The document discusses process capability analysis, which involves analyzing process data to determine if a process is capable of meeting specifications. It covers calculating capability indices like CP and PPM for normal distributions to quantify a process's capability. CP greater than 1.33 indicates a capable process that should produce less than 64 defective parts per million. The document provides examples of control charts and a sample capability analysis calculation.
This document discusses process capability analysis, which relates a production process's variability to customer specifications to determine if the process is capable of meeting requirements. It defines key terms like critical-to-quality characteristics, control charts, process capability indices Cp and Cpk. Cp measures a process's potential capability if centered on target, while Cpk considers deviation of the mean. For a process to be capable, its natural variation (control limits) must be narrower than specifications. If Cpk=1 the process is barely capable, and if Cpk<1 the process is incapable and requires improvement. Process capability analysis assumes an in-control, stable production process.
The document discusses statistical process control and data collection. It covers measuring performance, collecting and representing data, process control using statistical methods, sources of variation, and calculating process capability indices. Process capability analysis determines if a process is capable of meeting specifications and identifies when processes are in or out of control. Maintaining statistical control is important for process stability and minimizing non-conforming output.
The document discusses process capability and assessing whether a process is capable of meeting customer requirements. It provides definitions of key terms like capable process, process capability ratios (Cp and Cpk), and discusses the differences between short-term and long-term capability studies. Short-term studies look at random variation over days/weeks using 30-50 data points, while long-term studies examine non-random sources of variation over weeks/months using 100-200 data points. The document warns that capability assessments only indicate potential performance if the process is stable and in control.
The document discusses process capability and defines key terms related to process capability. It provides the standard formula for process capability using 6 sigma and explains how process capability is compared to specification limits. It then discusses different process capability indices including Cp, Cpk, and Cpm. It explains how these indices measure both potential and actual process capability. The document also discusses limitations of the Cp index and the use of Cpk to address process centering. It describes how to calculate confidence intervals for process capability ratios and discusses some key process performance metrics.
This document outlines statistical quality control techniques for evaluating manufacturing and service processes. It discusses measuring and controlling process variation using variables like mean, standard deviation and control charts. Key aspects covered include process capability analysis using metrics like Cpk, acceptance sampling plans to determine quality levels while balancing producer and consumer risks, and operating characteristic curves.
This document discusses process capability analysis. It defines specification and tolerance limits as boundaries that define conformance for manufacturing or service operations. Process capability indices like Cp, Cpk, CPU, CPL and Ppk are used to determine if a process's natural variation can meet specifications. Cp measures a process's potential to meet specifications based on its spread. Cpk incorporates both mean and standard deviation. CPU and CPL measure if the process mean is centered between the specification limits. Ppk indicates actual long-term process performance meeting specifications. Maintaining capable processes with indices above 1 ensures high quality and uniform output.
There are six measures that can define a process's capability: Z values, sigma, percentage out of specification, defects per million opportunities (DPMO), defects per number of opportunities, and Cpk. These measures represent both short-term and long-term capability. Short-term refers to random variation, while long-term also includes non-random sources of drift. To convert between them, a 1.5 sigma shift is typically used. Calculating percentage out of specification first is generally recommended to determine process capability without distributional assumptions.
This document discusses process capability analysis, which relates a production process's variability to customer specifications to determine if the process is capable of meeting requirements. It defines key terms like critical-to-quality characteristics, control charts, process capability indices Cp and Cpk. Cp measures a process's potential capability if centered on target, while Cpk considers deviation of the mean. For a process to be capable, its natural variation (control limits) must be narrower than specifications. If Cpk=1 the process is barely capable, and if Cpk<1 the process is incapable and requires improvement. Process capability analysis assumes an in-control, stable production process.
The document discusses statistical process control and data collection. It covers measuring performance, collecting and representing data, process control using statistical methods, sources of variation, and calculating process capability indices. Process capability analysis determines if a process is capable of meeting specifications and identifies when processes are in or out of control. Maintaining statistical control is important for process stability and minimizing non-conforming output.
The document discusses process capability and assessing whether a process is capable of meeting customer requirements. It provides definitions of key terms like capable process, process capability ratios (Cp and Cpk), and discusses the differences between short-term and long-term capability studies. Short-term studies look at random variation over days/weeks using 30-50 data points, while long-term studies examine non-random sources of variation over weeks/months using 100-200 data points. The document warns that capability assessments only indicate potential performance if the process is stable and in control.
The document discusses process capability and defines key terms related to process capability. It provides the standard formula for process capability using 6 sigma and explains how process capability is compared to specification limits. It then discusses different process capability indices including Cp, Cpk, and Cpm. It explains how these indices measure both potential and actual process capability. The document also discusses limitations of the Cp index and the use of Cpk to address process centering. It describes how to calculate confidence intervals for process capability ratios and discusses some key process performance metrics.
This document outlines statistical quality control techniques for evaluating manufacturing and service processes. It discusses measuring and controlling process variation using variables like mean, standard deviation and control charts. Key aspects covered include process capability analysis using metrics like Cpk, acceptance sampling plans to determine quality levels while balancing producer and consumer risks, and operating characteristic curves.
This document discusses process capability analysis. It defines specification and tolerance limits as boundaries that define conformance for manufacturing or service operations. Process capability indices like Cp, Cpk, CPU, CPL and Ppk are used to determine if a process's natural variation can meet specifications. Cp measures a process's potential to meet specifications based on its spread. Cpk incorporates both mean and standard deviation. CPU and CPL measure if the process mean is centered between the specification limits. Ppk indicates actual long-term process performance meeting specifications. Maintaining capable processes with indices above 1 ensures high quality and uniform output.
There are six measures that can define a process's capability: Z values, sigma, percentage out of specification, defects per million opportunities (DPMO), defects per number of opportunities, and Cpk. These measures represent both short-term and long-term capability. Short-term refers to random variation, while long-term also includes non-random sources of drift. To convert between them, a 1.5 sigma shift is typically used. Calculating percentage out of specification first is generally recommended to determine process capability without distributional assumptions.
Statistical process control (SPC) involves using statistical methods to monitor and control processes to ensure they produce conforming products. Variation exists in all processes, and SPC helps determine when variation is normal versus requiring correction. Key SPC tools include control charts, which graph process data over time to identify special causes of variation needing addressing. Process capability analysis also examines whether a process can meet specifications under natural variation. Together these tools help processes run at full potential with minimal waste.
This document discusses statistical process control (SPC) tools and their application in manufacturing. It describes how SPC involves collecting data about processes, studying the data to understand how actions affect outcomes, and using the data to control processes to achieve desired results dependent on customer and business factors. The document provides an overview of how SPC is used, including understanding processes and sources of variation, and eliminating significant sources of variation through data analysis and problem-solving.
Asq Auto Webinar Spc Common Questions WebWalter Oldeck
This document summarizes a webinar on statistical process control (SPC) that addressed common questions. The webinar covered whether different sources of variation can be on the same control chart, the difference between specifications and control limits, why control limits are needed even if specifications exist, and what the process capability indices Cp, Cpk, Pp and Ppk represent and how they can differ depending on how well a process is centered and stable over time. The webinar encouraged participants to ask questions in the chat and provided information on how to access the slides and video recording.
This document discusses process capability analysis. It introduces process capability, why it is studied, and how it is measured through graphs and calculations of metrics like Cp. Process capability determines if a process meets specifications and can help reduce variability. The principles of process capability are explained, such as predicting variability. Methods like analytical calculations and process capability ratios are covered. Advantages include process improvement, while disadvantages are that it is best for large companies. Control charts can also be used to monitor processes.
Six Sigma is a methodology that aims to improve processes by eliminating defects. It was developed at Motorola in 1986 and uses a "sigma level" to measure a process's capability and quality. There are two approaches: DMAIC focuses on improving existing processes, while DMADV creates new processes. Process capability indices like Cp, Cpk, Pp and Ppk are used to measure the variation within a process compared to specifications and determine if a process is capable and centered. These indices help identify issues to target for improvement and ensure stable, high quality processes.
This document provides an introduction to statistical process control (SPC). It defines SPC as a strategy that uses statistical techniques to evaluate processes, identify variability, and find opportunities for improvement. The goal of SPC is to make high-quality products the first time by reducing variability, rather than reworking defective products. It focuses on monitoring process behavior rather than just final product quality. SPC distinguishes between common cause variability that is always present and special cause variability that can be addressed to improve the process. It emphasizes identifying and addressing special causes first before adjusting process means. Control charts are used to monitor processes and determine if they are in control or need adjustment.
This document discusses statistical process control and monitoring validated processes. It describes determining what processes need to be monitored based on risk assessments and process validations. It also covers the basics of statistical process control including common and assignable cause variation, setting control limits, and using tools like X-bar and R charts to monitor processes. Western Electric rules for determining when a process is out of control are also summarized.
This document discusses process capability analysis. It defines tolerance as the maximum acceptable error for a component. For example, a shaft diameter between 49.5-50.5mm is acceptable. Errors can be common causes, reduced through improved machinery, or special causes that cannot be predicted. Data was collected from a food and drug company showing process testing and statistics. The Cp and Cpk indices measure process potential and centering, with values over 1 indicating a reliable process. Cpk is the minimum value to show where defects are prominent, such as towards the upper or lower specification limits. A centered process has Cp equal to Cpk.
The document discusses process capability analysis using Cp and Cpk metrics. Cp measures how well a process fits within specification limits, while Cpk also considers centering. A Cpk over 1 indicates a capable and centered process. Values over 1.33 represent 4-sigma capability. Cp over 1 but Cpk below 1 means the process is capable but not centered. Histograms and Cp, Cpk values are used to compare baseline and improved processes.
This document discusses process capability analysis and process analytical technology. It begins with an introduction to capability, including histograms and the normal distribution. It then covers capability indices like Cp, Cpk, Pp and Ppk and how to calculate sigma. It discusses using capability analysis with attribute data by calculating defects per million opportunities (DPMO). It concludes with a brief overview of process analytical technology (PAT).
This document provides an overview of Six Sigma and its methodology for process improvement. It defines key Six Sigma concepts like process capability and sigma levels. Six Sigma aims to reduce process variation and improve yields from 3 sigma to 6 sigma, resulting in fewer than 3.4 defects per million opportunities. It outlines the DMAIC methodology used, including defining problems, measuring baseline performance, analyzing sources of variation, improving the process, and controlling the gains. The goal is to understand and control all sources of variation to meet customer requirements.
This document provides an overview of Six Sigma and its methodology. It discusses key Six Sigma concepts like process capability and sigma levels. It outlines Motorola's development of Six Sigma to reduce process variation and improve customer satisfaction. The Six Sigma improvement methodology is then summarized in five steps - Define, Measure, Analyze, Improve, Control (DMAIC) which aims to understand problems, measure performance, analyze causes of variation, improve the process, and control gains. Various tools used at each step like QFD, DOE, control charts are also briefly explained.
Process capability is a measure of a process's ability to meet specifications for a product or service. It is determined by comparing the process variability, as measured by the standard deviation, to the tolerances between the nominal value and upper and lower specifications. The process capability ratio Cp measures the tolerance width relative to the process variability, while the process capability index Cpk considers whether the process mean is centered between the specifications. For example, in assessing an intensive care lab's turnaround time process, which has a standard deviation of 1.35 minutes and specifications of 20-30 minutes, the Cp is calculated as 1.23 but the Cpk is 0.94, indicating the process mean of 26.2 minutes is not centered
Statistical process control (SPC) is a method that uses statistical methods to monitor processes and ensure they operate efficiently. Key tools in SPC include control charts, which graph process data over time and establish upper and lower control limits to detect assignable causes of variation. Control charts come in two main types - variables charts that monitor quantitative measurements like weight or temperature, and attributes charts that count defects. The advantages of SPC include increased stability, predictability, and ability to detect attempts to improve processes. SPC has various applications in pharmaceutical manufacturing for monitoring characteristics like drug potency, fill weight, and microbial counts.
- Seven tools;
- Process variability;
- Important use of the control chart;
- Statistical basis of the control chart:
> Basic principles and type of control chart;
> Choice of control limits;
> Sampling size and sampling frequency;
> Average run length;
> Rational subgroups;
> Analysis of patterns on control charts;
> Sensitizing rules for control charts;
> Phase I and Phase II of control chart.
Statistical process control is defined as and use of statistical technique to control a process or production method .It is used in manufacturing or production process to measure how consistently a product perform according to its design specification.
The document discusses process capability and how to evaluate whether a manufacturing process is capable of producing parts within its specified tolerances. It defines process capability as the ability of a process to make a feature within its tolerance. It describes how to calculate process averages and standard deviations from sample measurements and use those values to determine a process's Cpk value. A good process should have a Cpk of at least 1.33 but ideally 2 or more, indicating that the process mean is at least 6 standard deviations from the nearest specification limit. Graphs and examples are provided to illustrate capable versus incapable processes.
This document provides an overview of process quality control using statistical process control (SPC) and statistical quality control (SQC) approaches. It defines SPC and SQC, noting that SPC focuses on controlling process inputs through variables while SQC monitors outputs through attributes. The document outlines key learning objectives around these topics. It also defines key terms like process quality control and discusses the difference between SPC and SQC. Additionally, it covers process capability analysis using Minitab and controlling process inputs and monitoring outputs. Overall, the document serves as training material on quality control tools and techniques with a focus on SPC and SQC.
Quality andc apability hand out 091123200010 Phpapp01jasonhian
The document outlines key concepts in quality management and Six Sigma methodology. It discusses definitions of quality, total quality management (TQM), and Six Sigma. Six Sigma aims to reduce defects through eliminating variation and achieving near zero defect levels. It uses a Define-Measure-Analyze-Improve-Control (DMAIC) methodology. Statistical process control charts and process capability indices are also introduced to measure quality performance. An example of Mumbai's successful lunch delivery system achieving over 5-sigma quality levels is provided.
The document discusses process capability and assessing whether a process is capable of meeting specifications. It defines key terms like Cp, Cpk, Pp, and Ppk which are used to measure process capability and performance. Cp and Cpk indicate if a process is capable based on short-term variability, while Pp and Ppk indicate performance based on long-term variability including shifts. Short and long-term studies are different, as are capability which ignores shifts versus performance. Statistical assumptions like a stable process are important. The document provides examples and guidelines for evaluating processes using these measures.
Statistical process control (SPC) involves using statistical methods to monitor and control processes to ensure they produce conforming products. Variation exists in all processes, and SPC helps determine when variation is normal versus requiring correction. Key SPC tools include control charts, which graph process data over time to identify special causes of variation needing addressing. Process capability analysis also examines whether a process can meet specifications under natural variation. Together these tools help processes run at full potential with minimal waste.
This document discusses statistical process control (SPC) tools and their application in manufacturing. It describes how SPC involves collecting data about processes, studying the data to understand how actions affect outcomes, and using the data to control processes to achieve desired results dependent on customer and business factors. The document provides an overview of how SPC is used, including understanding processes and sources of variation, and eliminating significant sources of variation through data analysis and problem-solving.
Asq Auto Webinar Spc Common Questions WebWalter Oldeck
This document summarizes a webinar on statistical process control (SPC) that addressed common questions. The webinar covered whether different sources of variation can be on the same control chart, the difference between specifications and control limits, why control limits are needed even if specifications exist, and what the process capability indices Cp, Cpk, Pp and Ppk represent and how they can differ depending on how well a process is centered and stable over time. The webinar encouraged participants to ask questions in the chat and provided information on how to access the slides and video recording.
This document discusses process capability analysis. It introduces process capability, why it is studied, and how it is measured through graphs and calculations of metrics like Cp. Process capability determines if a process meets specifications and can help reduce variability. The principles of process capability are explained, such as predicting variability. Methods like analytical calculations and process capability ratios are covered. Advantages include process improvement, while disadvantages are that it is best for large companies. Control charts can also be used to monitor processes.
Six Sigma is a methodology that aims to improve processes by eliminating defects. It was developed at Motorola in 1986 and uses a "sigma level" to measure a process's capability and quality. There are two approaches: DMAIC focuses on improving existing processes, while DMADV creates new processes. Process capability indices like Cp, Cpk, Pp and Ppk are used to measure the variation within a process compared to specifications and determine if a process is capable and centered. These indices help identify issues to target for improvement and ensure stable, high quality processes.
This document provides an introduction to statistical process control (SPC). It defines SPC as a strategy that uses statistical techniques to evaluate processes, identify variability, and find opportunities for improvement. The goal of SPC is to make high-quality products the first time by reducing variability, rather than reworking defective products. It focuses on monitoring process behavior rather than just final product quality. SPC distinguishes between common cause variability that is always present and special cause variability that can be addressed to improve the process. It emphasizes identifying and addressing special causes first before adjusting process means. Control charts are used to monitor processes and determine if they are in control or need adjustment.
This document discusses statistical process control and monitoring validated processes. It describes determining what processes need to be monitored based on risk assessments and process validations. It also covers the basics of statistical process control including common and assignable cause variation, setting control limits, and using tools like X-bar and R charts to monitor processes. Western Electric rules for determining when a process is out of control are also summarized.
This document discusses process capability analysis. It defines tolerance as the maximum acceptable error for a component. For example, a shaft diameter between 49.5-50.5mm is acceptable. Errors can be common causes, reduced through improved machinery, or special causes that cannot be predicted. Data was collected from a food and drug company showing process testing and statistics. The Cp and Cpk indices measure process potential and centering, with values over 1 indicating a reliable process. Cpk is the minimum value to show where defects are prominent, such as towards the upper or lower specification limits. A centered process has Cp equal to Cpk.
The document discusses process capability analysis using Cp and Cpk metrics. Cp measures how well a process fits within specification limits, while Cpk also considers centering. A Cpk over 1 indicates a capable and centered process. Values over 1.33 represent 4-sigma capability. Cp over 1 but Cpk below 1 means the process is capable but not centered. Histograms and Cp, Cpk values are used to compare baseline and improved processes.
This document discusses process capability analysis and process analytical technology. It begins with an introduction to capability, including histograms and the normal distribution. It then covers capability indices like Cp, Cpk, Pp and Ppk and how to calculate sigma. It discusses using capability analysis with attribute data by calculating defects per million opportunities (DPMO). It concludes with a brief overview of process analytical technology (PAT).
This document provides an overview of Six Sigma and its methodology for process improvement. It defines key Six Sigma concepts like process capability and sigma levels. Six Sigma aims to reduce process variation and improve yields from 3 sigma to 6 sigma, resulting in fewer than 3.4 defects per million opportunities. It outlines the DMAIC methodology used, including defining problems, measuring baseline performance, analyzing sources of variation, improving the process, and controlling the gains. The goal is to understand and control all sources of variation to meet customer requirements.
This document provides an overview of Six Sigma and its methodology. It discusses key Six Sigma concepts like process capability and sigma levels. It outlines Motorola's development of Six Sigma to reduce process variation and improve customer satisfaction. The Six Sigma improvement methodology is then summarized in five steps - Define, Measure, Analyze, Improve, Control (DMAIC) which aims to understand problems, measure performance, analyze causes of variation, improve the process, and control gains. Various tools used at each step like QFD, DOE, control charts are also briefly explained.
Process capability is a measure of a process's ability to meet specifications for a product or service. It is determined by comparing the process variability, as measured by the standard deviation, to the tolerances between the nominal value and upper and lower specifications. The process capability ratio Cp measures the tolerance width relative to the process variability, while the process capability index Cpk considers whether the process mean is centered between the specifications. For example, in assessing an intensive care lab's turnaround time process, which has a standard deviation of 1.35 minutes and specifications of 20-30 minutes, the Cp is calculated as 1.23 but the Cpk is 0.94, indicating the process mean of 26.2 minutes is not centered
Statistical process control (SPC) is a method that uses statistical methods to monitor processes and ensure they operate efficiently. Key tools in SPC include control charts, which graph process data over time and establish upper and lower control limits to detect assignable causes of variation. Control charts come in two main types - variables charts that monitor quantitative measurements like weight or temperature, and attributes charts that count defects. The advantages of SPC include increased stability, predictability, and ability to detect attempts to improve processes. SPC has various applications in pharmaceutical manufacturing for monitoring characteristics like drug potency, fill weight, and microbial counts.
- Seven tools;
- Process variability;
- Important use of the control chart;
- Statistical basis of the control chart:
> Basic principles and type of control chart;
> Choice of control limits;
> Sampling size and sampling frequency;
> Average run length;
> Rational subgroups;
> Analysis of patterns on control charts;
> Sensitizing rules for control charts;
> Phase I and Phase II of control chart.
Statistical process control is defined as and use of statistical technique to control a process or production method .It is used in manufacturing or production process to measure how consistently a product perform according to its design specification.
The document discusses process capability and how to evaluate whether a manufacturing process is capable of producing parts within its specified tolerances. It defines process capability as the ability of a process to make a feature within its tolerance. It describes how to calculate process averages and standard deviations from sample measurements and use those values to determine a process's Cpk value. A good process should have a Cpk of at least 1.33 but ideally 2 or more, indicating that the process mean is at least 6 standard deviations from the nearest specification limit. Graphs and examples are provided to illustrate capable versus incapable processes.
This document provides an overview of process quality control using statistical process control (SPC) and statistical quality control (SQC) approaches. It defines SPC and SQC, noting that SPC focuses on controlling process inputs through variables while SQC monitors outputs through attributes. The document outlines key learning objectives around these topics. It also defines key terms like process quality control and discusses the difference between SPC and SQC. Additionally, it covers process capability analysis using Minitab and controlling process inputs and monitoring outputs. Overall, the document serves as training material on quality control tools and techniques with a focus on SPC and SQC.
Quality andc apability hand out 091123200010 Phpapp01jasonhian
The document outlines key concepts in quality management and Six Sigma methodology. It discusses definitions of quality, total quality management (TQM), and Six Sigma. Six Sigma aims to reduce defects through eliminating variation and achieving near zero defect levels. It uses a Define-Measure-Analyze-Improve-Control (DMAIC) methodology. Statistical process control charts and process capability indices are also introduced to measure quality performance. An example of Mumbai's successful lunch delivery system achieving over 5-sigma quality levels is provided.
The document discusses process capability and assessing whether a process is capable of meeting specifications. It defines key terms like Cp, Cpk, Pp, and Ppk which are used to measure process capability and performance. Cp and Cpk indicate if a process is capable based on short-term variability, while Pp and Ppk indicate performance based on long-term variability including shifts. Short and long-term studies are different, as are capability which ignores shifts versus performance. Statistical assumptions like a stable process are important. The document provides examples and guidelines for evaluating processes using these measures.
This document summarizes a seminar on statistical process control. It discusses key topics like process capability, estimating inherent capability from control charts, and Juran's 10 steps for quality improvement. Process capability refers to a process's ability to achieve results within specifications. It is quantified using indices like Cp and Cpk which measure potential and performance based on process data. Assumptions like statistical control are important for accurately applying these indices. Quality improvement involves building awareness, setting goals, training, implementing projects, and maintaining momentum through regular systems.
This document provides an overview of total quality management (TQM) concepts for manufacturing, including standard operating procedures (SOP), statistical process control (SPC), process capability indices, and control charts. It discusses how SOPs and quality control process charts are used to standardize operations and check quality. Statistical process control tools like control charts help monitor processes for variation. Process capability indices like Cp and Cpk indicate if a process is capable of meeting specifications. Together, these TQM elements aim to reduce variation and improve quality in manufacturing operations and supply chains.
STATISTICAL PROCESS CONTROL satyam raj.pptxSatyamRaj25
This document provides an overview of statistical process control (SPC). It defines SPC as the application of statistical methods to measure and analyze variation in a process. The document discusses the importance of SPC in reducing waste and costs while improving quality and uniformity. It also describes key SPC tools like control charts and process capability analysis. Control charts help monitor processes for common and special causes of variation, while process capability analysis compares process performance to product specifications to ensure quality.
This document provides an overview of statistical quality control (SQC). It defines SQC as using statistical tools to control quality throughout the production process. It outlines the objectives of understanding variability, control charts, and other statistical process control tools. Control charts are discussed as a key SQC tool to detect assignable causes of variation and ensure a process is in statistical control. The document also covers the different types of control charts for variables and attributes.
This document provides guidance on calculating and interpreting the process capability index Cpk. It defines Cpk as a ratio that compares the specification tolerance to the process variation expressed in terms of standard deviations. It explains how to calculate Cpk and discusses factors that influence Cpk values such as sample size, process centering, and measurement uncertainty. The document also provides examples of the expected defective parts per million that correspond to different Cpk values and factors to consider when improving Cpk, such as machine, tooling, workholding, and workpiece variables.
Statistical process control (SPC) techniques apply statistical methods to measure and analyze variation in manufacturing processes. SPC uses control charts to distinguish between common cause variation inherent to the process and special cause variation that can be assigned to a specific reason. Control charts monitor process data over time against statistical control limits. Process capability analysis compares process variation to product specifications to determine if the process is capable of meeting specifications. Key metrics like Cp, Cpk and Cpm indices quantify a process's capability relative to the specifications. For a process to have a valid capability analysis, it must meet assumptions of statistical control, normality, sufficient representative data, and independence of measurements.
This document provides information on selecting appropriate statistical process control charts and implementing statistical process control. It discusses different types of control charts for variable and attribute data, factors to consider when selecting control charts such as the type of data and subgroup size. It also covers collecting and sampling data, calculating control limits, detecting special causes or assignable causes from control charts, and determining sampling frequency. The goal of statistical process control is to monitor process variation and detect when a process is out of control through the use of control charts, which plot process data over time and can indicate the presence of special causes of variation.
Process capability indices like Cp, Cpk, and Ppk compare the output of a process to specification limits to determine how well the process meets requirements. Cp and Cpk measure an in-control process, looking at the ratio between the process spread and the specification width. Ppk measures an initial setup process before statistical control. A capable process has most measurements within limits. Larger capability indices indicate less likelihood of items being outside specifications. Calculating sigma quality levels involves defining critical quality characteristics and determining defect probabilities relative to those characteristics.
Cpk indispensable index or misleading measure? by PQ SystemsBlackberry&Cross
Capability analysis is a set of calculations used to assess whether a system is able to meet a set of requirements. Customers, engineers, or managers usually set the requirements, which can be specifications, goals, aims, or standards.
The primary reason for doing a capability analysis is to answer the question: Can we meet customer requirements? To be more specific: Can our system produce consistently within tolerances required by the customer now and in the future?
Capability analysis involves two entities: 1) the producer and 2) the consumer. The consumer sets the requirements and the producer must be able to meet the requirements.
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PQ Systems is a partner of Blackberry&Cross, since 2006.
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Operations Management: Six sigma benchmarking of process capability analysis...FGV Brazil
Six sigma benchmarking of process capability analysis and mapping of process parameters.
Author: Jagadeesh Rajashekharaiah
Journal of Operations and Supply Chain Management
Vol 9, No 2 (2016)
FGV's Brazilian School of Public and Business Administration (EBAPE)
Abstract
Process capability analysis (PCA) is a vital step in ascertaining the quality of the output from a production process. Particularly in batch and mass production of components with specified quality characteristics, PCA helps to decide about accepting the process and later to continue with it. In this paper, the application of PCA using process capability indices is demonstrated using data from the field and benchmarked against Six Sigma as a motivation to improve to meet the global standards. Further, how the two important process parameters namely mean and the standard deviation can be monitored is illustrated with the help of what if analysis feature of Excel. Finally, the paper enables to determine the improvement efforts using simulation to act as a quick reference for decision makers. The global benchmarking in the form of Six Sigma capability of the process is expected to give valuable insight towards process improvement.
ANALYZING THE PROCESS CAPABILITY FOR AN AUTO MANUAL TRANSMISSION BASE PLATE M...ijmvsc
The industry today is working intensively on a goal-oriented way towards introducing regular studies in
manufacturing. The current study is part of a large overall spanning project aiming towards an increase in
productivity, i.e. more products produced per year with availability. In this paper we have analyze what
Process Capability is and how it is implemented on a current process. All the steps are listed out in an easy
to understand manner. In current scenario, specifications for products have been tightened due to
performance competition in market. Statistical tools like control charts, process capability analysis and
cause and effect diagram ensure that processes are fit for company specifications while reduce the process
variation and improve product quality characteristic. Process capability indices (PCIs) are used in the
manufacturing process to provide numerical measures on whether a process is capable of producing items
within the predetermined limits. For the analysis purpose MINITAB 16.0 is used and is found that the
process is placed exactly at the centre of the control limits. Analysis also shows that process is not
adequate. The cause and effect diagram is prepared to found out the root cause of variation in diameter of
work. In this study, a process-capability analysis was also carried out in a medium-sized company that
produces machine and spare parts.
This document provides an overview of quality control and control charts. It defines quality control as recognizing and removing causes of defects and variations from set standards. The objectives of quality control are to establish quality standards, identify flaws in materials and processes, evaluate production methods, determine quality deviations, and analyze causes of deviations. Control charts are a type of statistical quality control technique used to monitor processes over time by plotting data points on charts with upper and lower control limits. The document describes X-charts for variable data, P-charts for counting data, and S-charts for monitoring process variability.
This document summarizes key concepts in quality control and statistical process control. It discusses total quality management, the Malcolm Baldridge National Quality Award criteria, ISO 9000 standards, and Six Sigma methodology. It also describes different types of control charts used in statistical process control, including x-bar, R, p, and np charts. Control charts help determine whether process variation is due to common or assignable causes by comparing output to control limits. Interpreting point patterns on control charts indicates whether a process is in statistical control.
Similar to Cp cpk para distribuciones no normales (20)
PROMOTING GREEN ENTREPRENEURSHIP AND ECO INNOVATION FOR SUSTAINABLE GROWTH.docxnehaneha293248
: This study investigates the multi-faceted relationship between entrepreneurship, innovation, and sustainability across countries at different development levels. We construct a novel dataset combining measures of entrepreneurial activity, innovation outputs, and sustainability performance indicators related to economic, social, and environmental dimensions.Using country-level panel regression analysis, we find that entrepreneurship rates and attitudes are positively associated with social sustainability factors like education, gender equality, and institutional quality. However, high entrepreneurship levels do not necessarily correlate with better environmental sustainability outcomes, suggesting entrepreneurs may prioritize economic objectives over environmental ones.The results for innovation are more mixed. Greater innovation output is linked to higher economic development, but also associated with both positive and negative sustainability factors. This implies that while innovations drive economic progress, they may come with environmental costs without complementary policies. The findings suggest that entrepreneurship supports social sustainability, but pursuing entrepreneurship and innovation alone is insufficient for achieving environmental sustainability goals. We discuss policy implications, including strengthening education and skills, improving access to financing for sustainable ventures, incentivizing green innovation, and developing sustainability reporting standards. By aligning entrepreneurship and innovation with sustainability priorities, policymakers can harness these dynamic forces to create more sustainable, inclusive, and resilient economies.
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1. Process Capability Analysis
March 20, 2012
Andrea Span`o
andrea.spano@quantide.com
1 Quality and Quality Management
2 Process Capability Analysis
3 Process Capability Analysis for Normal Distributions
4 Process Capability Analysis for Non-Normal Distributions
Process Capability Analysis 2 / 68
2. Quality and Quality Management
1 Quality and Quality Management
2 Process Capability Analysis
3 Process Capability Analysis for Normal Distributions
4 Process Capability Analysis for Non-Normal Distributions
Process Capability Analysis 3 / 68
Quality and Quality Management Definitions and Implications
ISO 9000 is a family of standards related to quality management
systems and designed to help organisations ensure that they meet
the needs of customers and other stakeholders.
Up to the end of December 2009, at least 1’064’785 ISO 9001 (2000
and 2008) certificates had been issued in 178 countries and
economies. (ISO Survey 2009)
In Italy, 13’066 certificates had been issued. Italy is the European
leader and among the first in the world for number of ISO 9001
certificates. (ISO Survey 2009)
Process Capability Analysis 4 / 68
3. Quality and Quality Management Definitions and Implications
It can be defined:
quality: the degree to which a set of inherent characteristics fulfils
requirement;
management: coordinated activities to direct and control;
quality management system: to direct and control an organization
with regard to quality.
Monitoring and Measurement of Product: The organization shall
monitor and measure the characteristics of the product to verify
that product requirements have been met.
ISO/TR 22514-4:2007 describes process capability and
performance measures that are commonly used. (Statistical
methods in process management - Capability and performance - Part
4: Process capability estimates and performance measures)
Process Capability Analysis 5 / 68
Quality and Quality Management Definitions and Implications
Process capability: ability of the process to realize a characteristic
that will fullfil the requirements for that characteristic. (ISO 25517-4)
Specification: an explicit set of requirements to be satisfied by a
material, product or service. Specifications are mandatory if adopted
by a business contract. Specifications must be respected to avoid
sanctions.
Process Capability Analysis 6 / 68
4. Process Capability Analysis
1 Quality and Quality Management
2 Process Capability Analysis
3 Process Capability Analysis for Normal Distributions
4 Process Capability Analysis for Non-Normal Distributions
Process Capability Analysis 7 / 68
Process Capability Analysis
The capability analysis is carried out in the following steps:
1 select the process to be analysed and collection of data;
2 identify specific limits according to which capability analysis will be
evaluated;
3 verify the process is under statistical control;
4 analyse data distribution;
5 estimate capability indices.
For the capability analysis to be performed the process needs to be
under statistical control.
Specification limits can be: the Upper Specification Limit (USL), the Lower
Specification Limit (LSL) and eventually a target value. Specification limits
are usually provided from outside (production requirements, market
requirements). Specifications can either be two-sided (when USL and LSL
are both specified) or one-sided (either USL or LSL is specified).
Process Capability Analysis 8 / 68
5. Process Capability Analysis
To sum up, a process is capable when:
it is under statistical control;
it has a low variability rate compared to the range of specified limits;
process distribution is possibly centered on specification limits
(centering).
If a process respects specifications and is under statistical control it
can be foreseen that specifications will not change in the future. If a
process respects specifications but is not under statistical control,
specification could change in the future.
Process variability indicates the spread within which 99.73% of the
process distribution is contained. A normal distribution has a 6σ
width range centered on the mean (µ ± 3σ).
Process Capability Analysis 9 / 68
Process Capability Analysis
15 16 17 18 19 20 21 22 23 24 25
LSL USL
x − 3σ^ x + 3σ^
5 8 11 14 17 20 23 26 29 32 35
LSL USL
x − 3σ^ x + 3σ^
The two distributions have same mean and specification limits. However,
dispersion in the distribution on the right is higher. Therefore, the process
capability of the distribution on the right is lower than the process capability of
the distribution on the left.
Process Capability Analysis 10 / 68
6. Process Capability Analysis
5 8 11 14 17 20 23 26 29 32 35
LSL USL
x − 3σ^ x + 3σ^
5 8 11 14 17 20 23 26 29 32 35
LSL USL
x − 3σ^ x + 3σ^
The two distributions have the same characteristics as far as shape, position and
dispersion are concerned. The limit spread is the same. The process on the right
is not centered with respect to its specification limits. Therefore process
capability on the right will be lower.
Process Capability Analysis 11 / 68
Process Capability Analysis
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Process Spread
LSL USL
Specifications
The process is both under statistical
control and capable.
This process will produce conforming
products as long as it remains in
statistical control.
Process Capability Analysis 12 / 68
7. Process Capability Analysis
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Process Spread
LSL USL
Specifications
The process is under statistical
control but it is not capable.
If the specifications are realistic, an
effort must be immediately made to
improve the process (i.e. reduce
variation) to the point where it is
capable of producing consistently
within specifications.
Process Capability Analysis 13 / 68
Process Capability Analysis
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Process Spread
LSL USL
Specifications
The process is out of control but it is
capable.
The process must be monitored: it
cannot be expected it will respect
specifics in the future.
Process Capability Analysis 14 / 68
8. Process Capability Analysis
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Process Spread
LSL USL
Specifications
The process is both out of control
and it is not capable.
The process must be adjusted to be
under control, then the capability
analysis must be performed again.
Process Capability Analysis 15 / 68
Process Capability Analysis for Normal Distributions
1 Quality and Quality Management
2 Process Capability Analysis
3 Process Capability Analysis for Normal Distributions
4 Process Capability Analysis for Non-Normal Distributions
Process Capability Analysis 16 / 68
9. Process Capability Analysis for Normal Distributions CP Index
The CP index is the most widely used capacity index.
It can be calculated only when USL and LSL are both specified.
Its theoretical value is:
CP =
USL − LSL
6σ
if data is normally distributed.
CP can be seen as the ratio between the “acceptable” variability
spread and the process variability spread.
In practical terms, “real” σ values are never known and need to be
estimated according to one of the following estimation procedures.
Process Capability Analysis 17 / 68
Process Capability Analysis for Normal Distributions PPM Index
Another capacity index is Parts per Million (PPM). This index
indicates the ratio between the number of pieces exceeding the
specification limits and a million produced units.
For example, the following CP values produce the PPM shown below:
CP = 1 → PPM = 2700;
CP = 1.33 → PPM = 64;
CP = 1.5 → PPM = 7.
PPM can be estimated based on empirical data (i.e. the number of
exceeding elements over one million) or with the cumulative
distribution function of the theoretical distribution.
Process Capability Analysis 18 / 68
10. Process Capability Analysis for Normal Distributions CP and PPM interpretation
Process Spread
LSL USL
Specifications
Case 1: CP > 1.33
A fairly capable process
This process should produce less than 64
non-conforming PPM.
This process will produce conforming
products as long as it remains in statistical
control. The process owner can claim that
the customer should experience least
difficulty and greater reliability with this
product. This should translate into higher
profits.
This process is contained within four
standard deviations of the process
specifications.
Process Capability Analysis 19 / 68
Process Capability Analysis for Normal Distributions CP and PPM interpretation
Process Spread
LSL USL
Specifications
Case 2: 1 < CP < 1.33
A barely capable process
This process will produce greater than 64
PPM but less than 2700 non-conforming
PPM.
This process has a spread just about equal
to specification width. It should be noted
that if the process mean moves to the left
or the right, a significant portion of product
will start falling outside one of the
specification limits. This process must be
closely monitored.
This process is contained within three to
four standard deviations of the process
specifications.
Process Capability Analysis 20 / 68
11. Process Capability Analysis for Normal Distributions CP and PPM interpretation
Process Spread
LSL USL
Specifications
Case 3: CP < 1
A not capable process
This process will produce more than 2700
non-conforming PPM.
It is impossible for the current process to
meet specifications even when it is in
statistical control. If the specifications are
realistic, an effort must be immediately
made to improve the process (i.e. reduce
variation) to the point where it is capable of
producing consistently within specifications.
Process Capability Analysis 21 / 68
Process Capability Analysis for Normal Distributions CP and PPM interpretation
Process Spread
LSL USL
TARGET
Specifications
Case 4: CP < 1
A not capable process
This process will also produce more than
2700 non-conforming PPM.
The variability and specification width is
assumed to be the same as in case 3, but
the process average is off-center. In such
cases, adjustment is required to move the
process mean back to target. If no action is
taken, a substantial portion of the output
will fall outside the specification limit even
though the process might be in statistical
control.
Process Capability Analysis 22 / 68
12. Process Capability Analysis for Normal Distributions CP and PPM interpretation
The table presents some recommended guidelines for minimum values of
CP and PPM (source: AIAG):
One-Sided Two-Sided
Specifications Specifications
CP PPM CP PPM
Existing processes 1.25 88.42 1.33 66.07
New processes 1.45 6.81 1.50 6.80
Critical existing process 1.45 6.81 1.50 6.80
Critical new process 1.60 0.79 1.67 0.54
Process Capability Analysis 23 / 68
Process Capability Analysis for Normal Distributions CP and PPM interpretation
The Figure below shows control charts for a process.
S Chart
for x
Group
Groupsummarystatistics
1 4 7 10 13 16 19 22 25 28 31 34 37 40
0.0000.0050.0100.015
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LCL
UCL
CL
Number of groups = 40
Center = 0.008889821
StdDev = 0.009457401
LCL = 0
UCL = 0.01857082
Number beyond limits = 0
Number violating runs = 0
xbar Chart
for x
Group
Groupsummarystatistics
1 4 7 10 13 16 19 22 25 28 31 34 37 40
73.99574.01074.025
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LCL
UCL
CL
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Number of groups = 40
Center = 74.00434
StdDev = 0.009490542
LCL = 73.99161
UCL = 74.01707
Number beyond limits = 4
Number violating runs = 5
Process Capability Analysis 24 / 68
13. Process Capability Analysis for Normal Distributions CP and PPM interpretation
The capability analysis performed on firsts twenty batches returns a CP
greater than 1.33.
c(74.03, 73.995, 73.988, 74.002, 73.992, 74.009, 73.995, 73.985, 74.008, 73.998, 73.994, 74.004, 73.983, 74.006, 74.012, 74, 73.994, 74.006, 73.984, 74)
Density
73.94 73.96 73.98 74.00 74.02 74.04 74.06
010203040
LSL = 74 USL = 74LSL = 74 USL = 74
TARGET
Process Capability using normal distribution for batches 1−20
cp = 1.49
cpk = 1.47
cpkL = 1.47
cpkU = 1.51
A= 0.3
p= 0.55
n= 20
mean= 74
sd= 0.0112
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Quantiles from distribution distribution
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73.98 73.99 74.00 74.01 74.02
73.9974.0174.03
c(0.5,5)
Expected Fraction Nonconforming
pt
pL
pU
= 8.06687e−06
= 5.25966e−06
= 2.80721e−06
ppm
ppm
ppm
= 8.06687
= 5.25966
= 2.80721
c(0.5,5)
Observed
ppm = 0
ppm = 0
ppm = 0
Process Capability Analysis 25 / 68
Process Capability Analysis for Normal Distributions CP and PPM interpretation
The capability analysis performed on lasts twenty batches, where the
process goes out-of-control, returns a CP less than 1.33.
c(73.988, 74.004, 74.01, 74.015, 73.982, 74.012, 73.995, 73.987, 74.008, 74.003, 73.994, 74.008, 74.001, 74.015, 74.03, 74.001, 74.035, 74.035, 74.017, 74.028)
Density
73.94 73.96 73.98 74.00 74.02 74.04 74.06
0102030
LSL = 74 USL = 74LSL = 74 USL = 74
TARGET
Process Capability using normal distribution for batches 21−40
cp = 1.08
cpk = 0.9
cpkL = 1.26
cpkU = 0.9
A= 0.239
p= 0.746
n= 20
mean= 74
sd= 0.0155
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Quantiles from distribution distribution
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73.98 74.00 74.02 74.04
73.9874.0074.02
c(0.5,5)
Expected Fraction Nonconforming
pt
pL
pU
= 0.00370131
= 8.15584e−05
= 0.00361975
ppm
ppm
ppm
= 3701.31
= 81.5584
= 3619.75
c(0.5,5)
Observed
ppm = 0
ppm = 0
ppm = 0
Process Capability Analysis 26 / 68
14. Process Capability Analysis for Normal Distributions CPK Index
The CP alone does not provide thorough information about the
correspondence of the process to the production specifications. The
CP index is exclusively based on the ratio between the width of the
tolerance spread and the width of the variability spread. It is not
based on process centering, i.e. it does not take into consideration
the central tendency of the process with regards to specification
limits. In theory, processes with very good CP values can be obtained
but they would be positioned beyond specification limits.
The CPK index is used to take into consideration the process
centering. It is defined as:
CPK = min(CPL, CPU)
Process Capability Analysis 27 / 68
Process Capability Analysis for Normal Distributions CPK Index
CPL and CPU estimate the coherence of the process with regards to the
Lower Specification Limit (CPL) and the Upper Specification Limit (CPU)
respectively. In the event of a normal distribution formulas are as follows:
CPL =
µ − LSL
3σ
CPU =
USL − µ
3σ
CP and CPK indices are the same if the process is centered, i.e. if:
µ =
USL − LSL
2
CP index is defined only if USL and LSL are simultaneously defined. CPK,
being the minimum between CPL and CPU, is always defined.
CP index calculates the tolerance with respect to two-sided
specifications. CPL, CPU and CPK indices estimate the tolerance of
one-sided specifications.
Process Capability Analysis 28 / 68
15. Process Capability Analysis for Normal Distributions CPK Index
3σ
mu
USL
USL − mu
3σ
mu
LSL
mu − LSL
Process Capability Analysis 29 / 68
Process Capability Analysis for Normal Distributions CPM Index
The calculation of the CPM index is possible if and only if a target
value T, i.e. an ideal value of the production process, is specified.
The T value does not necessarily need to be equal to the midpoint
between the USL and LSL.
If both USL and LSL are specified, the CPM index is calculated with:
CPM =
min(T − LSL, USL − T)
3 ×
k
j=1
nj
i=1 (xij−T)2
k
j=1 nj−1
If only CPL or CPU is specified, the numerator of the
above-mentioned formula becomes T − LSL and USL − T
respectively.
Process Capability Analysis 30 / 68
16. Process Capability Analysis for Normal Distributions Parameters Estimation
The values of the µ and σ parameters are usually unknown and need
to be estimated with sample data.
In the event of normal distributed data, the variance process can be
estimated with:
The overall variance (overall capability). CP, CPK, CPU and CPL
capability indices computed using overall capability are also known as
PP, PPL, PPU and PPL performance indices.
The within variance (potential capability). The within variance is an
estimation of variability common causes.
Process Capability Analysis 31 / 68
Process Capability Analysis for Normal Distributions Parameters Estimation
Within standard deviation can be estimated using the pooled
standard deviation:
sW =
s
c4
=
k
j=1
nj
i=1 (xij−xk)2
k
j=1 (nk−1)
c4
The above formula numerator (s) is not an unbiased estimator of σ.
Can be shown that if the underlying distribution is normal, then s
estimates c4s where c4 is a constant that depends on a sample size
(n) and group numbers (k). s
c4
is so an unbiased estimator of σ.
CP and CPK capability indices are estimated from sample data.
Confidence intervals can be found.
Process Capability Analysis 32 / 68
17. Process Capability Analysis for Normal Distributions Parameters Estimation
Which standard deviation to use - overall or within?
Although both indices show similar information, they have slightly
different uses.
CPKσoverall
attempts to answer the question “does my current
production sample meet specification?”
On the other hand, CPKσwithin
attempts to answer the question “does
my process in the long run meet specification?” Process capability
can only be evaluated after the process is brought into statistical
control. The reason is simple: CPKσwithin
is a prediction, and only
stable processes can be predicted.
The difference between σoverall and σbetween represents the variability
due to special causes.
The difference between PPMσoverall
and PPMσwithin
represents the loss
of quality due to lack of statistical control.
Process Capability Analysis 33 / 68
Process Capability Analysis for Normal Distributions Example
BrakeCap data frame contains shoes soles hardness measurements
(Rockwell scale). 50 batches of 5 elements each have been sampled. The
LSL is 39, the USL is 43 and the target is the midpoint, 41.
> brakeCap = read.table("BrakeCap.TXT", header = TRUE, sep = "|")
Data frame has 250 observation and 4 variables. Three variables measure
different settings of the same process; the fourth variable identifies the
batch.
> str(brakeCap)
’data.frame’: 250 obs. of 4 variables:
$ Hardness : num 39.1 38.5 40.3 39.2 39.4 ...
$ Centering: num 40.9 38.8 41.1 41.8 40.1 ...
$ Quenching: num 40.7 41.1 40.6 40.5 41.6 ...
$ Subgroup : int 1 1 1 1 1 2 2 2 2 2 ...
Process Capability Analysis 34 / 68
18. Process Capability Analysis for Normal Distributions Example
Analysis requires three steps:
1 Verify that the process is under statistical control.
2 Identify the distribution.
3 Perform the capability analysis.
Process Capability Analysis 35 / 68
Process Capability Analysis for Normal Distributions Example
Control charts are used to verify if the process is under statistical control.
Data is contained in Hardness variable
> library(qcc)
> Hardness.group = qcc.groups(brakeCap$Hardness, brakeCap$Subgroup)
> qcc(Hardness.group, type = "S")
> qcc(Hardness.group, type = "xbar")
S Chart
for Hardness.group
Group
Groupsummarystatistics
1 4 7 11 15 19 23 27 31 35 39 43 47
0.00.51.01.5
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LCL
UCL
CL
Number of groups = 50
Center = 0.9116825
StdDev = 0.9698899
LCL = 0
UCL = 1.904503
Number beyond limits = 0
Number violating runs = 0
xbar Chart
for Hardness.group
Group
Groupsummarystatistics
1 4 7 11 15 19 23 27 31 35 39 43 47
39.040.041.0
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LCL
UCL
CL
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Number of groups = 50
Center = 40.2728
StdDev = 0.9583001
LCL = 38.9871
UCL = 41.55849
Number beyond limits = 0
Number violating runs = 1
Process Capability Analysis 36 / 68
25. Process Capability Analysis for Non-Normal Distributions
1 Quality and Quality Management
2 Process Capability Analysis
3 Process Capability Analysis for Normal Distributions
4 Process Capability Analysis for Non-Normal Distributions
Process Capability Analysis 49 / 68
Process Capability Analysis for Non-Normal Distributions
Capability analysis formulas are based on the percentiles of the distribution
of reference. Therefore, verifying if the empirical distribution can be
considered Gaussian is fundamental. For example, applying the formulas of
the measurement distribution of origin to very asymmetrical segments can
result in an incorrect estimation of the CP index.
When data is not normally distributed, possible approaches fall into two
categories:
A Identification of a distribution able to describe data;
B Transformation of data to obtain a normal distribution, e.g.
Box-Cox transformation.
Process Capability Analysis 50 / 68
26. Process Capability Analysis for Non-Normal Distributions Distribution Identification
Formulas of capability indices for non-normal distributed data are
based on percentiles. The first thing to do is identifying another
probability distribution which can be adapted to original data.
Goodness-of-fit tests, such as those of Anderson-Darling and
Kolmogorov-Smirnov, can help identify the best distribution for the
analysed sample.
Process Capability Analysis 51 / 68
Process Capability Analysis for Non-Normal Distributions Distribution Identification
If control charts indicate a stable process, even if not modelled by a
normal distribution, a useful model which is suitable for data needs to
be defined. A tool to identify the distribution can be used
assuming no previous knowledge of a reasonable model for the
examined process.
This tool tests the adaptation of data to different distributions and
successively selects a distribution according to the probability plot,
the results of the goodness-of-fit test and the physical and historical
knowledge about the process.
Process Capability Analysis 52 / 68
27. Process Capability Analysis for Non-Normal Distributions Distribution Identification
In this case, the calculation of CP and CPK indices is based on the
percentiles of the chosen distribution. The percentiles contain the same
“amount of data” as the µ ± 3σ interval of the normal distribution:
CP =
USL − LSL
x0.99865 − x0.00135
where x0.99865 is the 99.865th percentile.
CPK = min(CPL, CPU)
CPL =
x0.5 − LSL
x0.5 − x0.00135
CPU =
USL − x0.5
x0.99865 − x0.5
Process Capability Analysis 53 / 68
Process Capability Analysis for Non-Normal Distributions Data Transformation
For a capability analysis to be carried out, the process is assumed to be,
approximately if not thoroughly, describable by a theoretical distribution. A
suitable distribution ought to be used to create a significant inference, for
example on the ratio of pieces situated beyond specification limits.
Capability estimations of CP, CPK and so on tend to assume normality.
When normality is mistakenly assumed, the estimated ratio of points falling
beyond specification limits can be over- or underestimated.
An attempt to transform original data to render it more similar to normal
data is possible. The Box-Cox and Johnson transformations can often
result in an “acceptably Gaussian” distribution. Once the best
transformation has been chosen, it has to be applied to control limits as
well. Consequently, capability formulas for normal distributions can be used
on the sample and on transformed control limits.
Process Capability Analysis 54 / 68
28. Process Capability Analysis for Non-Normal Distributions Box-Cox Transformation
The Box-Cox transformation changes original data so that the normal
distribution for capability analysis can be used.
The Box-Cox transformation uses a power transformation for original data:
y(λ)
=
yλ − 1
λ
if λ = 0
ln y if λ = 0
When the λ parameter changes, different distributions of changed data are
obtained.
Process Capability Analysis 55 / 68
Process Capability Analysis for Non-Normal Distributions Box-Cox Transformation
Some of notable values of λ:
λ power transformation description
-1.0 1 − 1/y inverse function
0.0 ln y logarithmic function
0.5 2
√
y − 1 square root
We look for the λ value which transforms the data into a distribution as
similar as possible to a Gaussian distribution. The sought for λ value is the
value which stabilizes the variance (which coincides with the value of that
parameter for which the variance is lowest).
Process Capability Analysis 56 / 68
29. Process Capability Analysis for Non-Normal Distributions Box-Cox Transformation
Once the transformation has been performed, the adaptation of the
normal distribution to changed data is estimated by assessing the
probability plots and the results of the Anderson-Darling test.
The use of the highest p-value alone is not always the most intelligent
approach to choose the best model, especially when the suitability of
a particular distribution is historically and theoretically known.
Process Capability Analysis 57 / 68
Process Capability Analysis for Non-Normal Distributions Summary
Final remarks on capability analysis for non-normal distributed data:
To decide which transformation results are better it is useful to take
into consideration the validity of the selected model using:
the knowledge about the process;
the probability plot.
For a reliable process capability estimation to be obtained, the process
needs to be stable and data ought to follow the above-mentioned
distribution curve.
The use of goodness-of-fit test and probability plots can affect the
identification of the model but the knowledge about the process
plays a useful role too.
Process Capability Analysis 58 / 68
30. Process Capability Analysis for Non-Normal Distributions Example
Ceramic data frame contains data about ceramic isolators. Isolators are
cylinders with an hole in the center. 20 batches of 10 elements each have
been sampled. Specifics require that hole diameter is less than 30
micrometre (micron).
> ceramic = read.table("Ceramic.TXT", header = TRUE, sep = "|",
+ stringsAsFactors = F)
Data frame has 200 observation and 2 variables. The Concentricity
variable contains the distance of the hole from the required one, the
Date.Time variable contains the batch id.
> str(ceramic)
’data.frame’: 200 obs. of 2 variables:
$ Concentricity: num 14.62 9.23 7.89 3.18 10.38 ...
$ Date.Time : chr "1/5 10am" "1/5 10am" "1/5 10am" "1/5 10am" ...
Process Capability Analysis 59 / 68
Process Capability Analysis for Non-Normal Distributions Example
Control charts are used to verify if the process is under statistical control.
> library(qcc)
> concentr.group = qcc.groups(ceramic$Concentricity, ceramic$Date.Time)
> qcc(concentr.group, type = "S")
> qcc(concentr.group, type = "xbar")
S Chart
for concentricity.group
Group
Groupsummarystatistics
1/5 10am 1/5 7pm 1/6 1pm 1/6 7pm 1/7 4am
246810
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LCL
UCL
CL
Number of groups = 20
Center = 6.715974
StdDev = 6.904755
LCL = 1.905359
UCL = 11.52659
Number beyond limits = 0
Number violating runs = 0
xbar Chart
for concentricity.group
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Groupsummarystatistics
1/5 10am 1/5 7pm 1/6 1pm 1/6 7pm 1/7 4am
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UCL
CL
Number of groups = 20
Center = 11.44248
StdDev = 6.532115
LCL = 5.245577
UCL = 17.63939
Number beyond limits = 0
Number violating runs = 0
Process Capability Analysis 60 / 68
31. Process Capability Analysis for Non-Normal Distributions Example
The Anderson-Darling test verifies the hypothesis of normal distribution of
data.
> library(qualityTools)
> adTest = qualityTools:::.myADTest
> with(ceramic, {
+ qqnorm(Concentricity, main = "Concentricity")
+ qqline(Concentricity)
+ adTest(Concentricity, "normal")
+ })
Anderson Darling Test for normal distribution
data:
A = 6.7131, mean = 11.442, sd = 7.240, p-value < 2.2e-16
alternative hypothesis: true distribution is not equal to normal
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Theoretical Quantiles
SampleQuantiles
Process Capability Analysis 61 / 68
Process Capability Analysis for Non-Normal Distributions Example
Test rejects the hypothesis of normal distribution of the data. The Weibull
distribution of data can be tested.
> qqPlot(ceramic$Concentricity, "weibull")
> adTest(ceramic$Concentricity, "weibull")
Anderson Darling Test for weibull distribution
data:
A = 2.1104, shape = 1.707, scale = 12.913, p-value <= 0.01
alternative hypothesis: true distribution is not equal to weibull
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Q−Q Plot for "weibull" distribution
Quantiles from "weibull" distribution
Quantilesforceramic$Concentricity
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Process Capability Analysis 62 / 68
32. Process Capability Analysis for Non-Normal Distributions Example
Test rejects the hypothesis of Weibull distribution of the data. The
Log-Normal distribution of data can be tested.
> qqPlot(log(ceramic$Concentricity))
> adTest(log(ceramic$Concentricity))
Anderson Darling Test for normal distribution
data:
A = 0.3306, mean = 2.250, sd = 0.624, p-value = 0.5116
alternative hypothesis: true distribution is not equal to normal
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Q−Q Plot for "normal" distribution
Quantiles from "normal" distribution
Quantilesforlog(ceramic$Concentricity)
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Process Capability Analysis 63 / 68
Process Capability Analysis for Non-Normal Distributions Example
Data can be assumed to follow a log-normal distribution. Overall
capability analysis can be performed via data transformation.
> pcr.Concentricity = pcr(log(ceramic$Concentricity), usl = log(30))
1.87610061766181, 3.12746204423189, 2.83462411315923, 1.83418018511201, 3.05442561989321, 3.19006474301408, 1.87410721057681, 1.93730177451871, 2.49535190764954, 1.45348561066021, 2.38259703096343, 1.80286424896016, 3.24773535354875, 2.2083842975
8655, 2.54065666489249, 2.07115732209214, 1.97810071401816, 3.1349723406455, 1.98910646549825, 3.07869379426834, 1.25219137659641, 3.07352630099401, 2.22527198989138, 2.31589610705372, 3.09995714461152, 1.83130064384859, 2.36602926555058, 1.037445
03673688495002, 2.33243511496293, 2.16355288429849, 2.01969153727406, 1.87210982188171, 2.65647616555616, 1.66335775442353, 1.79009141212736, 3.03220884036401, 2.11529119457353, 2.3112474660727, 1.62806337655727, 2.6234359487123, 1.72419414973229
.51349410663991, 2.35356333775868, 2.69482986293136, 2.37360258022933, 3.10130776610263, 2.67318266577313, 1.87992296005736, 1.64942756065026, 1.73554178922983, 2.32678978169086, 2.0547645539674, 2.20033085895529, 2.28340227357727, 1.439598133142
931770302722541, 2.38342748083815, 2.42851264959184, 2.33708313340161, 2.02326793687389, 3.05692144186641, 1.59066275710777, 2.39297409272173, 1.8484548129046, 3.00775965376671, 1.5746394068914, 3.46732838412798, 1.68491637361001, 2.3106524640718
254094148791, 2.03234993414599, 3.18428438009858, 1.93340369696313, 2.63088113833108, 2.64886620774876, 2.38812038110031, 1.49917651817656, 1.92264152211596, 1.80959938735639, 2.63626772800651, 2.79654935968462, 2.18109519540634, 2.84908669321251
6996756837921, 2.30198491292201, 2.25737850015597, 2.57649780135983, 2.48131688093744, 3.57088389587623, 2.08156428706081, 3.15128197429496, 2.78278654721623, 1.12590314890401, 3.01817846238382, 0.868779749203103, 1.27703737518294, 3.290787100335
2.04601401556732, 3.1067368315969, 2.5865603035145)
Density
0 1 2 3 4
0.00.10.20.30.40.50.60.7
USL = 3.4
Process Capability using normal distribution for log(ceramic$Concentricity)
cp = *
cpk = 0.61
cpkL = *
cpkU = 0.61
A= 0.331
p= 0.512
n= 200
mean= 2.25
sd= 0.624
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Quantiles from distribution distribution
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c(0.5,5)
Expected Fraction Nonconforming
pt
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= 32536.2
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Observed
ppm = 0
ppm = 5e+06
ppm = 5e+06
Process Capability Analysis 64 / 68
33. Process Capability Analysis for Non-Normal Distributions Example
Box-Cox transformation seems confirm the log data transformation.
> boxcox(ceramic$Concentricity ~ 1)
−2 −1 0 1 2
−1100−1050−1000−950−900
λ
log−Likelihood
95%
Process Capability Analysis 65 / 68
Process Capability Analysis for Non-Normal Distributions Example
Capability analysis can be performed directly on log-normal data.
> usl = 30
> par(mfrow = c(1, 1))
> hist(ceramic$Concentricity, main = "Concentricity", col = "gray")
> abline(v = usl, col = "red", lwd = 2)
> concentricity.distpars = fitdistr(ceramic$Concentricity, "log-normal")$estimate
> meanlog = concentricity.distpars[1]
> sdlog = concentricity.distpars[2]
> cpk = usl/qlnorm(0.99865, meanlog, sdlog)
> ppm = (1 - plnorm(30, meanlog, sdlog) * 10^6
Concentricity
ceramic$Concentricity
Frequency
0 10 20 30 40
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Process Capability Analysis 66 / 68
34. Process Capability Analysis for Non-Normal Distributions Example
Results obtained directly on log-normal data can be compared with
transformed data.
> cpk # original data
[1] 0.4886823
> pcr.Concentricity$cpk # transformed data
[1] 0.6149203
> ppm # original data
[1] 32200.77
> pcr.Concentricity$ppt * 10^6 # transformed data
[1] 32536.18
Regardless the adopted methodology to estimate indices, the process is far
from been capable.
Process Capability Analysis 67 / 68
References
References
Montgomery, D.C. (1997). Introduction to Statistical Quality Control. Wiley.
Roth, T. (2010). Working with the qualityTools Package.
http://www.r-qualitytools.org
Roth, T. (2011). Process Capability Statistics for Non-Normal Distributions in R.
http://www.r-qualitytools.org/useR2011/ProcessCapabilityInR.pdf
Kapadia, M. Measuring Your Process Capability.
http://www.symphonytech.com/articles/processcapability.htm
Scrucca, L. (2004). qcc: an R package for Quality Control Charting and Statistical
Process Control. R News 4/1, 11-17.
Roth, T. (2011). qualityTools: Statistics in Quality Science. R package version
1.50.
Process Capability Analysis 68 / 68