 930 for the standardized CUSUM with k = 0.5 and h = 5, and 336 for h = 4, while Quality Gurus quotes about 465 for the same k and h. NIST’s table describes a one-sided scheme detecting an upward shift, while a two-sided CUSUM runs two such charts, so its false-alarm rates add: $1/ (1/930 + 1/930)$ = 465 (computed). The figures therefore agree once the configuration is stated; for routine two-sided monitoring, 465 is more likely the relevant value (my reading). The CUSUM’s strength is small shifts: at 1 sigma its ARL is 10.4 against 44.0 for Shewhart, while at 3 sigma Shewhart is faster, 2.00 against 2.57 (NIST CUSUM page). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, section 6.3.2.3.1, ‘Cusum Average Run Length’ (fetch-tool summary); Quality Gurus, ‘Understanding Average Run Length in Control Charts’ (fetch-tool summary).

Ques: How reliable is a Cpk estimate?

Less than its two decimals suggest. NIST’s example with limits 8 and 20, s = 2 and mean 16 gives Cp = 1.0 but Cpk = 0.667, because the process is off centre, and NIST wants Cpk at least 1.0. The approximate 95 per cent interval $\hat{C}_{p k} \pm 1.96 1/ (9 n) + \hat{C}_{p k}^{2} / (2 (n - 1))$ shows that an estimate of 1.33 spans 0.93 to 1.73 with 25 samples and 1.13 to 1.53 with 100 (computed). NIST requires at least 25 samples and generally 100. Its reject table lists 64 ppm at Cp = 1.33 and 0.6 ppm at 1.66; these correspond to exactly 4/3 and 5/3, while the rounded values give 66 and 0.64 ppm (computed). Measurement variation also lowers observed capability (Gauge R&R page). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, section 6.1.6, ‘What is Process Capability?’ (fetch-tool summary); Gauge Repeatability and Reproducibility (Gauge R&R) page of this vault.

References

MetricGate, “Shewhart X-bar / R Control Chart,” calculator page. Available: https://metricgate.com/calculator/shewhart-xbar-r-control-chart
NIST/SEMATECH, “6.3.2.3.1. Cusum Average Run Length,” e-Handbook of Statistical Methods. Available: https://itl.nist.gov/div898/handbook/pmc/section3/pmc3231.htm
Quality Gurus, “Understanding Average Run Length in Control Charts.” Available: https://www.qualitygurus.com/understanding-average-run-length-in-control-charts/
Fabrico, “X-bar R chart” guide. Available: https://www.fabrico.io/blog/xbar-r-chart/
NIST/SEMATECH, e-Handbook of Statistical Methods, section 6.3.2, “What are Variables Control Charts?” Available: https://itl.nist.gov/div898/handbook/pmc/section3/pmc32.htm
Montana State University, STAT 528 course notes, section 9.7, “Exponentially Weighted Moving Average Control Charts.” Available: https://math.montana.edu/jobo/st528/documents/chap9d.pdf
Calibration and Traceability page of this vault.
NIST/SEMATECH, “6.3.2.4. EWMA Control Charts,” e-Handbook of Statistical Methods. Available: https://itl.nist.gov/div898/handbook/pmc/section3/pmc324.htm
NIST/SEMATECH, “6.1.6. What is Process Capability?” e-Handbook of Statistical Methods. Available: https://itl.nist.gov/div898/handbook/pmc/section1/pmc16.htm
Gauge Repeatability and Reproducibility (Gauge R&R) page of this vault.

Quartz 5

Explorer

Statistical Process Control (SPC)