CPK calculation formula and steps

This post was last edited by ray1513 on 2011-6-27 at 15:57. Quality management techniques: 6 Sigma, calculation of CA/CP/CPK. 1. The Chinese definition of Cpk is: Process Capability Index, which represents a quantitative measure of the level of a certain process or engineering effort, and it serves as one of the indicators used for evaluating such processes. 2. Two parameters closely related to Cpk: Ca and Cp. Ca: Process accuracy. Cp: Process precision. 3. The relationship among Cpk, Ca, and Cp: Cpk = Cp * (1 – |Ca|). Cpk represents a combination of Ca and Cp; Ca reflects the positional relationship (trend of central tendency), while Cp reflects the dispersion relationship (trend of variability). 4. When choosing process stations for control using Cpk, cost should be considered as a key factor, along with the impact of quality characteristics on subsequent processes. 5. To be sufficiently representative, the sampled data should consist of at least 20 to 25 sets of data. 6. To calculate Cpk, in addition to collecting sampling data, it is also necessary to know the upper and lower specification limits (USL, LSL) for that quality characteristic in order to compute its value successfully. 7. First, the Excel function “STDEV” can be used to automatically calculate the standard deviation (σ) of the sampled data; thereafter, the specification tolerance (T) and the specification center value (u) can be determined. Specification tolerance = Upper specification limit – Lower specification limit ； Specification median = (Upper specification limit + Lower specification limit) / 2 ； 8. Using the formula: Ca = (X’ – U) / (T/2), the process accuracy is calculated; this gives the Ca value. 9. Using the formula: Cp = T/6Sigma, the process precision is calculated; this gives the Cp value. 10. Using the formula: Cpk = Cp * (1 – |Ca|), the process capability index is calculated; this gives the Cpk value. 11. Rating criteria for Cpk: (These criteria can be used to determine appropriate actions based on the calculated process capability index)
A++ grade: Cpk ≥ 2.0 – Excellent; cost reduction can be considered.
A+ grade: 2.0 > Cpk ≥ 1.67 – Good; this level should be maintained.
A grade: 1.67 > Cpk ≥ 1.33 – Fair; the process has good capability and stable performance, but efforts should be made to raise it to the A+ level.
B grade: 1.33 > Cpk ≥ 1.0 – Average; the process performance is average, and minor variations in process factors can lead to defects; various resources and methods should be used to improve it to the A level.
C grade: 1.0 > Cpk ≥ 0.67 – Poor; there are many defects in the process, and its capability needs to be improved.
D grade: 0.67 > Cpk – Unacceptable; the process capability is too poor, and it is necessary to reconsider and redesign the process. Now many customers want to know about the capacity of your production equipment and ask to see your Cpk value. What is the Cpk value? I’m sharing some introductions here; for a detailed understanding, you still need to refer to SPC. Explanations of SPC-related terms --- Cpk or Ppk: The customer requests a capability report for the products or processes you provide. You know that to calculate Cpk, product specifications, the average value, and Sigma are required. When collecting this information, someone might ask: Which Sigma do they need? Should I use the estimated Sigma or the calculated Sigma? Which one is more accurate? Naturally, most people want to use a Sigma value that makes the Cpk figure look better, but such a Sigma value may not reflect the production process as understood by the customer. To prevent confusion in Cpk calculations, a new index called Ppk was introduced—the process performance index. Ppk uses Sigma calculated from a single value. How should they be used? The capability values (Cp, Cpk, Cr) calculated using the estimated Sigma are used to measure a system’s potential ability to meet customer requirements. It is generally used to analyze the natural tendency of a system. The actual or calculated Sigma and related indices (Pp, Ppk, Pr) are used to measure a system’s performance or ability to meet customer requirements. It is generally used to analyze the actual performance of a process. ---Symmetry and Kurtosis: Skewness (also known as “skewness degree”): Measures the extent to which a distribution deviates from a normal distribution. If the distribution is asymmetric, it is called skewed. If one side of the distribution has more elements than the other (“tail”), it is considered “skewed”. If the “tail” tends to larger values, the distribution is said to be positively skewed or right-skewed ； If the “tail” tends toward a smaller value, the distribution is said to be negatively skewed or left-skewed. Kurtosis measures the sharpness of a distribution. A value of 0 indicates a normal distribution. If it is a positive value, it indicates that more values are concentrated around the mean ； If it is negative, it indicates that the curve has a peak that is sharper than that of a normal distribution. A brief introduction to Measurement System Analysis (MSA) Introduction: In the daily production processes in factories, we often need to analyze various measurement data in order to draw certain conclusions or take action. To ensure that the conclusions drawn or actions taken are correct, in addition to using proper analysis methods, it is necessary to pay attention to the quality of the measurement data. Quality of measurement data: A measurement system refers to the entire process of obtaining measurement results, which consists of operations, procedures, measuring tools, equipment, software, and the operators. An ideal measurement system should produce only “correct” measurement results each time it is used; however, there are hardly any measurement systems with such ideal statistical properties. The quality of measurement data is related to the statistical properties of multiple measurement results obtained from a measurement system operating under stable conditions; the most common statistical properties used to characterize data quality are bias and variance. The characteristic of bias refers to the position of the data relative to the standard value, whereas the characteristic of variance refers to the distribution of the data. One of the most common reasons for low-quality data is excessive variation in the measurements; much of the variation in a set of measurements is caused by the interaction between the measurement system and its environment. A measurement system with high variability may be inappropriate for analyzing a manufacturing process, as the variability of the measurement system could mask the variability in the manufacturing process itself. We should monitor and control the degradation of the measurement system; if the quality of the measurement data is unacceptable, the measurement system must be improved. Statistical characteristics of the measurement system: To obtain high-quality measurement data, a measurement system must possess the following characteristics: 1) The measurement system must be under statistical control, which means that the variation within the system can only be caused by common factors rather than special causes; this is referred to as statistical stability ； 2) The variation of the measurement system must be smaller than that of the manufacturing process ； 3) The variation of the measurement system should be less than the tolerance band ； 4) The measurement accuracy should be higher than the higher of the process variation and the tolerance band; generally, the measurement accuracy is one-tenth of the higher of these two values ； 5) If the statistical characteristics of the measurement system may change depending on the item being measured, then the maximum variation of the measurement system should be less than the smaller of the process variation and the tolerance band. Measurement system evaluation: When evaluating a measurement system, it is first necessary to determine whether it has sufficient resolution – that is, the ability of the measurement system to detect and accurately indicate even the smallest changes in the characteristic being measured. The resolution should be at most one-tenth of the total process’s 6Sigma (standard deviation). Systematic measurement errors can be divided into five types: bias, linearity, stability, repeatability, and relinearity. Bias refers to the difference between the observed average value of the measurement results and the reference value ； Bias = Observed average value – Baseline value. Linear: Linear refers to the difference in bias values within the expected operating range of the measuring instrument ； Selecting some parts within the operating range of the measuring instrument can determine linearity. The bias of these selected parts is determined by the difference between the baseline value and the average of the measured observations. Stability: Stability refers to the total variation in the measurement values obtained by a measurement system when measuring a single characteristic of the same reference or part over a certain period of time ； To determine statistical stability by using control charts, these charts provide a way to distinguish between the variation caused by factors affecting all measurement results and the variation resulting from special conditions. Repeatability and reproducibility (R&R): Repeatability refers to the variation in the measurements taken by one evaluator using the same measuring tool multiple times when assessing a certain characteristic of a part ； The repeatability of the measurement process means that the variation within the measurement system itself is consistent. Measurement errors caused by changes in the instrument itself and by the position of its components are two common causes of repeatability error. Reproducibility refers to the variation in the average values obtained when different evaluators use the same measuring tool to assess a certain characteristic of a part. The reproducibility of the measurement process indicates that the variability among evaluators is consistent, with this variability representing the incremental bias introduced by each evaluator. If such a bias truly exists, all the averages for each evaluator will be different. The acceptable criteria for the repeatability and reproducibility (R&R) of measuring tools are: an error of less than 10% – the measurement system is acceptable ； 10% to 30% error — may be acceptable depending on the importance of the application, the cost of the measuring instrument, and the costs associated with maintenance, etc ； Error greater than 30% — The measurement system needs improvement. 6 Sigma: 1 Sigma = 690,000 errors per million operations; 2 Sigma = 308,000 errors per million operations; 3 Sigma = 66,800 errors per million operations; 4 Sigma = 6,210 errors per million operations; 5 Sigma = 230 errors per million operations; 6 Sigma = 3.4 errors per million operations; 7 Sigma = 0 errors per million operations. What is 6 Sigma? “σ” is a Greek letter used as a statistical unit to measure the standard deviation within a set of values. The defect rate in typical companies is around 3 to 4 sigma; with 4 sigma, this means there are 6,210 errors per million opportunities. If a company continuously strives for quality improvement and achieves 6 sigma, its performance will be almost perfect in meeting customer requirements, with only 3.4 defects found out of one million opportunities. 6 Sigma evolved from a total quality management approach in the mid-1990s into a highly effective technique for designing, improving, and optimizing business processes, offering a range of new product development tools that are equally applicable to design, production, and services. Subsequently, moving in tandem with strategies such as globalization, product services, and e-commerce, it has become the most important strategic initiative for companies around the world striving for managerial excellence. 6 Sigma has gradually evolved into a quality management philosophy that focuses on the customer as the core for determining a company’s strategic goals and product development standards, striving for continuous improvement. The main principles of 6 Sigma (1) To achieve significant results by implementing 6 Sigma, companies must regard it as a management philosophy. In this philosophy, there are six key principles, and behind each principle are numerous tools and methods to provide support. The main principles of 6 Sigma (II)  Sincerely caring about customers. 6 Sigma puts the customer first. For example, when evaluating the performance of a department or an employee, it is necessary to think from the customer’s perspective. First, understand what the customers’ needs are, and then set corporate goals and measure performance based on those needs. Main Principles of 6 Sigma (III)  Management based on data and facts. In recent years, although knowledge management has gradually gained attention, most companies still make decisions based on opinions and assumptions. The primary rule of 6 Sigma is to determine exactly what measurements need to be taken in order to assess performance, and then to use that data and analysis to understand how far the company’s performance falls short of its goals. Main principles of 6 Sigma (IV)  Focus on processes. Whether it is designing products or improving customer satisfaction, 6 Sigma views processes as a means to success, an approach that delivers value to customers and creates competitive advantages. The main principles of 6 Sigma (V)  Proactive management. Companies must consistently take the initiative to do things that ordinary firms often overlook, such as setting ambitious goals and constantly reviewing them ； Set clear priorities ； Emphasize prevention rather than firefighting ； One often questions \"Why do we do this?\" rather than saying \"We all do it this way.\" 」 Main principles of 6 Sigma (VI)  Unrestricted cooperation. Improving the cooperative relationships among various departments within a company, as well as between the company and its suppliers and customers, can bring significant business opportunities to the enterprise. 6 Sigma emphasizes boundary-less collaboration, helping employees understand how they should contribute to the organization’s overall goals, and identifying the relationships between various activities across different departments within a company’s processes. , The main principles of 6 Sigma (VII)  Strive for perfection, but at the same time tolerate failure. In 6 Sigma companies, employees constantly seek ways to provide better service while reducing costs. Companies continuously strive for perfection, but they can also accept or deal with occasional setbacks and learn from their mistakes*. What is 6 Sigma quality? A company’s product quality is the result of its overall operations, with many complex factors influencing it. Motorola uses 6 Sigma quality standards to define its goals, making complex issues easier to understand. At Motorola, the meaning of the 6 sigma quality level is as follows:  1. 3.4 PPM (defect rate or number of defects per million is 3.4). 2. 99.99966% of the products have no defects. 3. Provide a benchmark for comparison with competitors, offering a measure for TQM. 4. It shows how far one is from being flawless. 6 Sigma quality has attracted many companies for the following reasons: 1. 6 Sigma quality provides a benchmark for relatively complex products or services. 2. 6 Sigma quality can be used to measure the quality gap between one’s own company and competitors. 3. Show progress toward perfection. 4. Provide a clear goal for each department. What is 6 Sigma Management (Part 1)  It is a comprehensive management system and development strategy for achieving and maintaining a company’s operational success and maximizing its performance. It is a business approach that enables companies to achieve rapid growth. Improvements in operational performance include: • An increase in market share • A rise in customer retention rates • Reduced costs • Shorter cycles • Lower defect rates • Faster product/service development • Changes in corporate culture. What is 6 Sigma Management (Part 2)?  It is a process improvement method that is led and driven from the top down by the company’s highest management. The top management sets the goals for improvement/innovation (which are closely related to the company’s development strategy and vision), as well as the resources and time frame. • This innovative approach is centered around a structured improvement process known as DMAIC: Define, Measure, Analyze, Improve, and Control. DMAIC is used for three basic improvement processes: 6 Sigma product/service implementation process improvement, 6 Sigma business process improvement, and 6 Sigma design SSDP. In terms of implementation, it is supported by a structured organization composed of four levels of trained personnel with clear responsibilities: Champions, Master Black Belts, Black Belts, and Green Belts. This innovative approach emphasizes the use of quantitative methods/tools, as well as a detailed definition and quantitative expression of customer needs/satisfaction. Each stage has clear objectives, which are supported by appropriate tools or methods. Why Use 6 Sigma Management (I): For Survival: “Why implement 6 Sigma management?” "Motorola’s answer is: to survive. From the 1970s to the 1980s, Motorola lost the markets for radios and televisions in competition with Japan, and later lost the markets for pagers and semiconductors as well. In 1985, the company faced bankruptcy. A Japanese company acquired Motorola’s television manufacturing company in the 1970s. After being modified by the Japanese, it was put into production quickly, with a defect rate of only 1/20 of that under Motorola’s management. They used the same personnel, technology, and design. Obviously, the problem lies in Motorola’s management. In the competitive market, the harsh realities of survival led Motorola’s executives to conclude that \"our quality is terrible.\" Under the leadership of its CEO, Motorola embarked on the path to 6 Sigma quality. Today, “Motorola” has become a world-famous brand. In 1998, Motorola received the American Baldridge **Quality Management Award**. The secret to their success is the 6 Sigma quality approach. It was 6 Sigma management that enabled Motorola to evolve from the brink of bankruptcy to a world-renowned company leading in quality and profitability. Why Use 6 Sigma Management (II): To Help Companies Gain Core Competencies:. . Whether a company can survive and succeed depends on the value it provides to the market/customers. According to economic theory: . . . . Q Quality V Value=-------- . . . . P Price. . 6 Sigma core capabilities: improve quality, reduce costs, maximize value, and achieve customer satisfaction/strong market competitiveness. How to promote 6 Sigma: There are many methods and tools available as companies strive to implement 6 Sigma. One important method is a five-phase improvement process called DMAIC (pronounced Deh-maik): Define, Measure, Analyze, Improve, and Control. Through these steps, the return on investment for the company will naturally increase.  Define the core processes and key customers; from the customer’s perspective, identify what is most important to them, that is, the \"Critical to Quality\" (CTQ) factors. Clarify the team charter and the core business processes.  Measurement: Identifying key metrics involves establishing basic steps for measuring the flaws in a process. Staff must receive training in basic probability and statistics, as well as courses in statistical analysis software and measurement analysis. To avoid placing a heavy burden on employees, it might be advisable to have those with six standard deviations of practical implementation experience train together with newcomers, helping them overcome difficulties. For complex calculation problems, automatic computing tools can be provided to reduce the time required for such calculations.  Analyze and investigate the root causes of errors. Using statistical analysis, identify the potential variables that affect the results in order to determine the most important causes of defects. The tools used include many statistical analysis tools.  Improve: Identify the best solution, then formulate an action plan and carry it out effectively. This step requires continuous testing to see if the improvement measures truly work to reduce errors.  Control to ensure that the improvements made are sustained. Measurement must not be interrupted to prevent errors from occurring again. In many process improvement initiatives in the past, the concept of control was often overlooked ； And among the six standard deviations, control is the key to its long-term improvement in quality and cost. If successful, 6 Sigma will help to change the inherent inertia of an organization, enabling employees to constantly ask questions and seek better solutions, thus keeping the company on an upward trajectory. Implementers of 6 Sigma  Some say that Mr. Welch, the CEO of GE, is an expert at developing talent. It was he who, through GE’s 6 Sigma management program, trained thousands of individuals who created value for the company. They are the ones known as warriors, dark belts, black belts, and green belts. These individuals come from various positions within the company; having received specialized 6 Sigma training, they provide organizational support for 6 Sigma management. Black belts and master black belts who are dedicated to 6 Sigma projects become a valuable asset to companies; many of these black belts take on leadership roles in the company after completing a two-year assignment on 6 Sigma projects. Warrior: Managers at the senior level of an organization who are responsible for implementing 6 Sigma. Responsible for deploying Sigma implementation and providing all support. Responsible for identifying or selecting 6 Sigma projects. Track or monitor the progress of 6 Sigma. Black Belt: Overall responsible for 6 Sigma implementation techniques. Assist warriors in selecting projects, developing implementation plans and timeline schedules, provide Black Belts with support in advanced 6 Sigma technical tools, and be responsible for mobilization, coordination, and communication. Black Belt: Coming from various departments within the company, they have received comprehensive training on 6 Sigma improvement processes and tools, are familiar with these processes, and are responsible for guiding or leading improvement projects. Provide training and guidance to green belts. Served as a full-time black belt for 2 years. One black belt completes 5-7 projects per year, resulting in cost savings of around 1 million. Green Belt: Individuals who have received training and are involved in 6 Sigma projects in their respective roles. Staff ratio: per 1,000 employees. . Black Belt: 1. . Black belts: 10. . Green Belt: 50–70 people; 6 Sigma support techniques.  Measurement techniques: • Methods for calculating DPMO • Process capability analysis techniques (including long/short-term process capability analysis)  Basic techniques: The old and new seven tools  Advanced techniques: • SPC – for measuring, analyzing, improving, and monitoring process variability • DOE/Monte Carlo methods – for optimizing process design in order to achieve optimal process capability • FMEA – a risk analysis technique used to identify areas that need improvement and to set improvement goals • QFD – a technique for analyzing customer requirements, helping to convert them into appropriate internal requirements • Error prevention – methods for preventing errors from occurring in the first place.  Soft skills: • Leadership • Improving team efficiency • Employee capabilities and empowerment • Communication and feedback. The impact of 6 Sigma management on corporate culture: When it comes to corporate culture, it may seem like an abstract concept. However, by closely observing the common beliefs, values, and behavioral norms among the people around you when dealing with even the simplest and most minor issues, it is not difficult to sense its presence. Simply put, corporate culture is “the way we do things here.” When you try to improve quality, especially by improving work processes (including manufacturing, service, administration, and management) in order to achieve the best possible product and service quality, culture poses a significant obstacle. Therefore, Mr. Hodgetts pointed out that: (1) when strategy and culture conflict, culture always prevails ； (2) When corporate culture is incompatible with the spirit of change, efforts to bring about change will fail. Mr. Hodgetts summarized 20 lessons learned from companies that won the U.S. Bowditch Award. By analyzing the lessons learned from these successful companies, particularly those related to corporate culture development at the highest levels, it is easy to see that successful companies take an extra step when implementing quality strategies compared to other companies. In other words, while striving to improve the quality of their products and services, they are willing to make significant efforts to transform their corporate culture, which is not in line with 6 Sigma quality standards, so that the beliefs, attitudes, values, and expectations of all employees are aligned with those of 6 Sigma quality. This has led to the creation of a strong corporate quality culture, ensuring the success of the 6 Sigma quality strategy. Success Story (1): At GE, Jack Welch’s 6 Sigma was the most important strategic development that the company has ever experienced. 6 Sigma was implemented at the end of 1995 ； Cost-benefit of implementing Sigma savings: 300 million dollars in 1997, 750 million dollars in 1998, 1.5 billion dollars in 1999 ； The profit margin increased from 13.6% in 1995 to 16.7% in 1998, and the value exceeded 3 trillion US dollars. Mr. Jack Welch established three strategic development plans for GE: 6 Sigma, product services, and globalization, which enabled GE to rapidly grow into the world’s largest and most successful diversified multinational corporation. Success Story (II): Lenox Company – Larry Bossidy introduced 6 Sigma in 1994. The benefits resulting from the implementation of 6 Sigma exceeded 2 billion dollars, with a profit margin of over 14%. In 1999, it was the company with the fastest growth in market value

The Complex Performance Index, Cpk, indicates the degree to which the central value (mean) of the equipment’s process results is close to the midpoint of the specification range. The formula for Cpk is Cpk = (1 – K)Cp = (1 – |Ca|)Cp. Here, K represents the offset; it is calculated as K = ε / (T/2) = 2ε / T = Ca. The offset refers to the deviation between the center of the distribution and the center of tolerance, with ε = |X – μ|, where X is the mean value and μ is the center of tolerance； Process accuracy Ca (Capability of Accuracy): The Ca value measures the degree of consistency between the actual average value of a process and the target value specified for it. It indicates the ability to control the difference between the average quality of the products produced and the midpoint of the acceptable specifications. The calculation for Ca is as follows:
Ca = (Actual center value – Target center value) / (Half of the specification tolerance) × 100% = (X – μ) / (T/2) × 100%
Where μ = (SU – SL)/2 = (Upper specification limit + Lower specification limit)/2, and T = SU – SL = Upper specification limit – Lower specification limit. Note: For one-sided specifications, since there is no target center value, Ca cannot be calculated.