Acceptable quality level

Acceptable quality level (AQL) is a statistical measure used in quality control to define the maximum percentage of defective units that can be considered acceptable when sampling a batch of products ^[1]. The concept is fundamental to acceptance sampling which is a method of inspecting a random sample from a lot to decide whether to accept or reject the entire batch. AQL provides a practical approach to quality management when 100 percent inspection is impractical or too costly.

Historical development

The foundations of acceptance sampling and AQL were laid at Bell Telephone Laboratories in the 1920s and 1930s. Walter A. Shewhart introduced control charts in the 1920s providing tools for monitoring process variation that informed later sampling methodologies ^[2]. Building on this work Harold F. Dodge and Harry G. Romig developed acceptance sampling tables in the 1930s and 1940s focusing on attribute inspection to balance inspection risks and costs.

The Dodge-Romig Sampling Inspection Tables published in 1940 introduced fundamental concepts such as consumer's risk, producer's risk, lot tolerance percent defective (LTPD) and average outgoing quality limit (AOQL) ^[3]. This work provided the theoretical basis for modern acceptance sampling procedures.

During World War II the United States military faced unprecedented demands for reliable munitions and equipment. With production scales overwhelming traditional 100 percent inspection methods the military adopted statistical sampling techniques to ensure quality while managing resources efficiently ^[4]. This led to the development of the Army Service Forces inspection tables in 1942 and 1943.

The first version of MIL-STD-105 was issued in 1950 as MIL-STD-105A and went through several revisions (B, C, D and E) until the final version MIL-STD-105E in 1989 ^[5]. Although the military standard was cancelled in 1995 its methodology persists through civilian adaptations including ANSI/ASQ Z1.4 and the international standard ISO 2859.

Definition and concept

According to ISO 2859 the acceptable quality limit is defined as the quality level that is the worst tolerable process average when a continuing series of lots is submitted for acceptance ^[6]. It represents the maximum percentage of defective units in a batch that is considered acceptable for the purposes of sampling inspection.

The AQL is not a guarantee that no defective units will be shipped. Rather it establishes a statistical threshold that balances the costs of inspection against the risks of accepting some defective items. Lots at or better than the AQL will be accepted approximately 95 percent of the time while lots worse than the AQL will be rejected more frequently.

Common AQL values

Different AQL values are used depending on the criticality of defects and the nature of the product ^[7]:

AQL 0.65% - Used for critical defects that could cause injury or safety hazards
AQL 1.0% - Applied to products with low risk tolerance such as medical supplies or precision instruments
AQL 2.5% - The most common level for general consumer products representing major defects
AQL 4.0% - Used for minor defects where some imperfections are acceptable

Sampling plans and inspection levels

ISO 2859 establishes a comprehensive system for determining sample sizes and acceptance criteria based on lot size, inspection level and the chosen AQL ^[8].

Inspection levels

Three general inspection levels are provided for routine use:

Level I - Provides less discrimination and smaller sample sizes. Used when less stringent inspection is needed or when the cost of testing is high
Level II - The default level used for most inspections. Provides a balance between sample size and discrimination
Level III - Requires larger samples and provides greater discrimination. Used when more rigorous inspection is necessary

Four special inspection levels (S-1, S-2, S-3 and S-4) are also available for situations where relatively small sample sizes are necessary and larger sampling risks can be tolerated.

Sample size determination

The sample size is determined through a two-step process. First the lot size and inspection level are used to find a code letter from a lookup table. Then the code letter is combined with the AQL to determine the sample size and the acceptance and rejection numbers.

Lot Size	Level I	Level II	Level III
2 to 8	A	A	B
9 to 15	A	B	C
16 to 25	B	C	D
26 to 50	C	D	E
51 to 90	C	E	F
91 to 150	D	F	G
151 to 280	E	G	H
281 to 500	F	H	J
501 to 1200	G	J	K

Acceptance and rejection criteria

For each combination of sample size code and AQL the standard provides an acceptance number (Ac) and a rejection number (Re) ^[9]. If the number of defective units found in the sample is less than or equal to the acceptance number the lot is accepted. If it equals or exceeds the rejection number the lot is rejected.

For example with a sample size of 125 units and an AQL of 2.5 percent the acceptance number might be 7 and the rejection number 8. This means:

Finding 7 or fewer defects results in lot acceptance
Finding 8 or more defects results in lot rejection

Producer's risk and consumer's risk

Two types of risk are inherent in any sampling plan ^[10]:

Producer's risk (Alpha)

The producer's risk is the probability of rejecting a lot that actually meets the quality requirements. It is also called the Type I error or false rejection rate. In standard sampling plans this risk is typically set at 5 percent meaning that good lots will be incorrectly rejected about 5 percent of the time.

Consumer's risk (Beta)

The consumer's risk is the probability of accepting a lot that should have been rejected because it contains too many defects. It is also called the Type II error or false acceptance rate. This risk is typically set at 10 percent at the lot tolerance percent defective (LTPD) level.

Lot tolerance percent defective

The LTPD represents a quality level that the sampling plan will routinely reject ^[11]. It is defined as the percent defective at which the probability of acceptance is only 10 percent. While the AQL represents acceptable quality the LTPD represents clearly unacceptable quality.

The relationship between AQL and LTPD defines the discrimination power of a sampling plan. A single sampling plan with sample size n=50 and acceptance number a=1 has an AQL of approximately 0.72 percent and an LTPD of approximately 7.6 percent. Lots between these two values are sometimes accepted and sometimes rejected.

Switching rules

ISO 2859 includes provisions for switching between normal, tightened and reduced inspection based on recent quality history ^[12]:

Normal to tightened

Tightened inspection is instituted when two out of five consecutive lots have been rejected during normal inspection. Tightened inspection uses stricter acceptance criteria making it harder for lots to pass.

Tightened to normal

Normal inspection is restored when five consecutive lots have been accepted during tightened inspection without any rejections.

Normal to reduced

Reduced inspection which uses smaller sample sizes can be instituted when production is steady and the preceding ten lots have been accepted. This reduces inspection costs when quality is consistently good.

Applications and industries

AQL-based sampling is widely used across many industries ^[13]:

Manufacturing - Inspection of components and finished goods before shipment
Import/export - Verification of product quality before goods leave the factory or clear customs
Retail - Supplier audits and incoming goods inspection
Electronics - Testing of electronic components and assembled products
Textiles and apparel - Inspection of fabrics and finished garments
Food and pharmaceuticals - Quality control of packaged goods

Advantages of AQL sampling

Reduces inspection costs compared to 100 percent inspection
Provides objective criteria for accept/reject decisions
Based on sound statistical principles with known risk levels
Internationally recognized through ISO 2859 standard
Flexible system adaptable to different product types and quality requirements
Encourages suppliers to maintain consistent quality

Limitations of AQL sampling

Does not guarantee zero defects in accepted lots
May accept lots with defect rates higher than the AQL
Requires proper training to implement correctly
Sample selection must be truly random for results to be valid
Not suitable for very small lot sizes
Does not address process improvement only product disposition

Acceptable quality level — recommended articles

Quality control — Quality management — Sampling — Statistical process control — Quality — Standard — Risk — Production — Product

References

ISO 2859-1:1999, Sampling procedures for inspection by attributes - Part 1: Sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection.
Dodge H.F., Romig H.G. (1959), Sampling Inspection Tables: Single and Double Sampling, John Wiley & Sons, 2nd edition.
Montgomery D.C. (2019), Introduction to Statistical Quality Control, John Wiley & Sons, 8th edition.
Schilling E.G., Neubauer D.V. (2017), Acceptance Sampling in Quality Control, CRC Press, 3rd edition.
Quality Magazine (2024), Brief History of ANSI/ASQ Z1.4.

Footnotes

ISO 2859-1:1999, Section 3.1
Montgomery D.C. (2019), pp. 645-650
Dodge H.F., Romig H.G. (1959), pp. 1-15
Quality Magazine (2024)
Quality Magazine (2024)
ISO 2859-1:1999, Section 3.1
Montgomery D.C. (2019), pp. 660-665
ISO 2859-1:1999, Section 9
Schilling E.G., Neubauer D.V. (2017), pp. 85-95
Montgomery D.C. (2019), pp. 655-658
Schilling E.G., Neubauer D.V. (2017), pp. 45-50
ISO 2859-1:1999, Section 10
Montgomery D.C. (2019), pp. 670-675

Author: Sławomir Wawak