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SOP Statistical Methods

Use this SOP Statistical Methods template to standardize and document your approach to statistical techniques, sampling plans, and data analysis for product development, process validation, and quality assurance activities. This template is essential for ensuring regulatory compliance, supporting risk-based decision-making, and justifying sample sizes and statistical methods during design verification, process capability studies, and supplier evaluations. Complete this SOP when implementing or revising statistical methods as part of your quality management system or prior to conducting verification, validation, or ongoing production monitoring.
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SOP Statistical Methods

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1. Purpose

This document provides guidance for the use of statistical techniques and tools at Consectetur adipiscing elit.

2. Scope

The procedure applies to the use of statistical techniques at Sed do eiusmod tempor during the development and evaluation of products or performance of other quality activities. The use of specific statistical techniques is not described as it is understood that many techniques are available and some may be more suited to a specific task than others.

It also applies for component and production activities performed by Incididunt ut labore to establish, control and verify the acceptability of process capability and product characteristics as well as relevant quality systems activities that require statistical analysis.

Component and production activities performed by contract manufacturers and other suppliers for establishing, controlling and verifying the acceptability of process capability and product characteristics will be addressed by established vendor procedures and/or agreements.

3. Responsibilities

  • Product and Software Teams - Responsible for identifying the need for statistical techniques and the application of those techniques.

4. References

  • ISO 13485:2016 Quality management systems — Requirements for regulatory purposes
  • Zero Acceptance Number Sampling Plans, Fifth Edition by Nicholas L. Squeglia

5. Records

Any statistical records generated will be filed and maintained with the relevant documentation, e.g., test protocol, etc.

6. Definitions

  • Attribute Data: Attribute measurements consist of noting the presence (or absence) of some characteristic or attribute in each of the units in the group under consideration, and counting how many units do (or do not) possess the quality attribute. Attribute data that are binary in nature and are classified in categories or groups. Examples of attribute data are pass/fail and yes/no. Data may only be placed in one category.
  • Characteristic: A distinguishing feature of a process or its output on which variables or attributes data can be collected.
  • Confidence: The certainty with which the results can be interpreted.
  • Reliability: The probability of performance without failures under stated conditions for stated time.
  • Variable Data: Variable measurements consist of measuring and recording the numerical magnitude of a quality characteristic for each of the units in the group under consideration. Data measured on a continuous scale or by counting (discrete). Examples include tensile strength (continuous) and things that are counted, such as errors in a process (discrete).

7. Procedure

  1. Statistical techniques will be employed as appropriate, in the development and maintenance of products and processes.
  2. Areas where statistical applications may be relevant are product development (design verification testing), and quality investigations/trending.
  3. Statistical techniques will be employed as appropriate to support manufacturing activities.
  4. When statistical techniques are identified as being necessary, teams will develop and document processes, procedures, and work instructions in order to implement such techniques, or describe or reference use of the technique, including a description of the collection and analysis of the data, e.g., in protocols or reports.
  5. When necessary, statistical training will be provided to employees responsible for the analysis of the data. Alternatively, consultants and other subject matter experts may be used to carry out statistical analysis or activities.

7.1 Sample Size Determination

Sample size determination is evaluated on a case-by-case basis. Each verification/validation plan should describe the sampling plan chosen, the sample size, and accept/reject criteria, as well as a justification. One of the following methods may be used:

  • Type test with acceptable justification: Acceptable justifications for using type testing include but are not limited to:
    • Type testing per standards or pre-existing data from sample population where the data is within existing tolerances.
    • For Deterministic Tests (e.g., a product characteristic or requirement that does not vary from product unit to product unit such as the presence or absence of a feature (software)), a sample size of 1 unit is acceptable.
  • Confidence/reliability criteria based severity:
    • Table 1. Severity Levels + Confidence and Reliability

Refer to Appendix A for Attribute Sample Sizes per Table 1 Confidence/Reliability.

7.2 Variable Data Set

Variable data with normal distribution can be evaluated using Appendix B to Appendix E regarding K-Factors for One-Sided and Two-Sided Tolerances. Sample size can be performed per appropriate standards, as necessary.

Various considerations may be used to establish the required sample size for a specific test/study. These include:

  • Observed or expected attributes of the sample or population, e.g., data average, distribution, the type of data (variable or attribute) and the specification against which the data will be assessed.
  • Resources such as industry standards, industry practices, regulatory bodies, recommendations by consultants or test labs.
  • Risk to the patient should be evaluated.
    • Example: The patient impact of sterility failure vs. cosmetic defects are significantly different in risk and therefore sample sizes should be evaluated differently.
  • Consistency of process or outputs.
    • Example: where for a specific input, the same specific output will result (all units will behave in the same manner or with limited variation as is typically seen in software testing or for electrical safety and EMC testing). Also, where an output is 100% verified as part of the production or release process, e.g., an electronic test, or the process has been validated.
  • Mitigation at other points in the process, e.g., the attribute or parameter is wholly or partially addressed elsewhere

Appendix A (Attribute Data)

Sample Size Needed to Show Compliance (Assuming Zero Failures for Attribute Data)

Sample size needed to show compliance (assuming 0 - 3 failures permissible for attribute data)
Attribute Data Sampling Plans

Appendix B: One-Sided Statistical Tolerance Limit Factors k for a Normal Distribution

One-sided factors cover X - ks or X + ks
where X and s are estimates of the mean and standard deviation computed from a sample size of n, and k is determined from the table below.

Appendix C: Two-Sided Statistical Tolerance Limit Factors k for a Normal Distribution

Two-sided factors cover X ± ks
where X and s are estimates of the mean and standard deviation computed from a sample size of n, and k is determined from the table below.

Appendix D: Guide to using K factors

K factors are used for variable data and the results are acceptable if the relevant upper or lower confidence limit (UCL or LCL, or both) meet product specifications.
The sample size needed to assess the capability of the product or process to meet the confidence limit(s) is determined as follows:

  • Based on the required reliability and confidence levels and whether the comparison(s) to specifications is either in one-direction (Appendix B) or both directions (Appendix C), determine the K factor (and associated sample size) such that the calculated UCL, LCL or both are within the required specifications. This may require repeated calculations to determine the maximum acceptable value of K (lowest sample size) that meets requirements.
  • Using a candidate K factor from Appendix B or C (as appropriate), perform the following calculation(s) using the values for the average and standard deviation expected in the test data to be collected.
    • UCL = average + (std deviation * K)
    • LCL = average - (std deviation * K)
  • If the UCL, LCL or both are outside the required specifications, choose a smaller K value (larger sample size) and repeat the above calculations until they are within specifications. Alternatively, a different confidence/reliability level may need to be used.
  • All quantitative data sets should be assessed for normality prior to the estimation of sample size and as part of the statistical analysis of the resulting sample data set. If there is evidence of non-normality and the sample size to be used is less than 30, then other methods of determining the appropriate sample size should be examined.

Appendix E: Sample Calculations Using K Factors

Data are planned to be collected to see if a process would meet a one-sided specification for curing time of 15.0 seconds (maximum time permitted). Requirements are 95% confidence and 95% reliability.

  • Sample size: 11
  • Expected Sample mean: 9.3 seconds
  • Expected Std. deviation: 1.97
  • K factor (from Appendix B): 2.815 (for sample size of 11)

Calculation

  • Formula: mean + (s * k)
  • Result: 9.3 + (1.97 * 2.815) = 14.86 seconds

A proposed sample size of 11 is the smallest sample size that results in an expected UCL that is within 15 seconds. The sample size may be increased beyond 11 to help ensure that final confidence levels are acceptable.

When the test has been completed and the data collected, the above calculation can be repeated with the observed sample mean and standard deviation to confirm that the UCL is within the specification.

If data is used for a two-sided specification with lower and upper control limits of 5 and 15 seconds respectively (the data is assessed at 95% confidence and 95% reliability):

  • Sample size: 25
  • Sample mean: 8.9 seconds
  • Std. deviation (s): 2.01
  • K factor (from table): 2.631 (for sample size of 25)

LCL Calculation

  • Formula: mean - (s * k)
  • Result: 8.9 - (2.01 * 2.631) = 3.61 seconds
  • Outcome: Fail at lower limit

UCL Calculation

  • Formula: mean + (s * k)
  • Result: 8.9 + (2.01 * 2.631) = 14.2 seconds
  • Outcome: Accept at upper limit

Appendix F: Zero Number Sampling Plan

The zero number sampling plan may be used to determine sample sizes for attribute inspections or inspections that are treated as attribute inspections. Note that rejects are not acceptable with this sampling plan, so a reject part indicates a rejection of the entire lot unless the lot is 100% inspected and sorted.
To build a Zero Acceptance Number Sampling Plan (C=0) based on the ANSI/ASQ Z1.4 standard, follow these steps. While I cannot reproduce the proprietary ANSI/ASQ Z1.4 tables directly, I can guide you through their methodology and provide a structure for how to determine your plan.

Steps to Create a C=0 Sampling Plan

  1. Determine Key Parameters
    • Lot Size (N): Total number of units in the batch.
    • Inspection Level: Choose between I (reduced), II (normal), or III (tightened).
    • Acceptance Quality Level (AQL): Desired quality threshold, typically expressed as a percentage.
  2. Refer to the Sampling Table
    • ANSI/ASQ Z1.4 tables provide the relationship between lot size, inspection level, and sample size code letters.
    • Match the lot size to a sample size code letter based on the inspection level.
  3. Find the Sample Size
    • Use the code letter from Step 2 to determine the actual sample size from the AQL table.
    • For a C=0 plan, the acceptance number is always zero.
  4. Define Acceptance Criteria
    • Inspect the sample. If any defects are found, the lot is rejected.

Example: Building a C=0 Sampling Plan

Let’s construct a hypothetical plan for a lot size of 1,200 units, an AQL of 1.5%, and a normal inspection level (Level II):

Step 1: Lot Size and Inspection Level

  • Lot size ( N = 1,200 )
  • Inspection level = Level II (Normal)

Step 2: Determine Sample Size Code Letter
Refer to the General Inspection Level table. For a lot size of 1,200 under Level II, the code letter might be L (based on ANSI/ASQ Z1.4).

Step 3: Find the Sample Size
Using the AQL table for 1.5%, the corresponding sample size for code letter L might be 80 units.

Step 4: Apply Zero Acceptance Criteria

  • Sample size ( n = 80 )
  • Acceptance number ( C = 0 ) (reject the lot if any defect is found).

C=0 Sampling Plan

Appendix G: Process Capability (Cpk and Ppk)

Process capability can be assessed for normal processes using the process capability formula below.

C_pk = min {(USL - x̄) / 3σ, (x̄ - LSL) / 3σ}

where:

  • USL = Upper Specification Limit
  • LSL = Lower Specification Limit
  • x̄ = process mean
  • σ = within subgroup standard deviation

P_pk = min {(USL - x̄) / 3s, (x̄ - LSL) / 3s}

where:

  • USL = Upper Specification Limit
  • LSL = Lower Specification Limit
  • x̄ = process mean
  • s = overall variation

Alternatively, this may be calculated using Minitab 17:

  • Stat -> Quality Tools -> Capability Analysis, or
  • Stat -> Quality Tools -> Capability Sixpack

Ppk for non-normal datasets may also be analyzed using Minitab by choosing non-normal analysis methods in the same menus.

Appendix H: Distribution Identification for Datasets

Identifying a distribution model for datasets is important for analysis. For example, checking for normality or other distribution is an important first step before deciding which statistical tool to use to analyze data.

To identify a distribution model in Minitab 17, use Stat -> Quality Tools -> Identify Individual Distribution Identification and enter your dataset attributes. Upon running the tool, the readout will provide p-values associated with different distributions. A p-value of greater than 0.050 is an indication that the dataset may fit that individual distribution and may be considered for analysis.

A Normality Test may be performed to confirm whether a dataset is normal or not. To perform a normality test in Minitab, select: Stat -> Basic Statistics -> Normality Test… and follow the prompt. Typically, a p-value greater than 0.050 indicates that the dataset may be considered to be normal for data analysis purposes.

Appendix I: Test and Inspection Method Validation using Minitab 17

Test Method Validations

Test Method Validations may be performed using gage studies. Gage Repeatability and Reproducibility studies may be set up in Minitab 17 by selecting: Stat -> Quality Tools -> Gage Study -> Create Gage R&R Study Worksheet…

Once complete, Gage R&R studies may be analyzed by selecting: Stat -> Quality Tools -> Gage Study -> Gage R&R Study (Crossed)…

Inspection Method Validations

Inspection Method Validations may be performed using Attribute Agreement Analysis tool by selecting: Stat -> Quality Tools -> Create Attribute Agreement Analysis Worksheet…

Once complete, an Attribute Agreement Analysis may be analyzed by selecting: Stat -> Quality Tools -> Attribute Agreement Analysis…

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