SPC Xbar / R Charts

1 Introduksi

1.1 Latar Belakang

Statistical Process Control (SPC) adalah keluarga metode untuk monitor stability proses produksi real-time dengan tujuan mendeteksi special cause variation (assignable cause) sedini mungkin sebelum batch defective dihasilkan. Diperkenalkan oleh Walter Shewhart (1931) di Bell Labs, Shewhart control chart (Xbar-R, Xbar-S, individual, p-chart, c-chart) menjadi standar GMP, ISO 9001, Six Sigma, dan ASTM (Montgomery, 2020; Wheeler, 2010).

Xbar-R chart adalah paired chart untuk continuous variable: Xbar chart memonitor process mean (centering); R chart memonitor process spread (range subgroup). Subgroup tipikal $n = 4{-}5$ unit, diambil per shift/jam/batch. Control limits ($\mu \pm 3\sigma$) dihitung dari historical capable process (Phase I baseline), lalu monitor Phase II.

1.2 Tujuan Modul

Modul SPC Xbar / R Charts di SQalytics ditujukan untuk:

Menerima subgroup data dari production (multi-batch, multi-time).
Menghitung control limits Xbar dan R berbasis $A_2$, $D_3$, $D_4$ factors (Shewhart, 1931).
Menerapkan Western Electric rules (Western Electric, 1956) untuk deteksi out-of-control patterns: 1 point > 3σ, 2/3 points > 2σ, 4/5 > 1σ, 8 in row same side.
Memvisualisasikan time-series chart dengan UCL/LCL/CL + flag rule violations.
Audiens: praktisi quality engineering, supervisor produksi, peneliti pengembangan proses.

1.3 Posisi di Antara Alternatif

Pilih SPC Xbar / R Charts untuk continuous variable subgroup monitoring. Untuk defective count, pakai p-chart / np-chart (akan datang). Untuk defect count per unit, pakai c-chart / u-chart. Untuk process capability ($C_p$, $C_{pk}$), pakai Cp / Cpk Capability. Untuk acceptance sampling, pakai Acceptance Sampling.

2 Metode

2.1 Dasar Teoretis

Subgroup statistics: untuk subgroup $i$ size $n$, hitung $\bar{X}_i$ (mean) dan $R_i$ (range = max − min).

Grand mean dan mean range dari $k$ subgroups:

$$\overline{\overline{X}} = \frac{1}{k} \sum_{i=1}^{k} \bar{X}_i, \quad \bar{R} = \frac{1}{k} \sum_{i=1}^{k} R_i$$

Control limits Xbar chart:

$$\text{UCL}_{\bar{X}} = \overline{\overline{X}} + A_2 \bar{R}, \quad \text{LCL}_{\bar{X}} = \overline{\overline{X}} - A_2 \bar{R}$$

Control limits R chart:

$$\text{UCL}_R = D_4 \bar{R}, \quad \text{LCL}_R = D_3 \bar{R}$$

$A_2$, $D_3$, $D_4$ factors (Shewhart 1931 table; ASTM E2587):

$n$ subgroup	$A_2$	$D_3$	$D_4$
2	1.880	0	3.267
3	1.023	0	2.574
4	0.729	0	2.282
5	0.577	0	2.114
6	0.483	0	2.004
7	0.419	0.076	1.924
8	0.373	0.136	1.864
10	0.308	0.223	1.777

Western Electric Rules (Western Electric Co., 1956) untuk deteksi pattern non-random:

Rule 1: 1 point di luar 3σ.
Rule 2: 2 dari 3 consecutive points > 2σ (same side).
Rule 3: 4 dari 5 > 1σ (same side).
Rule 4: 8 consecutive points di same side of centerline.
Rule 5 (Nelson): 6 increasing/decreasing.
Rule 6: 14 alternating up/down.

Bila salah satu trigger → out-of-control signal → investigasi assignable cause.

Estimated process $\sigma$:

$$\hat{\sigma} = \bar{R} / d_2$$

dengan $d_2$ factor: 1.128 (n=2), 1.693 (n=3), 2.059 (n=4), 2.326 (n=5).

2.2 Persamaan Inti

Xbar UCL/LCL: $\overline{\overline{X}} \pm A_2 \bar{R}$
R UCL/LCL: $D_4 \bar{R}$ dan $D_3 \bar{R}$
Process $\sigma$: $\hat{\sigma} = \bar{R} / d_2$
Western Electric rules untuk deteksi pattern.

2.3 Asumsi & Batas Validitas

Asumsi	Konsekuensi jika dilanggar	Cara cek di SQalytics
Subgroup rational (homogeneous source)	Within-vs-between confounded	Define rational subgrouping
Subgroup independent (no autocorrelation)	False alarm meningkat	Cek autocorrelation residual
Process Phase I stable (baseline)	UCL/LCL bias	Pakai capable historical data
$n \geq 4{-}5$ per subgroup	Estimasi noisy	Modul flag
Variable distribution approximately normal	False alarm	Transform atau pakai EWMA
$\geq 25$ subgroups untuk baseline	Limit tidak reliable	Modul flag

3 Cara Kerja

3.1 Step-by-Step di SQalytics

Buka SPC Xbar / R Charts dari domain Mutu dan Analisis Lanjutan.
Muat tabel: kolom Subgroup_ID (atau time/batch), Sample_1, Sample_2, ..., Sample_n (per row = 1 subgroup).
Atur subgroup size $n$.
(Opsional) Pisahkan Phase I (baseline) dan Phase II (monitoring).
Atur rules: default WE 1-4, tambah Nelson optional.
Klik Generate SPC Charts.
Tinjau hasil: Tab Xbar Chart — time-series + UCL/LCL/CL + rule violations.
Cek Tab R Chart, Tab Process Summary ($\bar{\bar{X}}$, $\bar{R}$, $\hat{\sigma}$, $C_p$ estimate), Tab Rule Violations Log.

3.2 Template Tabel Input + Contoh Data Sintetis

Kolom	Tipe	Wajib	Catatan
`Subgroup_ID`	category	✓	Time/batch label
`Sample_1` ... `Sample_n`	numeric	✓	Pengukuran

Contoh data sintetis (filling line, $n = 5$ per subgroup, 25 subgroups, fill weight g):

Subgroup	S1	S2	S3	S4	S5
1	252.1	250.5	251.8	249.9	251.2
2	250.8	251.5	250.2	251.8	250.5
3	249.5	250.2	251.0	250.5	249.8
… (subgroups 4–17: stable Phase I baseline)
18	253.5	254.2	253.8	254.5	253.0
… (subgroups 19–25: Phase II monitoring)

SYNTHETIC Filling line data: $n = 5$ per subgroup, 25 subgroups. Phase I baseline (subgroups 1-15) used untuk establish UCL/LCL. Phase II (subgroups 16-25) monitored — upward shift visible di subgroup 18. CSV setara: docs/assets/example-data/id/quality-advanced/template_quality_spc.csv.

3.3 Contoh Luaran

Baseline statistics (Phase I, subgroups 1-15):

Statistic	Value
$\overline{\overline{X}}$	250.8 g
$\bar{R}$	1.85 g
$A_2$ (n=5)	0.577
$D_3$, $D_4$	0, 2.114
UCL Xbar	251.87
LCL Xbar	249.73
UCL R	3.91
LCL R	0
$\hat{\sigma}$	0.80 g

Rule violations (Phase II, subgroups 16-25):

Subgroup	Violation	Action
18	Rule 1: Xbar = 253.8 > UCL	Investigate assignable cause
19	Rule 2: 2/3 > 2σ same side	Confirm trend
20	Rule 4 (cumulative)	Process shifted; reset baseline

SPC Xbar / R Charts — figure 01 — Gambar 1. *Panel* (a) Xbar chart dengan UCL/LCL dan rule violation flags; *panel* (b) R chart — subgroup fill weight data (n=5/subgroup, 25 subgroups, Phase I *baseline* + Phase II monitoring).

Kesimpulan ringkas: "SPC Xbar-R chart fill weight (n=5/subgroup, 25 subgroups) menunjukkan process stable di Phase I (subgroups 1-15) dengan $\overline{\overline{X}}$ = 250.8 g, $\hat{\sigma}$ = 0.80 g. Out-of-control signal terdeteksi pada subgroup 18: Xbar = 253.8 g > UCL 251.87 g (Rule 1 violation). Pattern subsequent confirms upward shift (Rule 2 + Rule 4). Assignable cause investigation needed: cek filler pump calibration, ingredient temperature, atau operator change. Action: (i) stop production untuk root cause analysis, (ii) reset baseline setelah corrective action, (iii) re-validate dengan Phase I baru. Lanjut ke Cp / Cpk Capability untuk uji apakah process meet spec setelah correction."

4 Kesimpulan

4.1 Relevansi Real-World

Filling weight control — package weight per shift.
Temperature monitoring — sterilization, fermentation, drying.
pH control — fermentasi yogurt, juice.
Brix monitoring — sirup, jus, jam production.
Color consistency — biskuit baking, beer brewing.
Pharmaceutical content uniformity — tablet weight.

4.2 Where to Go from Here

Cp / Cpk Capability — uji process meet spec.
Acceptance Sampling — lot release dengan SPC capable.
ANOVA Studio — uji statistical antar shift / operator.
Montgomery (2020) untuk SPC textbook lengkap.

⚙ Troubleshooting Cepat

Frequent false alarms. Cek autocorrelation; pakai EWMA atau CUSUM untuk autocorrelated process.

No OOC signal walau visible drift. Tambah sensitivity (Nelson rules); periksa subgroup size.

R chart erratic. Within-subgroup variation tidak homogeneous; revise rational subgrouping.

i Riwayat Revisi

Tanggal	Revisi	Penulis
2026-05-12	Draft v2 publikasi (KaTeX Xbar-R + $A_2$/$D_3$/$D_4$ + WE rules + APA Shewhart/Montgomery/Western Electric)	Claude
2026-05-12	Konversi MD → HTML (W5 quality-advanced batch)	Claude

4 Referensi

Shewhart, W. A. (1931). Economic control of quality of manufactured product. D. Van Nostrand.
Western Electric Company. (1956). Statistical quality control handbook. Western Electric Co.
Montgomery, D. C. (2020). Introduction to statistical quality control (8th ed.). John Wiley & Sons.
Wheeler, D. J. (2010). Advanced topics in statistical process control (2nd ed.). SPC Press.
Nelson, L. S. (1984). The Shewhart control chart — Tests for special causes. Journal of Quality Technology, 16(4), 237–239. https://doi.org/10.1080/00224065.1984.11978921
ASTM International. (2016). ASTM E2587: Standard practice for use of control charts in statistical process control. ASTM International.