Module 02: AQL 2.5 Sampling & Inspection Standards
Downloads (2)Double & Multiple Sampling Plan Procedures & SOPs
Reading Time: 35 minLesson 5 of 18Comprehensive Master Manual
Lesson Contents Overview
Learning Objectives
Understand the statistical theory and Average Sample Number (ASN) curves of Single, Double, and Multiple Sampling Plans.
Differentiate between the operational advantages and administrative complexities of Single vs. Double Sampling in apparel factories.
Master the step-by-step decision tree of Double Sampling: First Sample ($n_1, Ac_1, Re_1) and Cumulative Second Sample ($n_1+n_2, Ac_2, Re_2).
Extract exact Double Sampling plan values from ISO 2859-1 Table 3-A for standard apparel Code Letters (K, L, M, N) across AQL 2.5 and 4.0.
Calculate the economic labor and inspection time savings generated by Double Sampling for high-quality production lines.
Understand the multi-stage architecture of Multiple Sampling Plans (ISO 2859-1 Table 4-A) and their applications in physical lab testing.
Execute a standardized factory SOP for carton quarantine, representative second-sample drawing, defect logging, and final certificate issuance.
Comprehensive Technical GuidanceTextbook & Training SOP
Double & Multiple Sampling Plan Procedures & SOPs
Commercial Process Workflow
═══════════════════════════════════════════════════════════════════════════════ MODULE 02: AQL 2.5 SAMPLING & INSPECTION STANDARDS LESSON 02.03: DOUBLE & MULTIPLE SAMPLING PLAN PROCEDURES & SOPS ═══════════════════════════════════════════════════════════════════════════════ [Course: AQL 2.5 Quality Inspection & QA Masterclass] [⏮️ Previous: 02.02 Switching Rules]
1. Introduction: The Statistical Logic of Multi-Stage Sampling
In international apparel manufacturing and export quality control, Single Sampling is the most widely recognized protocol. However, international standards—including ISO 2859-1, ANSI/ASQ Z1.4, and MIL-STD-105E—also provide formalized Double Sampling Plans (Table 3-A) and Multiple Sampling Plans (Table 4-A).
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐
│ WHY USE DOUBLE SAMPLING IN APPAREL AUDITS? │
│ │
│ 1. LOWER AVERAGE SAMPLE NUMBER (ASN) │
│ Clean lots pass on a smaller first sample (saving 20% to 35% time). │
│ │
│ 2. PSYCHOLOGICAL FAIRNESS ("THE SECOND CHANCE") │
│ Borderline lots are not rejected on a single borderline check; │
│ a second independent sample confirms true process quality. │
│ │
│ 3. MATHEMATICAL INTEGRITY │
│ Maintains the exact same Producer and Consumer risk profiles │
│ as standard single sampling plans. │
└─────────────────────────────────────────────────────────────────────────────┘Double sampling gives suppliers and buyers an optimal balance: fast clearance for high-quality production and rigorous verification for borderline batches.
2. The Step-by-Step Decision Architecture of Double Sampling
In a Double Sampling Plan, the inspection proceeds through two distinct stages:
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐ │ DOUBLE SAMPLING COMPLETE DECISION TREE │ │ │ │ [DRAW FIRST SAMPLE: n1] │ │ │ │ │ ┌────────────────────┼────────────────────┐ │ │ ▼ ▼ ▼ │ │ d1 <= Ac1 Ac1 < d1 < Re1 d1 >= Re1 │ │ [ACCEPT LOT] [INCONCLUSIVE] [REJECT LOT] │ │ (Audit Ends) │ (Audit Ends) │ │ ▼ │ │ [DRAW SECOND SAMPLE: n2] │ │ (From Fresh Unopened Boxes) │ │ │ │ │ ▼ │ │ [CALCULATE CUMULATIVE DEFECTS] │ │ D = d1 + d2 │ │ │ │ │ ┌───────────┴───────────┐ │ │ ▼ ▼ │ │ D <= Ac2 D >= Re2 │ │ [ACCEPT LOT] [REJECT LOT] │ │ (Audit Ends) (Audit Ends) │ └─────────────────────────────────────────────────────────────────────────────┘
Stage 1: The First Sample ($n_1)
- Draw $n_1 garments randomly across opened cartons.
- Tally defective garments ($d_1):
- If $d_1 le Ac_1$: The lot is ACCEPTED immediately. Inspection terminates.
- If $d_1 ge Re_1$: The lot is REJECTED immediately. Inspection terminates.
- If $Ac_1 < d_1 < Re_1$: The result is INCONCLUSIVE. Proceed to Stage 2.
Stage 2: The Second Sample ($n_2)
- Draw $n_2 garments randomly from a fresh set of unopened cartons.
- Tally defective garments in the second sample ($d_2).
- Calculate Cumulative Defective Garments: $D_{total} = d_1 + d_2.
- If $D_{total} le Ac_2$: The lot is ACCEPTED.
- If $D_{total} ge Re_2$: The lot is REJECTED (where $Re_2 = Ac_2 + 1).
3. Double Sampling Master Table: ISO 2859-1 Table 3-A (Normal Level II)
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐ │ ISO 2859-1 TABLE 3-A: DOUBLE SAMPLING PLANS (NORMAL LEVEL II) │ ├───────┬──────────┬──────────┬──────────┬──────────────┬─────────────────────┤ │ CODE │ SAMPLE │ SAMPLE │ CUMULAT. │ AQL 2.5 │ AQL 4.0 │ │LETTER │ STAGE │ SIZE (n) │ SIZE (N) │ (Major) │ (Minor) │ │ │ │ │ │ Ac / Re │ Ac / Re │ ├───────┼──────────┼──────────┼──────────┼──────────────┼─────────────────────┤ │ H │ First │ 32 │ 32 │ 1 / 3 │ 2 / 5 │ │(281- │ Second │ 32 │ 64 │ 4 / 5 │ 6 / 7 │ │ 500) │ │ │ │ │ │ ├───────┼──────────┼──────────┼──────────┼──────────────┼─────────────────────┤ │ J │ First │ 50 │ 50 │ 2 / 5 │ 3 / 6 │ │(501- │ Second │ 50 │ 100 │ 6 / 7 │ 8 / 9 │ │ 1200) │ │ │ │ │ │ ├───────┼──────────┼──────────┼──────────┼──────────────┼─────────────────────┤ │ K │ First │ 80 │ 80 │ 3 / 6 │ 5 / 8 │ │(1201- │ Second │ 80 │ 160 │ 9 / 10 │ 12 / 13 │ │ 3200) │ │ │ │ │ │ ├───────┼──────────┼──────────┼──────────┼──────────────┼─────────────────────┤ │ L │ First │ 125 │ 125 │ 5 / 9 │ 7 / 11 │ │(3201- │ Second │ 125 │ 250 │ 12 / 13 │ 18 / 19 │ │10000) │ │ │ │ │ │ ├───────┼──────────┼──────────┼──────────┼──────────────┼─────────────────────┤ │ M │ First │ 200 │ 200 │ 7 / 11 │ 11 / 16 │ │(10001-│ Second │ 200 │ 400 │ 18 / 19 │ 26 / 27 │ │35000) │ │ │ │ │ │ └───────┴──────────┴──────────┴──────────┴──────────────┴─────────────────────┘
4. Average Sample Number (ASN) Analysis: Why Double Sampling Saves Money
The Average Sample Number (ASN) represents the expected number of sample garments inspected per lot across a continuous production run.
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐ │ AVERAGE SAMPLE NUMBER (ASN) COMPARISON │ │ │ │ [SINGLE SAMPLING: CODE L (8,000 PCS)] │ │ └── Constant sample: 200 garments inspected EVERY TIME. │ │ │ │ [DOUBLE SAMPLING: CODE L (8,000 PCS)] │ │ ├── Clean Production (Defect rate ≤ 1.5%): │ │ │ └── 85% of lots pass on First Sample (n1 = 125 pcs). │ │ │ └── Expected ASN = (0.85 × 125) + (0.15 × 250) = 143.7 pcs. │ │ │ └── SAVINGS: 56.3 garments per inspection (28.1% labor reduction!). │ │ │ │ │ └── Poor Production (Defect rate ≥ 6.0%): │ │ └── 90% of lots fail on First Sample (n1 = 125 pcs). │ │ └── Fast rejection without wasting time on second sample. │ └─────────────────────────────────────────────────────────────────────────────┘
5. Multiple Sampling Plans (ISO 2859-1 Table 4-A)
A Multiple Sampling Plan extends the logic of double sampling across 5 to 7 successive sample stages.
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐ │ MULTIPLE SAMPLING PLAN STRUCTURE (CODE L) │ ├─────────┬──────────────┬──────────────────┬──────────────┬──────────────────┤ │ STAGE │ SAMPLE SIZE │ CUMULATIVE SIZE │ Ac / Re (2.5)│ OUTCOME │ ├─────────┼──────────────┼──────────────────┼──────────────┼──────────────────┤ │ Stage 1 │ 50 pcs │ 50 pcs │ # / 4 │ Accept or Next │ │ Stage 2 │ 50 pcs │ 100 pcs │ 1 / 6 │ Accept/Rej/Next │ │ Stage 3 │ 50 pcs │ 150 pcs │ 4 / 8 │ Accept/Rej/Next │ │ Stage 4 │ 50 pcs │ 200 pcs │ 7 / 10 │ Accept/Rej/Next │ │ Stage 5 │ 50 pcs │ 250 pcs │ 10 / 12 │ Final Pass/Fail │ └─────────┴──────────────┴──────────────────┴──────────────┴──────────────────┘ * (# means acceptance not permitted at Stage 1 for this AQL level)
Why Multiple Sampling is Rare in Apparel Final Random Inspections (FRI):
- Warehouse Chaos: Opening and tracking cartons across 5 separate rounds creates massive logistical clutter in export packing areas.
- Time Unpredictability: Auditors cannot predict whether an inspection will take 1 hour (Stage 1 pass) or 6 hours (Stage 5 pass).
- Application in Apparel: Used primarily in laboratory physical/chemical testing (e.g., fabric fiber blends, formaldehydes, colorfastness) where destructive sample costs are extremely high.
6. Real Garment Case Study: Complete Double Sampling Execution
Order Specification:
- Garment: Men’s 100% Cotton Denim Jeans
- Order Quantity: 6,000 pcs (packed 20 pcs/carton = 300 cartons)
- Sampling Protocol: ISO 2859-1 Level II Double Sampling
- Quality Standard: Critical = 0.0, Major = 2.5, Minor = 4.0
Commercial Process Workflow
STEP-BY-STEP CALCULATION: 1. LOT SIZE: 6,000 pcs -> Code Letter "L" 2. DOUBLE SAMPLING PLAN (Table 3-A): ├── First Sample (n1) = 125 pcs | Major 2.5: Ac1=5, Re1=9 | Minor 4.0: Ac1=7, Re1=11 └── Second Sample (n2) = 125 pcs | Major 2.5: Ac2=12, Re2=13 | Minor 4.0: Ac2=18, Re2=19
Inspection Findings & Decision Flow:
| Inspection Phase | Sample Inspected | Critical Defects | Major Defects Found | Minor Defects Found | Immediate Decision |
|---|---|---|---|---|---|
| First Sample ($n_1) | 125 pcs (10 cartons) | 0 | 7 defective jeans | 5 defective jeans | INCONCLUSIVE on Major ($5 < 7 < 9). Must draw $n_2. |
| Second Sample ($n_2) | 125 pcs (10 fresh cartons) | 0 | 4 defective jeans | 8 defective jeans | Cumulative Tally Required |
| Cumulative Total | 250 pcs | 0 | 11 Major ($7 + 4) | 13 Minor ($5 + 8) | FINAL DECISION: PASSED! |
Evaluation:
- Critical: 0 found $le 0 (PASS).
- Cumulative Major: $11 le Ac_2 (12)$ (PASS).
- Cumulative Minor: $13 le Ac_2 (18)$ (PASS).
Final Disposition: INSPECTION PASSED. LOT CLEARED FOR SHIPMENT.
7. Factory SOP: Double Sampling Carton Management
Commercial Process Workflow
┌─────────────────────────────────────────────────────────────────────────────┐ │ FACTORY DOUBLE SAMPLING CARTON WORKFLOW │ │ │ │ [1. PALLET QUARANTINE] │ │ └── Stage entire shipment in designated QA audit zone. │ │ │ │ [2. INITIAL CARTON DRAWING] │ │ └── Draw √(Cartons)+1 master cartons for First Sample (n1 = 125 pcs). │ │ │ │ [3. AUDIT FIRST SAMPLE] │ │ └── Clockwise inspection under 1,000 lux. Log defect count d1. │ │ │ │ [4. DECISION GATE] │ │ ├── If d1 <= Ac1 -> Release lot. │ │ ├── If d1 >= Re1 -> Reject & quarantine lot for 100% rework. │ │ └── If Ac1 < d1 < Re1 -> Lock first cartons; draw fresh cartons for n2. │ │ │ │ [5. AUDIT SECOND SAMPLE & CUMULATIVE TALLY] │ │ └── Inspect n2 (125 pcs); sum d1+d2; compare against Ac2/Re2. Issue IC. │ └─────────────────────────────────────────────────────────────────────────────┘
8. Summary & Key Takeaways
- Double Sampling Plans evaluate an initial sample ($n_1) and, if defect counts are borderline ($Ac_1 < d_1 < Re_1), draw a second sample ($n_2) for cumulative evaluation.
- Lower Average Sample Number (ASN): Clean factory production lines save 20% to 35% of sample inspection labor compared to single sampling.
- Cumulative Defect Decision: Disposition on the second sample is strictly based on cumulative defects ($d_1 + d_2 le Ac_2 to accept).
- Fresh Carton Mandate: The second sample ($n_2) must always be drawn from fresh, unopened cartons to preserve statistical independence.
- Critical Defects: Always remain Zero Tolerance ($Ac_1=0, Re_1=1) across all sampling plans.
1. Single vs. Double Sampling Comparison (AQL 2.5 Major & AQL 4.0 Minor)
| Code Letter (Lot Size) | Single Sampling (Sample: Ac / Re) | Double Sampling 1st Sample (Sample: Ac / Re) | Double Sampling 2nd Sample (Cum. Sample: Ac / Re) | Average Sample Savings (Clean Lots) |
|---|---|---|---|---|
| Code H (281–500 pcs) | 50 pcs (Maj: 3/4, Min: 5/6) | 32 pcs (Maj: 1/3, Min: 2/5) | 64 pcs (Maj: 4/5, Min: 6/7) | 36.0% sample reduction on 1st pull |
| Code J (501–1,200 pcs) | 80 pcs (Maj: 5/6, Min: 7/8) | 50 pcs (Maj: 2/5, Min: 3/6) | 100 pcs (Maj: 6/7, Min: 8/9) | 37.5% sample reduction on 1st pull |
| Code K (1,201–3,200 pcs) | 125 pcs (Maj: 7/8, Min: 10/11) | 80 pcs (Maj: 3/6, Min: 5/8) | 160 pcs (Maj: 9/10, Min: 12/13) | 36.0% sample reduction on 1st pull |
| Code L (3,201–10,000 pcs) | 200 pcs (Maj: 10/11, Min: 14/15) | 125 pcs (Maj: 5/9, Min: 7/11) | 250 pcs (Maj: 12/13, Min: 18/19) | 37.5% sample reduction on 1st pull |
| Code M (10,001–35,000 pcs) | 315 pcs (Maj: 14/15, Min: 21/22) | 200 pcs (Maj: 7/11, Min: 11/16) | 400 pcs (Maj: 18/19, Min: 26/27) | 36.5% sample reduction on 1st pull |
| Code N (35,001–150,000 pcs) | 500 pcs (Maj: 21/22, Min: 21/22) | 315 pcs (Maj: 11/16, Min: 15/20) | 630 pcs (Maj: 26/27, Min: 32/33) | 37.0% sample reduction on 1st pull |
2. Average Sample Number (ASN) & Operational Trade-Offs
| Feature / Metric | Single Sampling Plan | Double Sampling Plan | Multiple Sampling Plan (5–7 Stages) |
|---|---|---|---|
| Average Sample Number (ASN) | Constant (100% of sample size) | Variable: 65% to 80% of Single Sample for clean lots | Lowest ASN for extreme lots (50% to 70% of Single) |
| Decision Speed for Clean Lots | Moderate (Requires full sample) | Fastest (Passes on smaller 1st sample) | Fast for extreme lots, slow for borderline lots |
| Administrative Complexity | Lowest (Very simple to manage) | Moderate (Requires staging & 2nd pull protocol) | Highest (Complex record keeping across 5+ pulls) |
| Psychological Vendor Impact | Definitive / Strict ("One shot") | High acceptance ("Fair second chance") | Often perceived as tedious and confusing |
| Primary Apparel Application | Universal default for 90%+ RMG PSI | Large volume repeat orders & vendor audits | Laboratory physical/chemical textile testing |
3. Master Decision Tree Matrix for Double Sampling (Code L: Lot 3,201 to 10,000 pcs)
| Inspection Stage | Sample Drawn | Cumulative Units | Major 2.5 Decision Criteria | Action / Next Step |
|---|---|---|---|---|
| Stage 1 (First Pull) | 125 garments | 125 units | 0 to 5 Major Defects | PASS LOT IMMEDIATELY (Audit Complete) |
| Stage 1 (First Pull) | 125 garments | 125 units | 9 or More Major Defects | REJECT LOT IMMEDIATELY (Audit Complete) |
| Stage 1 (First Pull) | 125 garments | 125 units | 6, 7, or 8 Major Defects | INCONCLUSIVE: Draw Second Sample (125 pcs) |
| Stage 2 (Second Pull) | 125 garments | 250 units | Cumulative Defects <= 12 | PASS LOT (Issue Inspection Certificate) |
| Stage 2 (Second Pull) | 125 garments | 250 units | Cumulative Defects >= 13 | REJECT LOT (Issue Formal Quarantine Report) |
Practical Commercial Flow
Double Sampling Audit on 12,000 pcs Men’s Chino Trousers
A third-party QA auditor conducted a Final Random Inspection on 12,000 pcs Men’s Woven Chino Trousers (packed 20 pcs/carton = 600 cartons) under ISO 2859-1 Double Sampling Level II (Code M) at AQL 2.5 Major / AQL 4.0 Minor:
Factory Real-Life Scenario
Scenario: An apparel exporter in Gazipur producing 30,000 pcs of Fleece Jackets faced persistent friction with buyer inspectors. Under Single Sampling (Code M: 315 pcs, Ac=14, Re=15), borderline lots with 15 defects were instantly rejected, resulting in 100% carton re-opening and shipment delays.
Solution: The factory and buyer agreed to adopt ISO 2859-1 Double Sampling Level II (Code M: $n_1=200, Ac_1=7, Re_1=11$; $n_2=200, Cumulative 400 pcs, $Ac_2=18, Re_2=19). In addition, the factory added an automated stitch sensor on fleece pocket overlock machines.
Impact: Over the subsequent 20 shipments, 14 lots passed cleanly on the first sample ($n_1=200), saving 1,610 garment audit checks. 5 borderline lots were cleared fairly on the second sample ($n_1+n_2), and only 1 defective lot was rejected. The factory saved $48,000 in unnecessary sorting and air-freight delay penalties.
Commercial Officer Best Practice
For large export orders (Code M: 315 pcs single vs $125+125 double), deploying Double Sampling allows a clean factory to clear inspection after auditing just 125 garments instead of 315 garments, saving over 4 hours of inspection table time.
Top Beginner Mistakes to Avoid
- Evaluating the second sample ($n_2) independently without adding the defects from the first sample ($d_1).
- Drawing the second sample ($n_2) from the same cartons that were already opened during the first sample pull.
- Confusing Single Sampling Ac/Re thresholds with Double Sampling first-stage thresholds (which are significantly lower).
- Using Double Sampling for Critical safety defects (Critical defects always remain Zero Tolerance: $Ac_1=0, Re_1=1).
- Failing to document both $n_1 and $n_2 findings separately on the official inspection report.
- Allowing factory personnel to sort or touch the shipment between the first and second sample pulls.
Important Commercial & Trade Terminology (6 Terms)
Double Sampling Plan
A statistical sampling procedure in which the decision to accept or reject a lot may be reached after inspecting a first sample ($n_1) or, if inconclusive, after inspecting a second sample ($n_2).
💡 Note: Standardized under ISO 2859-1 Table 3-A.First Sample Size ($n_1)
The initial quantity of units drawn from the lot during the first stage of a double sampling plan.
💡 Note: Typically equal to approximately 60% of the single sampling size.Second Sample Size ($n_2)
The secondary quantity of units drawn from fresh cartons when the first sample defect count falls between $Ac_1 and $Re_1.
💡 Note: Must be drawn randomly from unopened cartons.Cumulative Defects ($d_1 + d_2)
The total number of nonconforming garments found across both the first and second samples combined.
💡 Note: Compared against $Ac_2 and $Re_2 to reach the final disposition.Average Sample Number (ASN)
The average number of sample units inspected per lot over a sequence of lots under a specific sampling plan.
💡 Note: Double sampling provides an ASN 20%–35% lower than single sampling for high-quality lots.Multiple Sampling Plan
A multi-stage sampling procedure (typically 5 to 7 sample stages) where decisions are made iteratively at each stage.
💡 Note: Found in ISO 2859-1 Table 4-A; used primarily in laboratory testing.Interview Preparation Questions & Answers (4 Questions)
Master these high-frequency interview questions for Commercial Officer, Merchandising, and Export Executive roles.
Suggested Professional Answer:
An apparel buyer chooses Double Sampling primarily for two reasons: (1) Economic efficiency: For high-quality factories with low defect rates, Double Sampling reduces the Average Sample Number (ASN) by 20% to 35%, cutting inspection time and auditor fees. (2) Psychological fairness: It gives suppliers a mathematically sound "second chance" when a lot is borderline on the first sample, preventing unnecessary commercial disputes while maintaining strict statistical quality control.
Frequently Asked Questions (3 FAQs)
Clear, beginner-friendly answers to the most common questions about the Commercial Department.
The main drawback is administrative complexity. If a lot requires a second sample, the auditor must spend additional time selecting and opening a fresh set of cartons, calculating cumulative scores, and explaining the multi-stage procedure to factory management.
Key Takeaways & Summary Points
1Double Sampling provides two distinct opportunities to reach a disposition: clean lots are accepted on the first sample ($n_1), very poor lots are rejected on $n_1, and only borderline lots require a second sample ($n_2).
2On average, Double Sampling reduces the Average Sample Number (ASN) by 20% to 35% compared to Single Sampling for lots with low defect levels, saving significant auditor man-hours.
3In Double Sampling, $n_1 and $n_2 are usually equal in size (e.g., for Code L, $n_1 = 125 and $n_2 = 125, compared to Single Sampling $n = 200).
4On the second sample, the decision is based on CUMULATIVE defects ($d_1 + d_2): if $d_1 + d_2 le Ac_2, the lot passes; if $d_1 + d_2 ge Re_2, the lot fails.
5In the final stage of any double or multiple sampling plan, the rejection number is always strictly $Ac_2 + 1 ($Re_2 = Ac_2 + 1), eliminating any inconclusive state.
6Multiple Sampling involves 5 to 7 successive sample stages; while statistically efficient, it is rarely used for garment visual audits due to warehouse administrative overhead.
Executive Lesson Summary
Double and Multiple Sampling Plans under ISO 2859-1 offer a sophisticated statistical alternative to standard Single Sampling. By reducing average sample sizes for high-quality production runs and providing a rigorous second-stage evaluation for borderline lots, Double Sampling optimizes QA labor efficiency, reduces commercial friction, and maintains uncompromising quality standards across the apparel supply chain.
Lesson Downloads & SOP Templates
PDF
ISO 2859-1 Double Sampling Master Tables & Decision SOP
PDF SOP • 540 KB • Free Download
XLSX
Double Sampling Cumulative Defect Calculator & Audit Log
Excel Tool • 470 KB • Free Download
Related Calculator
AQL 2.5 Sampling Calculator
Calculate Single, Double, and Multiple sampling thresholds.
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Topic Keywords: double sampling plan apparel, ISO 2859-1 Table 3-A, multiple sampling plans, Average Sample Number ASN, cumulative defect calculation, AQL double sampling Ac Re, garment inspection second sample, ANSI ASQ Z1.4 double sampling