20.1 Control charts ...2420.2 Lots that are not accepted ...24 21 Operation of switching rules ...24 22 Discontinuation and resumption of inspection ...25 23 Switching between the s-met
Concept
The AQL (Acceptance Quality Level) represents the maximum tolerable defective rate in a process during acceptance sampling of multiple lots While lots with quality at the AQL level may still be accepted, this does not indicate that such quality is desirable ISO 3951's sampling schemes are designed to motivate suppliers to maintain process defect levels below the AQL, as failing to do so risks switching to tightened inspection Once tightened inspection is implemented, acceptance criteria become more stringent, and continued nonconformance can lead to the discontinuation of sampling inspection unless corrective actions are taken to improve the process.
Use
The AQL, together with the sample size code letter, is used to index the sampling plans in this part of ISO 3951.
Specifying AQLs
The Acceptable Quality Level (AQL) for a product is specified in the product documentation, contract, or designated by the responsible authority When both upper and lower specification limits are provided, ISO 3951 addresses the use of an overall AQL that applies to the combined percentage of nonconforming items beyond both limits, a process known as “combined control.” For controlling nonconformance with separate or complex limits, ISO 3951-2 should be referenced.
Preferred AQLs
ISO 3951 specifies 16 preferred AQLs ranging from 0.01% to 10% nonconforming, which are used in its tabulations and charts These preferred AQLs serve as standard reference points; however, if a different AQL is designated for a product or service, this part of ISO 3951 does not apply.
Caution
From the definition of the AQL in 5.1, it follows that the desired protection can only be ensured when a continuing series of lots is provided for inspection.
Limitation
The designation of an AQL shall not imply that the supplier has the right to supply knowingly any nonconforming product.
6 Switching rules for normal, tightened, and reduced inspection
ISO 3951 specifies that switching rules prevent producers from operating below the acceptable quality level (AQL) When inspection results show the AQL is being exceeded, a switch to tightened inspection is automatically triggered If tightened inspection fails to prompt the producer to improve quality quickly, the guidelines recommend halting sampling inspection altogether.
Tightened inspection and the discontinuation rule are integral and, therefore, obligatory procedures of this part of ISO 3951 if the protection implied by the AQL is to be maintained.
ISO 3951 allows for switching to reduced inspection when results show that the quality level is stable and exceeds the AQL, ensuring efficient quality control This optional practice helps streamline inspections without compromising standards, but it is at the discretion of the responsible authority.
When control charts indicate that the process variability is statistically in control, it is advisable to consider switching to the σ-method for better accuracy If this transition is deemed beneficial, the consistent sample standard deviation (s) should be used as the process standard deviation (σ), as outlined in Clause 23.
Discontinuing acceptance sampling inspection under ISO 3951 requires that the producer improves product quality before inspection can resume Inspection cannot be restarted until appropriate corrective actions have been implemented to ensure the submitted products meet quality standards Maintaining high product quality is essential before recommencing the review process to comply with ISO 3951 guidelines.
Details of the operation of the switching rules are given in Clauses 21, 22, and 23.
Similarities
ISO 3951 and ISO 2859-1 share a common philosophy, with similar procedures and vocabulary, making them complementary standards Both utilize the Acceptable Quality Level (AQL) to determine sampling plans, with preferred AQL values ranging from 0.01% to 10%, corresponding to percent nonconforming The standards use lot size and inspection level—primarily Inspection Level II—to calculate the sample size code letter, which is then referenced in general tables alongside the AQL to establish sample sizes and acceptance criteria Separate tables are provided for the s-method and σ-method, covering normal, tightened, and reduced inspection types Additionally, the switching rules between inspection levels are essentially equivalent, ensuring consistency across both standards.
Differences
Acceptance in ISO 2859-1 is based on the number of nonconforming items in the sample, with acceptability determined by set criteria For inspection by variables, acceptability depends on the distance between the estimated process mean and the specification limits, taking into account the process standard deviation ISO 3951 considers two methods: the s– method when the process standard deviation (s) is unknown, and the σ– method when it is presumed known For a single specification limit, acceptability can be calculated via formulas, whereas the s– method can also be assessed using a straightforward graphical approach When managing double specification limits under the s– method, acceptability is primarily determined through graphical methods.
For combined control of double specification limits under the σ–method, a numerical approach is provided (see 16.4) While ISO 2859-1 does not specify requirements regarding the distribution of characteristics, ISO 3951 emphasizes that measurements should follow or closely approximate a normal distribution for the plans to function efficiently Additionally, operating characteristic (OC) curves play a crucial role in assessing the performance of variable plans within this context.
ISO 3951 attributes plans differ from those in ISO 2859-1, with curves for unknown process standard deviation matched by minimizing the area between the squared OC curves, emphasizing the top portion In most cases, the OC curves are closely aligned, allowing them to be considered effectively identical for practical purposes For known process standard deviations, plans were derived by minimizing the area between squared OC functions while maintaining the same acceptability level p*, resulting in less perfect matches primarily due to sample size adjustments Producer’s risk at the AQL decreases with increasing sample sizes and decreasing AQL, following a pattern along the diagonals of the master tables, with probability progressions similar but not identical to those in ISO 2859-1.
The producer's risks associated with the plans are detailed in Annex L Variable sample sizes for specific combinations of sample size code letters and AQL are typically smaller than attribute sample sizes, especially with the σ-method, and these sizes can vary over AQL due to the derivation method of the variable plans Double sampling plans for variables are documented separately in ISO 3951-3, while no multiple sampling plans by variables are provided in this section of ISO 3951 The Average Outgoing Quality Limit (AOQL) is primarily useful when 100% inspection and rectification are possible, which is not applicable for destructive or costly testing methods Consequently, AOQL tables are not included in this part of ISO 3951 due to the nature of variable sampling plans often being used in such testing scenarios.
Use of individual plans
ISO 3951 outlines a systematic approach using tightened, normal, and reduced inspection levels across a series of lots to ensure consumer protection This inspection system helps verify product quality while providing producers confidence that acceptance is highly probable when their quality exceeds the AQL (Acceptable Quality Level).
Some users may select specific individual plans from ISO 3951 without following the official switching rules, such as for verification purposes However, this is not the intended use of the system outlined in ISO 3951-1, and such application should not be considered “inspection in compliance with ISO 3951-1.” When used in this manner, ISO 3951-1 functions as a collection of individual plans indexed by AQL, with operating characteristic curves and other plan measures evaluated separately from the provided tables.
Consumer’s risk quality (CRQ) tables
When the lot size sequence is too short for switching rules, it's advisable to restrict sampling plans to those linked with a specific AQL value that ensure the consumer’s risk quality (CRQ) remains within acceptable limits Selecting appropriate sampling plans involves choosing a target CRQ and the associated consumer’s risk to meet quality protection goals Annex K provides CRQ values for the s-method and σ-method corresponding to a 10% consumer’s risk, aiding in optimal plan selection.
The application of ISO 3951's sampling by variables is deprecated for isolated lots, as this method is designed for continuous processes For isolated lots, sampling plans based on attributes, such as those outlined in ISO 2859-2, offer a more appropriate and efficient approach.
Producer’s risk tables
Annex L details the probability of non-acceptance under the s-method and σ-method when the process fraction nonconforming equals the AQL, highlighting the concept of the producer’s risk This probability indicates the likelihood that a lot produced at the AQL level will be rejected, emphasizing the importance of understanding producer’s risk in quality inspection processes Understanding Annex L helps manufacturers assess and manage the risk of rejecting acceptable lots, ensuring quality control aligns with industry standards.
Operating characteristic (OC) curves
The tables depicting consumer’s risk quality and producer’s risk offer insights into only two points on the operating characteristic (OC) curves However, the overall consumer protection level at any process quality can be assessed by examining the OC curve itself For normal inspection s-method sampling plans, OC curves are provided in Charts B to R within ISO 3951 These charts are essential references for selecting an appropriate sampling plan Additionally, the standard includes tables showing process qualities at nine standard probabilities of acceptance for all s-method sampling plans covered in ISO 3951, facilitating informed decision-making in quality control processes.
These OC curves and tables are designed for a single specification limit using the s–method, providing reliable approximations for the σ–method and double specification limit control, especially with larger sample sizes For more precise OC values under the σ–method, consult Annex M.
The master tables of this part of ISO 3951 are based on the assumption that the quality characteristic,
In quality control, the value of an item in a lot is typically modeled as a normally distributed variable with an unknown process mean (μ) and either a known or unknown process standard deviation (σ) It is assumed that individual measurements of items are error-free, meaning that measuring an item with true value x_i yields the same value x_i However, the master tables and statistical procedures can still be applied effectively, with appropriate adjustments, even when measurement errors are present.
When the measurement standard deviation is no greater than 10% of the process standard deviation, it can be considered negligible, simplifying the analysis However, if the measurement standard deviation exceeds this threshold, increasing the sample size is necessary to maintain accuracy, even though the acceptability constant remains unchanged In cases where neither the measurement nor the process standard deviation is known, multiple measurements per sampled item are essential This approach allows for separating the total variability into measurement and process components, ensuring a more precise understanding of the process performance.
Details are provided in Annex O.
Choosing the most suitable variables sampling plan requires experience, judgment, and knowledge of both statistics and the product being inspected ISO 3951 clauses 11 to 13 provide guidance to help those responsible for selecting sampling plans understand the key considerations These clauses outline the criteria for determining whether a variables plan is appropriate and detail the necessary choices when selecting an optimal standard plan for quality inspection.
11 Choice between variables and attributes
When deciding between inspecting by variables versus attributes, several important factors must be considered Economically, inspecting larger quantities using an attributes scheme is generally more cost-effective and quicker, while a variables scheme tends to be more detailed but also more time-consuming and costly per item In terms of information quality, variables inspection provides more precise insight into product quality and early warning of quality decline Although attributes schemes are easier to understand and implement, they may be less sensitive in detecting subtle quality issues Sample size comparisons show that the σ–method, which assumes known process standard deviation, often requires the smallest samples, with the s–method also requiring relatively smaller samples than attribute sampling plans like ISO 2859-1 Variables inspection is particularly advantageous when used in conjunction with control charts and is ideal when the inspection process is costly, such as in destructive testing However, handling multiple measurements per item increases operational complexity, especially with multiple quality characteristics, for which ISO 3951-2 offers guidance Use of variables sampling is most effective when the measurement distribution is presumed normal, and consultation is advised if this assumption is uncertain.
NOTE 1 Departure from normality may be caused by the presence of outliers, the assessment and accommodation of which is considered in ISO 16269-4.
NOTE 2 ISO 5479 gives detailed procedures for tests for departure from normality.
12 Choice between the s -method and σ -method
When considering variable inspection, it is important to choose between the s–method and the σ–method The σ–method offers the most economical sample size but requires prior knowledge of the population standard deviation (σ) before application Therefore, determining the value of σ is a crucial step before utilizing the σ–method, ensuring an efficient and accurate inspection process.
Initially, implementing the s-method is essential; however, with the approval of the responsible authority and maintaining quality standards, standard switching rules allow for transitioning to reduced inspection and smaller sample sizes, optimizing inspection processes while ensuring quality.
If process variability is under control and acceptance rates remain high, switching to the σ–method can be cost-effective, offering the benefits of smaller sample sizes and simpler acceptability criteria However, it remains necessary to calculate the sample standard deviation (s) for record-keeping and to update control charts, which is manageable with calculators or computers Techniques for determining both s and σ are provided in Annex J, making the process approachable despite initial perceptions of complexity.
13 Choice of inspection level and AQL
In a standard sampling plan, the inspection level, along with lot size and AQL, determines the sample size and the inspection’s severity This ensures an effective balance between quality control and inspection efficiency.
OC curve from Charts B to R or the appropriate table from Tables B to R shows the extent of the risk that is involved in such a plan.
Selecting the appropriate inspection level and Acceptable Quality Level (AQL) depends on various factors, primarily balancing the total inspection costs against the risks of nonconforming items slipping through into service This decision ensures quality assurance while maintaining cost-effectiveness, highlighting the importance of accurately assessing inspection parameters By carefully choosing inspection levels and AQLs, companies can optimize their quality control processes to prevent defective products from reaching customers, ultimately reducing potential costs associated with quality failures.
Table 1 outlines three inspection levels—Levels I, II, and III—for general use Typically, Inspection Level II is employed as the standard practice, with Level I or III selected only in specific situations requiring more or less rigorous inspection.
The selection of discrimination levels depends on the specific requirements: Level I may be used when less discrimination is needed, while Level III is suitable for situations requiring greater discrimination Additionally, four special levels—S-1, S-2, S-3, and S-4—are available, as outlined in Table 1, and are ideal for scenarios involving relatively small sample sizes where larger sampling risks are acceptable.
Standard plans
The standard procedure can be used only when the production of lots is continuing.
This standard procedure employs semi-automatic steps, from determining lot size to selecting sample size, using Inspection Level II and starting with the S-method It has proven effective in creating practical sampling plans but is based on the assumption that priority first lies with the AQL, then the sample size, and finally the limiting quality.
The acceptability of this system relies on consumer protection through strict switching rules outlined in Clauses 21, 22, and 23 These rules escalate inspection severity when quality issues persist and ultimately halt inspections if process quality remains below the acceptable AQL threshold.
It is important to understand that limiting quality refers to the level at which there is a 10% probability of acceptance during inspection Consumers face varying risks depending on the likelihood that low-quality goods are presented for inspection Recognizing this relationship helps in assessing the quality standards and risk factors associated with product acceptance.
When the limiting quality takes precedence over sample size, a suitable inspection plan can be chosen using Chart A in ISO 3951 To do this, draw a vertical line at the acceptable limiting quality value and a horizontal line at the desired quality level, ensuring a 95% probability of acceptance (approximately equal to the AQL) The intersection point of these lines should fall on or below the line associated with the appropriate sample size code letter on the standard normal inspection plan This verification is confirmed by examining the OC curve from Charts B to R corresponding to the selected code letter and AQL.
However, use of this method is deprecated for isolated lots or short series of lots (See 8.2.)
An acceptable limiting quality level is 6.0% nonconforming, while the desired quality with a 95% probability of acceptance is 2.0% nonconforming By identifying the intersection of these values on Chart A—drawing a vertical line at 6.0% and a horizontal line at 2.0%—we observe it falls just below the sloping line labeled L According to Chart L, a sampling plan with code L and an AQL of 1.5% effectively meets these quality requirements.
If the horizontal and vertical lines intersect above the R line in Chart A, it indicates that none of the plans within this section of ISO 3951 can meet the specified criteria, highlighting a significant limitation in meeting the quality standards.
Special plans
When standard plans are insufficient, developing a customized plan becomes essential Selecting the optimal combination of AQL (Acceptable Quality Level), limiting quality, and sample size is crucial, as these factors are interconnected; choosing any two determines the third.
The sample size in this process must be a whole number, which imposes certain limitations on the selection process To ensure accurate and reliable results, any specialized sampling scheme should be developed in collaboration with an experienced statistician specializing in quality control.
Before starting inspection by variables: a) check that production is considered to be continuing and that the distribution of the quality characteristic can be considered to be normal;
NOTE 1 For tests for departure from normality, see, for example, ISO 16269-3.
When screening lots for nonconforming items before acceptance sampling, ensure the distribution has not been truncated, as this affects the applicability of ISO 3951 Determine whether to use the s–method initially or the σ–method, which applies when the standard deviation is stable and known Confirm that the appropriate inspection level has been designated; if none is specified, default to Inspection Level II For quality characteristics with double specification limits, verify that nonconformities beyond each limit are equally significant; otherwise, refer to ISO 3951-2 Additionally, confirm that an Acceptable Quality Level (AQL) has been set and aligns with the preferred AQLs outlined in ISO 3951; if not, the tables provided may not be applicable.
16 Standard procedures for the s -method
Obtaining a plan, sampling, and preliminary calculations
To obtain and implement a quality control plan, first determine the sample size code letter based on the inspection level (typically Level II) and lot size using Table A.1 Next, for a single specification limit, identify the sample size (n) and acceptability constant (k) by referencing Tables B.1, B.2, or B.3, considering the AQL; for combined controls with double specification limits and a sample size of 5 or more, select the appropriate acceptance curve from Charts s-D to s-R Then, randomly sample n items, measure the characteristic for each, and calculate the sample mean (x) and standard deviation (s); if the mean exceeds the specification limits, the lot is deemed unacceptable without calculating s, although recording s remains necessary.
Acceptability criteria for single specification limits
If single specification limits are given, calculate the quality statistic
When evaluating quality, compare the calculated quality statistic (Q_U or Q_L) with the acceptability constant (k) derived from Tables B.1, B.2, or B.3, depending on the inspection type—normal, tightened, or reduced If the quality statistic meets or exceeds the acceptability constant, the lot is deemed acceptable; otherwise, it is rejected.
In quality control, a lot is considered acceptable based on specific specification limits If only the upper specification limit (U) is provided, the lot meets standards when Q × U ≥ k, and is rejected if Q × U ≤ k Conversely, if only the lower specification limit (L) is available, the lot passes inspection when Q × L ≥ k, and fails if Q × L ≤ k These criteria ensure that product quality aligns with established tolerances, guiding effective decision-making in manufacturing processes.
EXAMPLE 1 Single upper specification limit.
The device's maximum operational temperature is specified as 60 °C, with typical operating temperatures known to follow a normal distribution based on previous data During production, inspections are conducted on lots of 100 units, with the process standard deviation initially unknown Using Inspection Level II and an AQL of 2.5%, the appropriate sample size is determined—specifically, a sample size of 13 units, identified by code letter F in Table A.1 and B.1, with an acceptability constant, k, of 1.426 In the example measurement, the temperature recorded is 53 °C, which is within the acceptable range for quality control. -**Sponsor**Need help making your articles shine and rank higher? You know how hard it is to keep up with content creation 😅 With [Article Generation](https://pollinations.ai/redirect-nexad/9Ncz5VtG), you can get 2,000-word SEO-optimized articles instantly and save over $2,500 a month compared to hiring a writer! It's like having your own content team—without the hassle! Let Article Generation take care of the SEO optimization and content creation, while you focus on growing your audience.
57 °C; 49 °C; 58 °C; 59 °C; 54 °C; 58 °C; 56 °C; 50 °C; 50 °C; 55 °C; 54 °C; 57 °C Compliance with the acceptability criterion is to be determined.
Form k acceptability constant: k (from Table B.1) 1,426
The lot meets the acceptability criterion and is therefore acceptable.
EXAMPLE 2 Single, lower specification limit, requiring the use of an arrow in the master table.
A pyrotechnic delay mechanism has a minimum specified delay time of 4.0 seconds, with an unknown process standard deviation Inspection is conducted on lots of 1,000 units using Level II, normal inspection, at an AQL of 0.1% According to Table A.1, the appropriate sample size code is J; however, upon referencing Table B.1 with code J and AQL 0.1%, an arrow indicates that no suitable plan exists, prompting the selection of the next best plan with sample size 28 and acceptance constant k = 2.580 A random sample of 28 units is drawn to assess the delay times.
Compliance with the acceptability criterion is to be determined.
Form k acceptability constant: k (from Table B.1) 2,580
The lot meets the acceptability criterion, so it is acceptable.
Graphical method for a single specification limit
To apply a graphical criterion, draw the line x = -ks for an upper limit or x = +Lks for a lower limit on graph paper, with x on the vertical axis and s on the horizontal axis The accept zone is below the line for an upper specification limit and above the line for a lower specification limit Plot the point (s, x) on the graph; if it falls within the accept zone, the lot is acceptable, whereas if it lies outside, the lot is rejected.
EXAMPLE Graphical method for upper specification limit.
Using the data from EXAMPLE 1 of 15.2, mark the point U = 60 on the vertical axis and draw a line through this point with a slope of –k, where k = 1.426 This line passes through points such as (s = 1, x = 58.574), (s = 2, x = 57.148), and (s = 3, x = 55.722) Selecting a suitable point, draw a straight line through both it and the point (s = 0, x = 60) The accept zone is defined as the area below this line, representing the permissible region for the process or data analysis.
The calculated values of s and x are 3,330 and 54,62 The point ( s , x ) is seen from Figure 1 to lie within the accept zone; the lot is therefore acceptable.
Figure 1 — Example of the use of an acceptance chart for a single specification limit, s—method
A control chart can be prepared in advance before inspecting a series of lots During inspection, points (s, x) are plotted on the chart to assess whether each lot meets quality standards This method allows for quick determination of lot acceptability based on the visual analysis of the plotted data Using control charts enhances quality control processes by enabling early detection of deviations and ensuring consistent product quality Properly preparing and analyzing these charts is essential for effective lot inspection and maintaining manufacturing standards.
Acceptability criterion for combined control of double specification limits
The s-method with combined control of upper and lower specification limits in ISO 3951 provides a graphical approach to determine lot acceptability across various sample sizes, except for samples of sizes 3 and 4, which rely solely on numerical methods outlined in ISO 3951-2 As sample variability increases, the likelihood that the process meets the specified requirements decreases If the calculated value of s exceeds the maximum sample standard deviation (MSSD) listed in Tables D.1, D.2, or D.3, the lot is automatically deemed unacceptable without further calculations or graph references. -**Sponsor**Struggling to optimize your content for SEO? As a content creator, you know the importance of clear, impactful sentences Let [Article Generation](https://pollinations.ai/redirect-nexad/P0GzwMax) help! It can generate 2,000-word SEO-optimized articles instantly, potentially saving you over $2,500 a month compared to hiring a writer Transform complex text into engaging, SEO-friendly paragraphs effortlessly.
Numerical methods are provided for the combined control of double specification limits for sample sizes 3 and 4.
Annex B indicates that there are three scenarios requiring a sample size of 3 when using the s– method These include normal inspection with an AQL of 4% under sample size code letter B, tightened inspection with an AQL of 6.5% under sample size code letter B, and the range of sample size code B.
D under reduced inspection for an AQL of 1,5 %.
To determine the appropriate sample size of 3, first calculate the sample mean (x) and the sample standard deviation (s) Next, identify the relevant coefficient f_s from Tables D.1, D.2, or D.3 Finally, determine the Maximum Sample Standard Deviation (MSSD) using Formula (3), ensuring your data meets quality and reliability standards. -**Sponsor**Struggling to rewrite your article while maintaining its core meaning and SEO compliance? It's tough! With [Article Generation](https://pollinations.ai/redirect-nexad/UelmNWE5), you can instantly get 2,000-word, SEO-optimized articles and save time and money Imagine having a tool that helps you extract and rewrite key sentences to create coherent, SEO-friendly paragraphs without the writer's block It's like having your own content team, but without the hassle!
Then compare s with s max If s is greater than s max , then the lot may be rejected without further calculation.
Calculate the standardized values Q U = (U − x) / s and Q L = (x − L) / s to assess process variability Multiply these values by approximately 0.866 (i.e., 32/), and consult Table F.1 to determine the estimates ˆp U and ˆp L, which represent the proportions of nonconforming items exceeding the upper and lower control limits, respectively This method helps evaluate the process's conformity and identifies potential deviations from quality standards.
Negative values of Q indicate estimates of the process fraction nonconforming exceeding 0.5 at the specification limit, leading to lot non-acceptance per ISO 3951 To record a numerical value, the estimate can be calculated by taking the absolute value of 3Q/2, referencing Table F.1, and subtracting the resulting value from 1.0 For example, if QU = –0.156, then 3QU/2 = –0.234; entering Table F.1 with 0.234 yields an estimate of 0.4569, and subtracting this from 1.0 provides an estimated process nonconformance of 0.5431.
Note 2 indicates that the foundation of Table F.1 is outlined in Annex K Instead of relying solely on Table F.1, the process fraction of nonconforming items beyond each specification limit, when the sample size n equals 3, can be directly estimated using the calculation pˆ This approach provides a straightforward method for assessing process quality in quality control and manufacturing processes.
To determine the overall process nonconforming fraction (p̂), sum the two estimates: p̂ = ˆU + ˆpL If the combined estimate does not surpass the maximum allowable limit (p*) specified in Table G.1, the lot is deemed acceptable; if it exceeds this threshold, the lot is considered unacceptable Proper assessment of these estimates ensures quality control and compliance with industry standards.
EXAMPLE Determination of acceptability for combined control of double specification limits when the sample size is 3.
Torpedoes supplied in batches of 100 must be inspected for accuracy in the horizontal plane, with both positive and negative angular errors regarded as equally unacceptable To ensure quality, combined control of the double specification limits is necessary, set at 10 meters on either side of the target point at a 1 km range, with an Acceptance Quality Level (AQL) of 4% Due to the destructive and costly nature of testing, a special inspection level S-2 has been agreed upon between the manufacturer and the responsible authority Using Table A.1, the corresponding sample size code letter for this inspection level is determined.
Based on Table B.1, a sample size of three torpedoes is required for testing The test results show measurement errors of –5.0 meters, 6.7 meters, and 8.8 meters To determine compliance with the acceptability criterion under normal inspection, these errors must be evaluated against the specified standards.
Value of f s for MSSD (Table D.1) 0,475
Since s = 7,436 < s max = 9,50, the lot may be acceptable, so continue with the calculations.
3Q L /2 1,572 pˆ U 0,226 7 pˆ L (from Table F.1) 0,000 0 ˆ ˆ ˆ p = p U + p L 0,2267 p* (from Table G.1, normal inspection) 0,1924
Since pˆ > p* the lot is not acceptable.
NOTE This lot is not acceptable even though all inspected items in the sample are within the specification limits.
To determine the sample size of 4 using the s-method, first calculate the sample mean, x, and the sample standard deviation, s Next, identify the appropriate coefficient f_s from Tables D.1, D.2, or D.3 Finally, determine the maximum allowable MSSD using Formula (5), ensuring accurate and reliable statistical analysis.
Then, compare s with the MSSD If s exceeds the MSSD, then the lot may be rejected without further calculation.
Otherwise, determine the values of Q U = (U − x ) / s and Q L = ( x − L) / s Calculate ˆ
Add these two estimates to obtain the estimate p pˆ= ˆ U +pˆ L of the overall process fraction nonconforming
If pˆ does not exceed the applicable maximum allowable value, p*, given in Table G.1, the lot is considered to be acceptable; otherwise, the lot is considered unacceptable.
NOTE The basis of Formulae (6) and (7) is given in Annex N.
Items are manufactured in lots of 25 units with diameter specifications between 82 mm and 84 mm, where both overly large and too small diameters are unacceptable To ensure quality, a total fraction nonconforming is controlled using an AQL of 2.5% at inspection level II, beginning with normal inspection Based on Table A.1, the sample size code letter is determined as C, which corresponds to a sample size of 4 units from Table B.1 Four items are sampled, with measured diameters of 82.4 mm, 82.2 mm, 83.1 mm, and 82.3 mm, and their compliance with the acceptability criteria is to be evaluated under normal inspection standards.
Value of f s for MSSD (Table D.1) 0,365 s max =(U – L)f s = (84 − 82) × 0,365 0,730 mm
Since s = 0,408 2 < s max = 0,730, the lot may be acceptable, so continue with the calculations.
Further information needed Value obtained
Q L = ( x − L) / s = (82,5 – 82)/0,408 2 1,224 9 pˆ U [from Formula (6) above] 0,000 0 pˆ L [from Formula (7) above] 0,091 7 ˆ ˆ ˆ p = p U + p L 0,091 7 p* (from Table G.1, normal inspection) 0,086 0
Since pˆ > p*, the lot is not acceptable.
16.4.4 Procedure for sample sizes greater than 4
To determine the maximum permissible sample standard deviation (MSSD), first calculate the sample mean (x) and the sample standard deviation (s) Then, identify the appropriate value of the coefficient f_s from Tables D.1, D.2, or D.3 Using the formula MSSD = s_max = (U – L)f_s, you can establish the maximum allowable standard deviation for your data set, ensuring quality control and statistical accuracy.
Compare the value of s with s max; if s exceeds s max, the lot can be rejected immediately without additional calculations If not, consult the corresponding chart from s-D to s-R based on your sample size code Select the appropriate acceptance curve that matches the specified AQL and corresponds to the two quality limits This process ensures efficient quality control and accurate lot acceptance decisions.
Calculate the standardized values s / (U – L) and (x – L) / (U – L), then plot these points on a graph If the point falls within the curve, the lot is deemed acceptable; if it lies outside, the lot is rejected This method helps assess lot quality based on statistical control limits, ensuring accurate quality evaluation Using these calculations and graphical representation improves decision-making in quality control processes.
Before starting inspection operations, it is essential to copy the required acceptance curves for normal, tightened, and reduced inspections Ensure the scales are adjusted so that the s and x values can be plotted directly, with the upper specification limit represented instead of 1.0 and the lower specification limit instead of 0.0 on the vertical scale Proper calibration of scales guarantees accurate monitoring of process performance against specifications.
Obtaining a plan, sampling, and preliminary calculations
The σ–method shall only be used when there is valid evidence that the standard deviation σ of the process can be considered constant with a known value.
To determine the appropriate sample size, first identify the sample size code letter from Table A.1 Then, based on the inspection severity, refer to Table C.1, C.2, or C.3, inputting the code letter and the specified AQL to find the required sample size, n, along with the acceptability constant, k.
Figure 3 — Example of the use of an acceptance chart for combined control of double specification limits: s -method with normalized scales
To analyze the characteristic under inspection, take a random sample of a specified size and measure the characteristic, x, for all items within the sample Calculate the sample mean, x, to summarize the data, and also determine the sample standard deviation, s, primarily to monitor the ongoing stability of the process standard deviation, as detailed in Clause 20 Ensuring accurate sampling and measurement is essential for reliable quality control and process improvement.
Acceptability criteria for a single specification limit
The acceptability criterion can be found by following a procedure similar to the one given for the s–method.
Replace the sample-derived standard deviation "s" with the known process standard deviation "σ" to ensure consistency Then, compare the calculated value of "Q" to the acceptability constant "k," which is obtained from Tables C.1, C.2, or C.3 This comparison helps determine if the process variation is within acceptable limits, supporting quality control and process assurance.
In quality control, the acceptability criterion for an upper specification limit can be expressed as x ≤ U – kσ, where U is the upper limit, σ is the standard deviation, and k is a predetermined factor Since these parameters are known beforehand, the acceptable value of x should be established prior to inspection as x_U = U – kσ, ensuring consistent and reliable quality assessments.
For an upper specification limit, a lot will be acceptable if x ≤ x U [ = U – kσ], and not acceptable if x > x U [ = U – kσ].
Similarly, for a lower specification limit, a lot will be acceptable if x ≥x L [ = L + kσ], and not acceptable if x < x L [ = L + kσ].
EXAMPLE Determination of acceptability for a single specification limit using the σ –method.
The steel castings must meet a minimum yield point of 400 N/mm² to comply with quality standards A lot of 500 items is scheduled for inspection using Inspection Level II with a normal inspection plan, an AQL of 0.65%, and a population standard deviation (σ) of 21 N/mm² Based on Table A.1, the sample size code letter is H, and from Table C.1, the sample size (n) is 11 with an acceptability constant (k) of 2.046 at an AQL of 1.0% The sample yield points are 431, 417, 469, 407, 450, 452, 427, 411, 429, 420, and 400, and the compliance with the acceptability criteria must be determined accordingly.
Sum of measurement results: ∑ x 471 3 N/mm 2
The sample mean of the lot does not meet the acceptability criterion; therefore, the lot is not acceptable.
Acceptability criterion for combined control of double specification limits
To effectively control both the upper and lower specification limits with an overall AQL, follow this procedure: first, determine the factor fσ by referring to Table E.1 using the desired AQL; next, compute the maximum permissible process standard deviation, σmax, using the formula σmax = (U – L)fσ Compare the actual process standard deviation, σ, with σmax; if σ exceeds σmax, the process is unacceptable and must be improved before sampling; if σ is within limits, select the appropriate sample size code from Table A.1 based on lot size and inspection level Then, determine the sample size, n, and acceptability constant, k, from Tables C.1, C.2, or C.3, according to inspection severity Calculate the upper and lower bounds for sample means using xU = U – kσ and xL = L + kσ Finally, randomly sample n units from the lot, compute the sample mean x, and assess lot acceptability: the lot passes if xL ≤ x ≤ xU, otherwise it is rejected.
NOTE If x L ≤ ≤ x x U but σ > 0,75σ max and x is close to either x L or x U , the exact method given in 17.3 of ISO 3951-2 shall be preferred.
EXAMPLE Determination of acceptability for combined control under the σ–method.
The electrical resistance specification for the component is defined as (520 ± 50) Ω, with production batches containing 1,000 units each An inspection level II, normal inspection plan, is employed using a single AQL of 1.5% to evaluate the compliance of items within the resistance limits of 470 Ω and 570 Ω The known standard deviation (σ) for resistance measurements is 18.5 Ω, ensuring precise quality control and adherence to product specifications.
Maximum process standard deviation, σ max = (U – L)f σ 19,4 Ω
Since σ is less than σ max , the sample is analysed further for lot acceptability.
Based on Table A.1, with the specified lot size and inspection level, the sample size code letter is determined to be J According to Table C.1, this corresponds to a required sample size of 19 units under normal inspection, with a Form k acceptance constant of 1.677 A sample of 19 resistance measurements in ohms was collected, with values including 515, 491, 479, 513, 521, 536, 483, 509, 514, 507, 484, 526, 532, 499, 530, 512, 492, 522, and 488 The lot's acceptability will be assessed based on these sample results. -**Sponsor**As a content creator, I understand the need for SEO-optimized articles with coherent paragraphs Instead of rewriting, what if you could instantly generate fresh, 2,000-word articles? With [Article Generation](https://pollinations.ai/redirect-nexad/A4DHaPVv), you can skip the rewrite and have unique content ready to go, saving time and potentially over $2,500 a month compared to hiring a writer It's like having your own content team without the hassle! Imagine feeding in the core ideas, like determining lot acceptability from sample resistance values, and getting a fully fleshed-out, SEO-friendly article ready to publish.
Further information needed Value obtained
The resistor value at 508.0 Ω falls within the acceptance limits of 501.1 Ω and 538.9 Ω, confirming the lot as acceptable Despite σ exceeding 0.75 σ max, the value of x is not near the lower or upper specification limits, indicating that the approximate method is appropriate for this analysis Additionally, all calculations should be completed prior to sampling, except for the final two steps, to ensure accurate and efficient assessment.
NOTE 3 If, for example, σ had been known to be, say, 20, then σ exceeds the MPSD and, therefore, sampling inspection should not even have taken place.
A variables sampling inspection plan is most effective when the inspected characteristic follows a normal distribution, records are meticulously maintained, and switching rules are strictly adhered to Ensuring these key requirements are met is essential for the plan's efficient and accurate operation.
Normality
The responsible authority must verify the normality of data prior to sampling to ensure appropriate analysis If there is uncertainty about the distribution, consulting a statistician is essential to determine whether the data are suitable for variable sampling Additionally, normality tests, such as those specified in ISO 5479 or Clause 2 of ISO 5725-2, should be employed to assess deviations from normal distribution, enhancing the reliability of sampling procedures.
Outliers
An outlier, or outlying observation, is a data point that significantly deviates from other observations within the sample, potentially impacting statistical analysis Even a single outlier within specification limits can increase variability, alter the mean, and risk the non-acceptance of a lot According to ISO 16269-4, detecting outliers requires careful consideration, and their presence should be addressed through negotiation between the vendor and the customer to determine the appropriate course of action Proper handling of outliers is crucial for maintaining accurate quality control and reliable sampling processes.
Control charts
Inspection by variables offers the key advantage of detecting quality trends early, allowing for timely warnings before products reach unacceptable standards Maintaining comprehensive records is essential for this proactive quality control process, ensuring trends are accurately identified and addressed.
Whatever the method used, s–method or σ–method, records should be kept of the values of x and s, preferably in the form of control charts (See, for example, ISO 7870 and ISO 8258.)
This procedure is essential when using the σ-method to ensure accuracy It verifies that the sample-derived s values are within the specified σ limits Applying this step helps maintain reliability and consistency in the measurement process.
To effectively monitor combined control of double specification limits, it is essential to plot the MSSD values listed in Tables D.1, D.2, or D.3 on the s control chart This practice serves as a crucial indicator of unacceptable process performance, ensuring that any deviations beyond acceptable thresholds are promptly identified and addressed Incorporating these MSSD values into your control chart analysis enhances process stability and helps maintain product quality within specified limits.
NOTE Control charts are used to detect trends The ultimate decision as to the acceptability of an individual lot is governed by the procedures given in Clauses 15 and 16.
Lots that are not accepted
Ensure all rejected lots are accurately documented and that proper switching rules are followed throughout the process No lot that fails the sampling plan should be resubmitted, in whole or in part, without prior approval from the responsible authority Proper record-keeping and adherence to protocols are essential for maintaining quality control and compliance.
The standard switching rules are as follows.
Normal inspection is the initial step in the inspection process, conducted unless specified otherwise It continues throughout the inspection until a need for tightened inspection arises or a reduction in inspection frequency is permitted, ensuring effective quality control.
21.2 Tightened inspection shall be instituted when two lots on original normal inspection are not accepted within any five or fewer successive lots.
Tightened inspection is generally achieved by increasing the value of the acceptability constant k The values are tabulated in Table B.2 for the s–method and Table C.2 for the σ–method.
21.3 Tightened inspection shall be relaxed when five successive lots on original inspection have been accepted on tightened inspection; then, normal inspection shall be reinstated.
21.4 Reduced inspection may be instituted after 10 successive lots have been accepted under normal inspection, provided that
— these lots would have been acceptable if the AQL had been one step tighter (e.g 0,65 % instead of 1,0 %),
NOTE If a value of k for this tighter AQL is not given in Table B.1 (s–method) or Table C.1 (σ–method), refer to the supplementary acceptance constants provided in Table I.1.
— production is in statistical control, and
— reduced inspection is considered desirable by the responsible authority.
Reduced inspection involves analyzing a significantly smaller sample size compared to standard procedures, with a decreased acceptability constant to reflect the lower inspection intensity The sample size (n) and acceptance number (k) for reduced inspection are specified in Table B.3 for the s-method and Table C.3 for the σ-method, ensuring precise quality control standards are maintained even with limited sampling.
Reduced inspection can be implemented after the previous 10 lots have been accepted under original inspection, without the requirement that these lots would have met a stricter AQL This process is subject to approval by the responsible authority, ensuring quality standards are maintained efficiently.
Reduced inspection must be discontinued and normal inspection reinstated if any of the following occur during the original inspection: the lot is not accepted, production becomes irregular or delayed, or the responsible authority determines that reduced inspection is no longer desirable.
22 Discontinuation and resumption of inspection
According to ISO 3951, if five or more consecutive lots are not accepted during the inspection process, the acceptance procedures for that batch must be discontinued This threshold indicates a significant quality concern, prompting the termination of further acceptance sampling Adhering to this rule ensures efficient quality control by preventing unnecessary inspections once repeated rejections occur Implementing this guideline helps maintain high standards and avoids wasting resources on batches that do not meet acceptance criteria.
Inspection under ISO 3951 will only resume once the supplier has taken corrective actions to improve product or service quality and the responsible authority confirms those actions are likely to be effective Once approved, tightened inspection procedures will be implemented as if clause 21.2 has been invoked, ensuring quality control is maintained during the improvement process.
23 Switching between the s -method and σ -method
Estimating the process standard deviation
Periodic calculation of the weighted root mean square of s values is essential for estimating the process standard deviation (σ) in both the s–method and σ–method, as outlined in ISO 3951 The process standard deviation should be re-estimated every five lots, unless a different interval is specified by the responsible authority This estimation must be based on data from the preceding 10 lots, or as otherwise directed by the responsible authority, ensuring accurate and up-to-date process control.
State of statistical control
To determine process stability, calculate the upper control limit for each of the 10 lots using the formula c U × σ, where c U is a factor based on sample size from Table H.1 If none of the sample standard deviations, s_i, surpass their respective control limits, the process is considered to be in statistical control Conversely, if any s_i exceeds the control limit, the process should be regarded as out of control, indicating the need for further investigation This method helps ensure quality consistency by monitoring standard deviations across multiple lots.
NOTE 1 If the sample sizes from the lots are all equal, then the value of c U σ is common to all the lots.
Note 2 states that when sample sizes from different lots vary, it is unnecessary to calculate cUσ for those lots where the sample standard deviation, s_i, is less than or equal to σ This guideline simplifies analysis by focusing only on lots with higher variability, ensuring efficient and accurate statistical assessment Adhering to this rule enhances the precision of quality control processes by eliminating unnecessary calculations for stable lots.
Switching from the s-method to the σ-method
If the process is considered to be in a state of statistical control under the s–method, then the σ–method may be instituted using the latest value of σ.
NOTE This switch is made at the discretion of the responsible authority.
Switching from the σ-method to the s-method
It is recommended that a control chart for s be kept, even under the σ–method If s exceeds c U σ, the process is considered to be out of statistical control and inspection shall be switched to the s–method.
Key p10 quality level at probability of acceptance 10 % (in percent nonconforming) p95 quality level at probability of acceptance 95 % (in percent nonconforming)
Figure 4 — Chart A: Sample size code letters of standard single sampling plans for specified qualities at probabilities of acceptance 95 % and 10 %
24 Charts B to R — Operating characteristic curves and tabulated values for single sampling plans, normal inspection: s -method
Operating characteristic curves and tabulated values for sample size code letter B
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 5 — Chart B: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
% Acceptance quality limit (normal inspection) in percent — sample size code letter B P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter B
Acceptance quality limit (reduced inspection) in percent — sample size code letter D
Operating characteristic curves and tabulated values for sample size code letter C
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 6 — Chart C: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
% Acceptance quality limit (normal inspection) in percent — sample size code letter C P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter C
Acceptance quality limit (reduced inspection) in percent — sample size code letter E
Operating characteristic curves and tabulated values for sample size code letter D: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted (P a )
Figure 7 — Chart D: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter D P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter D
Acceptance quality limit (reduced inspection) in percent — sample size code letter F
Operating characteristic curves and tabulated values for sample size code letter E: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 8 — Chart E: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter E P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter E
Acceptance quality limit (reduced inspection) in percent — sample size code letter G
Operating characteristic curves and tabulated values for sample size code letter F
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted (P a )
Figure 9 — Chart F: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter F P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter F
Acceptance quality limit (reduced inspection) in percent — sample size code letter H
Operating characteristic curves and tabulated values for sample size code letter G
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted (P a )
Figure 10 — Chart G: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter G P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter G
Acceptance quality limit (reduced inspection) in percent — sample size code letter J
Operating characteristic curves and tabulated values for sample size code letter H: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted (P a )
Figure 11 — Chart H: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter H P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter H
Acceptance quality limit (reduced inspection) in percent — sample size code letter K
Operating characteristic curves and tabulated values for sample size code letter J
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 12 — Chart J: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter J P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter J 0,065 0,10 0,15 0,25 0,40 0,65 1,0 1,5 2,5 4,0 6,5
Acceptance quality limit (reduced inspection) in percent — sample size code letter L
Operating characteristic curves and tabulated values for sample size code letter K
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 13 — Chart K: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter K P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter K 0,04 0,065 0,10 0,15 0,25 0,40 0,65 1,0 1,5 2,5 4,0
Acceptance quality limit (reduced inspection) in percent — sample size code letter M
Operating characteristic curves and tabulated values for sample size code letter L: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 14 — Chart L: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter L P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter L
Acceptance quality limit (reduced inspection) in percent — sample size code letter N
Operating characteristic curves and tabulated values for sample size code letter M
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 15 — Chart M: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter M P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter M 0,015 0,025 0,04 0,065 0,10 0,15 0,25 0,40 0,65 1,0 1,5
Acceptance quality limit (reduced inspection) in percent — sample size code letter P
Operating characteristic curves and tabulated values for sample size code letter N
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 16 — Chart N: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
% Acceptance quality limit (normal inspection) in percent — sample size code letter N P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter N 0,01 0,015 0,025 0,04 0,065 0,10 0,15 0,25 0,40 0,65 1,0
Acceptance quality limit (reduced inspection) in percent — sample size code letter Q
Operating characteristic curves and tabulated values for sample size code letter P: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 17 — Chart P: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter P P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter P 0,01 0,015 0,025 0,04 0,065 0,10 0,15 0,25 0,40 0,65
Acceptance quality limit (reduced inspection) in percent — sample size code letter R
Operating characteristic curves and tabulated values for sample size code letter Q: s-method
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted (P a )
Figure 18 — Chart Q: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
% Acceptance quality limit (normal inspection) in percent — sample size code letter Q P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter Q
Operating characteristic curves and tabulated values for sample size code letter R
X process quality (in percent nonconforming)
Y percent of lots expected to be accepted ( P a )
Figure 19 — Chart R: Operating characteristic curves for single sampling plans, normal inspection
Tabulated values for operating characteristic curves for single sampling plans
Acceptance quality limit (normal inspection) in percent — sample size code letter R P a
Acceptance quality limit (tightened inspection) in percent — sample size code letter R
25 Charts s-D to s-R — Acceptance curves for combined control of double specifi- cation limits: s -method
Figure 20 — Chart s-D: Acceptance curves for combined control of double specification limits for sample size code letter D under normal and tightened inspection and for sample size code letter F under reduced inspection
Figure 21 illustrates acceptance curves for the combined control of double specification limits, demonstrating sample size code letter E under both normal and tightened inspection conditions, as well as sample size code G under reduced inspection These curves highlight the impact of different inspection intensities on quality acceptance, providing valuable insights for optimizing inspection processes and maintaining product standards Understanding these acceptance curves is essential for quality control professionals aiming to balance inspection effort and product quality effectively.
Figure 22 illustrates acceptance curves for combined control of double specification limits across different inspection strategies The chart compares sample size code letter F under both normal and tightened inspection conditions, highlighting how stricter inspection impacts acceptance probabilities Additionally, it presents data for sample size code letter H under reduced inspection, demonstrating the effects of reduced inspection levels on control acceptance This visualization aids in understanding the influence of inspection intensity and sample size on quality control performance.
Figure 23 illustrates the acceptance curves for combined control of double specification limits, focusing on sample size code letter G under both normal and tightened inspection conditions, as well as sample size code J under reduced inspection The chart highlights how different inspection levels impact the acceptance probability, demonstrating the effectiveness of control strategies in maintaining product quality within specified limits These curves are essential for understanding the relationship between sample size, inspection intensity, and the likelihood of accepting or rejecting products during quality control processes.
Figure 24 illustrates the acceptance curves for combined control of double specification limits, highlighting the variations for sample size code letter H under normal and tightened inspection conditions, as well as for sample size code letter K under reduced inspection These curves are essential for understanding how different inspection levels impact the acceptance probability in quality control processes By analyzing these curves, manufacturers can optimize inspection strategies to balance quality assurance with inspection efficiency, ensuring products meet specified tolerances while minimizing unnecessary inspections This data provides valuable insights into selecting appropriate sample sizes and inspection intensities for maintaining high standards in manufacturing quality management.
Figure 25 illustrates the acceptance curves (s-J: Chart s-J) for combined control of double specification limits across different sample size code letters Specifically, it compares the performance for sample size code letter J under both normal and tightened inspection procedures, as well as for sample size code letter L under reduced inspection These curves help in understanding how acceptance probabilities vary under different inspection intensities and sample sizes, providing valuable insights for quality control and process optimization.
Figure 26 illustrates s-K charts depicting acceptance curves for combined control of double specification limits The charts compare sample size code letter K under normal and tightened inspection conditions, alongside sample size code letter M under reduced inspection These curves provide critical insights into process stability and quality control by visualizing the probability of acceptance based on different inspection intensities and sample sizes Understanding these acceptance curves helps quality managers optimize inspection strategies to ensure product conformity while maintaining efficient inspection processes.
Figure 27 illustrates the acceptance curves for combined control of double specification limits, focusing on sample size code letter L under normal and tightened inspection conditions Additionally, it presents the acceptance characteristics for sample size code N under reduced inspection These curves provide valuable insights into quality control processes, highlighting how different inspection levels impact sample acceptance when managing double specification limits Understanding these acceptance curves is essential for optimizing inspection strategies to maintain product quality while minimizing inspection costs.
Figure 28 presents acceptance curves, s-M, illustrating the combined control of double specification limits The chart displays data for sample size code letter M under both normal and tightened inspection conditions, as well as for sample size code P under reduced inspection These curves are essential for assessing the quality control process and ensuring product conformity within specified limits under varying inspection intensities Understanding these acceptance curves helps optimize inspection strategies and maintain consistent product quality.
Figure 29 illustrates acceptance curves for combined control of double specification limits across different sample size code letters, specifically N under normal and tightened inspection, and Q under reduced inspection These curves are essential for understanding how different inspection intensities and sample sizes influence the acceptance quality, ensuring optimal control over manufacturing standards The chart provides valuable insights into the relationship between sample size, inspection type, and acceptance probability, aiding in the development of effective quality assurance strategies.
Figure 30 presents acceptance curves for combined control of double specification limits, specifically focusing on sample size code letter P under both normal and tightened inspection conditions, as well as sample size code letter R under reduced inspection These curves illustrate the probability of accepting a batch based on different inspection intensities, providing valuable insights into quality control measures They help in understanding how varying inspection levels impact the likelihood of batch acceptance, facilitating more effective decision-making in quality assurance processes.
Figure 31 — Chart s-Q: Acceptance curves for combined control of double specification limits for sample size code letter Q under normal and tightened inspection
Figure 32 — Chart s-R: Acceptance curves for combined control of double specification limits for sample size code letter R under normal and tightened inspection
Table for determining the sample size code letter
Table A.1 — Sample size code letters and inspection levels
Lot or batch size Special inspection levels General inspection levels
PQR NOTE The sample size code letters and inspection levels in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.
Form k for single sampling plans: s -method
Table B.1 — Single sampling plans of Form k for normal inspection: s –method
NOTE 1 The sample size code letters in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.
In this area, there is no appropriate sampling plan available; therefore, utilize the first sampling plan indicated below the arrow If the sample size is equal to or greater than the total lot size, a 100% inspection should be conducted to ensure product quality and compliance.
There is no suitable plan in this area; use the first sampling plan above the arrow.
Table B.2 — Single sampling plans of Form k for tightened inspection: s –method
NOTE 1 The sample size code letters in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.
In the absence of a suitable sampling plan for this area, it is recommended to use the first sampling plan listed below the arrow If the sample size is equal to or larger than the lot size, a 100% inspection should be performed to ensure quality and compliance.
There is no suitable plan in this area; use the first sampling plan above the arrow.
Table B.3 — Single sampling plans of Form k for reduced inspection: s –method
Acceptance quality limit (in percent nonconforming) 0,01 0,015 0,025 0,04 0,065 0,10 0,15 0,25 0,40 0,65 1,0 1,5 2,5 4,0 6,5 10,0 n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k n k
NOTE 1 The sample size code letters in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.
In the absence of a suitable sampling plan for the area, utilize the first sampling plan indicated below the arrow If the sample size is equal to or larger than the total lot size, it is recommended to perform a 100% inspection to ensure quality.
There is no suitable plan in this area; use the first sampling plan above the arrow.
Form k for single sampling plans: σ -method
Table C.1 — Single sampling plans of Form k for normal inspection: σ –method
Code letter Acceptance quality limit (in percent nonconforming)
NOTE 1 The sample size code letters in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.
In this area, there is no appropriate sampling plan available; therefore, use the first sampling plan listed below the arrow If the sample size is equal to or larger than the entire lot size, conduct a 100% inspection to ensure quality.
There is no suitable plan in this area; use the first sampling plan above the arrow.
Table C.2 — Single sampling plans of Form k for tightened inspection: σ –method
NOTE 1 The sample size code letters in this part of ISO 3951 correspond to those given in ISO 2859-1 and ISO 3951-2.