Glossary

15.2 Process Capability Analysis

3 min read•july 23, 2024

Process capability analysis helps businesses ensure their products meet quality standards. It involves calculating indices like and , which measure how well a process meets specifications. These indices compare the spread of data to the allowable range.

Interpreting capability indices is crucial for quality control. Values above 1 indicate a capable process, while values below 1 suggest improvements are needed. The analysis assumes , but methods exist for non-normal data too.

Process Capability Analysis

Calculation of capability indices

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  • quantify how well a process meets specifications
    • Cp: Process Capability Index measures the potential capability of a process to meet specifications
      • Formula: Cp=[USL](https://www.fiveableKeyTerm:usl)−[LSL](https://www.fiveableKeyTerm:lsl)6σCp = \frac{[USL](https://www.fiveableKeyTerm:usl) - [LSL](https://www.fiveableKeyTerm:lsl)}{6\sigma}
        • USL: Upper Specification Limit (maximum allowable value)
        • LSL: Lower Specification Limit (minimum allowable value)
        • σ\sigma: Process standard deviation (measure of variability)
    • Cpk: Process Capability Index adjusted for process centering measures the actual capability of a process to meet specifications
      • Formula: Cpk=min⁡(USL−μ3σ,μ−LSL3σ)Cpk = \min(\frac{USL - \mu}{3\sigma}, \frac{\mu - LSL}{3\sigma})
        • μ\mu: Process mean (average value)
      • Cpk considers the process center relative to the (how well the process is centered between USL and LSL)

Interpretation of capability indices

  • Cp and Cpk values indicate the process capability
    • Cp and Cpk > 1: Process is capable of meeting specifications (producing parts within the allowable range)
      • Higher values indicate greater capability (more room for error)
    • Cp and Cpk < 1: Process is not capable of meeting specifications (producing out-of-spec parts)
    • Cp > Cpk: Process is not centered within the specification limits (off-center, closer to one limit)
    • Cp = Cpk: Process is centered within the specification limits (equal room for error on both sides)
  • Minimum acceptable values for Cp and Cpk depend on the industry and criticality of the product
    • Typical minimum values range from 1.33 (less critical) to 2.00 (highly critical, like aerospace)

Process capability for normal distributions

  • Process capability analysis assumes that the data follows a normal distribution (bell-shaped curve)
  • Steps to determine process capability for normally distributed data:
    1. Collect data on the process characteristic of interest (measurements, dimensions)
    2. Test the data for normality using graphical methods or statistical tests
      • Graphical methods: (shape), normal probability plot (straight line)
      • Statistical tests: Anderson-Darling, Shapiro-Wilk (p-value > 0.05 indicates normality)
    3. If the data is normally distributed, calculate the process mean (μ\mu) and standard deviation (σ\sigma)
    4. Calculate Cp and Cpk using the formulas provided
    5. Interpret the results based on the calculated values and the specific requirements of the process

Process capability for non-normal data

  • If the data is not normally distributed, process capability analysis can still be performed with some adjustments
  • Methods for assessing process capability with non-normal data:
    • Transform the data to achieve normality
      • Box-Cox transformation (power transformation to stabilize variance and make data more normal)
      • After transformation, follow the steps for normally distributed data
    • Use non-parametric methods, such as percentile-based indices
      • : Index
        • Formula: Pp=USL−LSL6sPp = \frac{USL - LSL}{6s}, where ss is the sample standard deviation
      • : Process Performance Index adjusted for process centering
        • Formula: Ppk=min⁡(USL−x~3s,x~−LSL3s)Ppk = \min(\frac{USL - \tilde{x}}{3s}, \frac{\tilde{x} - LSL}{3s}), where x~\tilde{x} is the sample median
    • Use distribution-specific capability indices
      • for Weibull distribution (used for reliability analysis)
  • Interpret the results based on the chosen method and the specific requirements of the process

Key Terms to Review (25)

Binomial Distribution: A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success or failure. This distribution is characterized by two parameters: the number of trials (n) and the probability of success in each trial (p). Understanding binomial distribution is essential for making predictions about events based on basic probability concepts, applying it to real-world situations, and assessing process capability in various fields.
Common cause variation: Common cause variation refers to the inherent, natural fluctuations that occur in a process over time due to factors that are part of the process itself. This type of variation is consistent and predictable, resulting from the system's design, materials, and environmental conditions. Understanding common cause variation is essential for evaluating process capability and identifying areas for improvement.
Continuous improvement: Continuous improvement is an ongoing effort to enhance products, services, or processes over time by making incremental improvements. This concept is crucial in achieving higher efficiency and effectiveness, ensuring that organizations adapt to changing environments and meet customer expectations. Continuous improvement relies on data-driven decision-making and the involvement of all employees, fostering a culture of innovation and accountability.
Control Charts: Control charts are graphical tools used to monitor the stability of a process over time by displaying data points against predetermined control limits. They help in identifying variations in processes, distinguishing between common causes of variation, which are inherent to the process, and special causes, which indicate that something unusual is affecting the process. This distinction is vital for making informed decisions that enhance quality and efficiency in various business operations.
Control Limits: Control limits are the thresholds set on control charts that determine the acceptable range of variation in a process. They help in identifying whether a process is in a state of statistical control or if it is exhibiting signs of variation that may indicate potential issues. By analyzing data points in relation to control limits, businesses can make informed decisions regarding process performance and quality management.
Cp: Cp, or process capability index, is a statistical measure that assesses a process's ability to produce output within specified limits. It essentially evaluates how well a process can meet its specifications compared to the natural variability of the process. A higher Cp value indicates that a process is more capable of producing items within desired tolerances, making it essential for quality control and improvement efforts.
Cpk: Cpk, or Process Capability Index, is a statistical measure that assesses how well a process can produce output within specified limits. It quantifies the capability of a process by comparing the width of the process variation to the width of the specification limits. A higher Cpk value indicates a more capable process that is producing fewer defects, while a lower Cpk suggests that the process may need improvement to meet quality standards.
Cpm: CPM, or Cost Per Mille, is a metric used in advertising that denotes the cost of acquiring 1,000 impressions on a digital ad. This measure helps advertisers evaluate the efficiency of their ad spending by providing a clear understanding of how much they are paying for exposure to potential customers. It connects closely with the concepts of reach and frequency in marketing, as it allows businesses to estimate how many people are being exposed to their advertisements relative to the costs incurred.
Defective Rate: The defective rate is a measure of the proportion of defective items produced in a manufacturing process, typically expressed as a percentage. It helps organizations assess their production quality and efficiency by quantifying the number of products that fail to meet quality standards. A lower defective rate indicates better process capability, reflecting more effective manufacturing practices.
Histogram: A histogram is a graphical representation of the distribution of numerical data that uses bars to show the frequency of data points within specified ranges, known as bins. It provides a visual interpretation of data that helps to identify patterns such as central tendency, dispersion, and the shape of the distribution, making it a fundamental tool in understanding data characteristics.
Lsl: The term lsl stands for 'lower specification limit,' which represents the minimum acceptable value in a process or measurement. In process capability analysis, lsl is critical as it helps determine if a process is capable of producing output that meets quality standards. A lower specification limit sets a boundary that indicates the lowest performance level acceptable for a product or service, aiding in quality control and improvement efforts.
Mu: Mu (μ) is the symbol commonly used to represent the population mean in statistics, which is the average value of a set of measurements or observations within a population. This term is significant in understanding how data behaves and provides a central point around which values tend to cluster. It serves as a fundamental parameter in various statistical analyses, helping to measure process performance and variability.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This characteristic forms a bell-shaped curve, which is significant in various statistical methods and analyses.
Pareto Chart: A Pareto chart is a type of bar graph that visually represents the frequency or impact of problems or causes in a process, helping to identify the most significant factors. The bars are arranged in descending order of height, while a cumulative line graph overlays the bars to show the total impact of the factors cumulatively. This tool is essential in identifying the 'vital few' issues that contribute most to an overall problem, allowing businesses to focus their efforts on the areas that will yield the greatest improvement.
Pp: In process capability analysis, 'pp' stands for process performance, which is a measure of how well a process can produce output that meets specifications over the long term. It is often calculated using the formula $$pp = \frac{USL - LSL}{6\sigma}$$ where USL and LSL are the upper and lower specification limits, respectively, and $$\sigma$$ represents the process standard deviation. This term helps to assess whether a process is capable of producing within specified limits and plays a crucial role in quality control and improvement initiatives.
Ppk: Ppk, or Process Performance Index, is a statistical measure used to evaluate how well a process is performing in relation to its specified limits. It provides insights into the inherent capability of a process by comparing the actual performance with the desired specifications, reflecting both the process mean and variation. A higher ppk value indicates a more capable process that produces fewer defects and meets customer requirements more consistently.
Process Capability Indices: Process capability indices are statistical measures used to assess the ability of a manufacturing process to produce products within specified limits. These indices evaluate how well a process can meet set specifications, typically involving target values and tolerances, which helps in identifying potential improvements in quality and performance.
Process performance: Process performance refers to the measure of how effectively a process operates in terms of producing quality outputs while meeting specified requirements. It encompasses various metrics that assess efficiency, effectiveness, and capability in relation to the intended outcomes of a process. Understanding process performance is essential for identifying areas for improvement, ensuring consistency, and achieving operational excellence.
Process stability: Process stability refers to the consistent performance of a process over time, where the process operates within defined limits without significant variations. This concept is crucial for ensuring that a process can reliably produce products or services that meet quality standards, and it connects closely with the evaluation of how capable a process is in fulfilling specifications.
Quality Assurance: Quality assurance is a systematic process designed to determine if a product or service meets specified requirements and standards. This proactive approach focuses on preventing defects and ensuring that processes are in place to maintain consistent quality over time. Quality assurance emphasizes the importance of improving operational processes, which relates directly to methodologies that analyze and enhance production quality, as well as evaluating a process's capability to meet customer expectations.
Sigma: Sigma, often represented by the Greek letter 'σ', is a statistical term that refers to the standard deviation of a population. It is a measure of the amount of variation or dispersion in a set of values, indicating how much individual data points differ from the mean. In the context of process capability analysis, sigma helps assess how well a process can produce output within specified limits.
Six Sigma: Six Sigma is a data-driven methodology aimed at improving processes by identifying and removing defects and minimizing variability. It focuses on using statistical tools and techniques to enhance product quality and operational efficiency, making it a crucial approach for organizations seeking to make informed decisions based on data analysis and performance metrics.
Special cause variation: Special cause variation refers to the variation in a process that can be traced to specific, identifiable factors that are not part of the inherent process. Unlike common cause variation, which is always present and predictable, special cause variation indicates that something unusual has occurred that disrupts the normal functioning of a process. Identifying and addressing these special causes is crucial for improving quality and maintaining process stability.
Specification Limits: Specification limits are the defined boundaries within which a product or process must operate to meet quality requirements. These limits indicate the acceptable range for a product's characteristics, ensuring that it meets customer expectations and regulatory standards. Understanding specification limits is crucial for process capability analysis, as they help determine if a process can consistently produce products that conform to these requirements.
Usl: The upper specification limit (USL) is the maximum acceptable value for a quality characteristic in a process or product. It is an essential component of process capability analysis as it helps determine how well a process can produce products that meet quality standards. By setting the USL, organizations can assess whether their processes are capable of consistently producing items within desired specifications.
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