SPC Blog

In Statistical Process Control (SPC), specification limits and control limits are two core concepts. Although they both play important roles in quality management and process monitoring, they differ in their definitions, sources, and applications. This article will explain these two concepts in detail and mainly discuss their potential to be one-sided or two-sided, as well as their impact on SPC metrics.

Specification Limits

Specification limits are the acceptable range of product or process characteristics specified by customers, design specifications, or industry standards. They define the quality requirements that a product or service must meet. Specification limits typically include the Upper Specification Limit (USL) and Lower Specification Limit (LSL), used to assess whether the product meets the expected quality standards. If product characteristics exceed the specification limits, they are considered nonconforming.

Specification limits can be one-sided or two-sided:

• If the concern is only whether a characteristic value is not less than a minimum value, only the Lower Specification Limit (LSL) is set.
• If the concern is only whether a characteristic value does not exceed a maximum value, only the Upper Specification Limit (USL) is set.

Control Limits

Control limits are derived from the statistical analysis of process data and are used to monitor process stability. Control limits typically consist of the Upper Control Limit (UCL) and Lower Control Limit (LCL), reflecting the natural variation range of most data points in a process under normal conditions (normal distribution).

Control limits are usually two-sided because their primary function is to detect process stability and abnormalities. The calculation of control limits is typically based on the 6σ principle under a normal distribution, where data points within the 3σ range are considered normal process variations, and data points beyond this range are considered abnormal signals (under a normal distribution, the probability of exceeding the control limits is less than 1%).

Regardless of whether the specification limits are one-sided or two-sided, control limits are used to determine process stability and should include both upper and lower control limits to assess stability.

For example, even if the specification limit only requires the characteristic value to be greater than a minimum value (e.g., greater than 100), control limits will still be calculated based on the data (e.g., UCL = 200, LCL = 50):

• The role of UCL and LCL: The Upper Control Limit (UCL) and Lower Control Limit (LCL) are used to detect abnormalities in the process. If process data points exceed these control limits, it indicates potential process anomalies that require further investigation. Even if a data point greater than 200 is acceptable based on the specification limit (as long as it is greater than 100), it may indicate process instability according to the control limit (UCL = 200), prompting further investigation into the cause of this anomaly.

• Abnormality and Stability: Even if the process characteristics meet the specification limits, significant process variation (e.g., data points exceeding UCL or LCL) may indicate process instability. Control limits help identify this potential instability.

If the specification limit is one-sided (e.g., only LSL), control limits will still be two-sided. Exceeding the control limits will still trigger alarms, but data points exceeding LCL should receive particular attention, while data points exceeding UCL can be analyzed or ignored as needed.

Impact of Specification Limits and Control Limits on SPC Metrics

When the specification limit is one-sided, it affects certain SPC metrics, especially those related to capability indices such as Ca, Ppk, and Cpk.

• Ca (Capability Index) measures the degree of deviation between the process mean and the specification center. For one-sided specification limits, Ca cannot be calculated due to the lack of a reference center value.

• Ppk and Cpk measure process performance and capability. For one-sided specification limits, Ppk and Cpk calculations only consider the direction of the existing specification limit. For example:

  • Only USL:
  • Only LSL:

In these cases, one-sided specification limits only assess process capability in one direction, potentially leading to an incomplete evaluation of the process. Particularly with one-sided specification limits, it is essential to use two-sided control limits to monitor process stability comprehensively.

You may have noticed that when using Minitab to create Xbar-R and Xbar-S control charts, each is composed of a pair of charts:

• Xbar-R Control Chart includes: Xbar Control Chart and R Control Chart

• Xbar-S Control Chart includes: Xbar Control Chart and S Control Chart

So the questions arise:

1. For the same data source, are the Xbar Control Charts in the Xbar-R and Xbar-S Control Charts the same?
2. When should we use the Xbar-R Control Chart, and when should we use the Xbar-S Control Chart?
3. How are the control limits of these control charts calculated?

This article provides the most detailed explanation available on the internet.

To clarify, the Xbar Control Charts in the Xbar-R and Xbar-S Control Charts are not the same. The use of Xbar-R Control Chart and Xbar-S Control Chart is conditional; we should not use Xbar-R and Xbar-S simultaneously on the same set of inspection data. Therefore, we do not need to worry about whether the Xbar Control Charts in Xbar-R and Xbar-S are the same, because we will not be using Xbar-R and Xbar-S at the same time.

For subgroup sizes ≤10

we need to use the Xbar-R Control Chart. The control limits for the Xbar Control Chart and R Control Chart are calculated as follows:

• Rbar: The average of the ranges of each subgroup

For subgroup sizes >10

we need to use the Xbar-S Control Chart. The control limits for the Xbar Control Chart and S Control Chart are calculated as follows:

• Sbar: The average of the standard deviations of each subgroup (Note: The standard deviation should be calculated with the denominator $n-1$)

The SPC constants such as A2, D4, A3, B4, etc., used in these formulas are as follows:

Let's say we have a batch of 3000 pieces, with each piece required to weigh 200g ± 5g. Based on previous experience, the pass rate is about 98%. Using this pass rate, we estimate that we need to sample 60 pieces. For this example, let's assume we sample 100 pieces.

We have sampled 100 pieces and measured their weights, resulting in 100 data points. How can we determine if the weights of this batch are consistent with our standard of 200g?

The data is as follows:

201.67, 202.33, 196.55, 197.94, 199.76, 195.77, 198.74, 199.81, 197.87, 198.49, 198.32, 199.14, 197.74, 200.36, 199.34, 197.67, 200.29, 200.98, 200.75, 202.73, 200.11, 201.47, 200.47, 201.23, 201.76, 204.01, 203, 200.3, 201.34, 197.02, 198.01, 196.63, 200.96, 201.84, 199.06, 201.19, 196.05, 198.24, 198.34, 201.16, 199, 199.12, 202.25, 200.77, 198.83, 201, 200.1, 199.7, 199.93, 199.86, 202.2, 198.8, 201.31, 200.96, 199.83, 202.44, 198.76, 197.26, 197.17, 201.26, 200.59, 197.6, 201.03, 203.05, 199.63, 197.48, 200.34, 200.42, 197.59, 198.16, 197.9, 198.05, 199.36, 202.68, 198.53, 201.11, 197.29, 200.38, 200.02, 201.64, 199.89, 199.5, 195.33, 203.19, 199.45, 199.66, 202.58, 201.08, 198.01, 199.08, 200.82, 197.92, 199.55, 198.81, 201.74, 201.54, 199.58, 198.09, 197.81, 201.56

SPC Analysis

If we use SPC (Statistical Process Control) analysis, we encounter the following issues:

1. Mean Value: The mean of the data is 199.77, very close to 200. But how do we determine if this is consistent? What defines "close"?
2. Control Limits: The data does not exceed the upper and lower three-sigma limits, but these limits are calculated based on the sample's mean and standard deviation, not the specification center.
3. Capability Index (Ca): The Ca value is -0.04, indicating the data is close to the specification center, but it doesn't define what is considered "close."

SPC analysis is typically based on time order, and in this case, the sequence of samples affects the control chart analysis and judgment. Therefore, if we cannot determine the order of the sampled items, using SPC control charts might lead to inaccurate conclusions.

Recommended Method: One-Sample T-Test

Instead of SPC, we should use a one-sample T-test. This test can determine if the sample mean is significantly different from the target value. The null hypothesis of the test is that the sample mean is equal to the target value (200g), and the alternative hypothesis is that the sample mean is not equal to the target value.

Results:

• Sample Mean: 199.77
• Sample Standard Deviation: 1.83
• Sample Size: 100
• T-Statistic: -1.26
• P-Value: 0.2116

The target value of 200.0 is within the confidence interval (199.4078, 200.1328), indicating no significant difference between the sample mean and the target value of 200.0.

The sample mean does not significantly differ from the target value of 200.0, so we cannot reject the null hypothesis (the sample mean is consistent with the target value).

The T-Test Calculation Process

Suppose we want to test a sample mean , sample standard deviation ( s ), and sample size ( n ) against a target mean ( μ ) of 200g. The t-value can be calculated using the following formula:

Confidence Interval Formula:

where is the critical value from the t-distribution with n-1 degrees of freedom.

T-Statistic and P-Value:

• If the p-value is less than the significance level (e.g., 0.05), reject the null hypothesis, indicating the sample mean is significantly different from 200g.
• If the p-value is greater than the significance level, fail to reject the null hypothesis, indicating no significant difference between the sample mean and 200g.

Confidence Interval:

• If 200g is within the confidence interval, it indicates no significant difference between the sample mean and 200g.
• If 200g is outside the confidence interval, it indicates a significant difference between the sample mean and 200g.

Using both the t-statistic/p-value and the confidence interval, we can determine if the sample weight is consistent with the target value of 200g.

Conclusion

SPC is not suitable for determining if a sample is consistent with the specification center; it is a tool for anomaly analysis. For determining if a sample is consistent with the specification center, the recommended method is the one-sample T-test.

When discussing quality anomaly analysis, almost everyone familiar with the field will mention SPC (Statistical Process Control) control charts. Indeed, SPC control charts are currently the most widely used tool for quality anomaly analysis (here, "widely used" refers to companies that already utilize quality analysis, though most manufacturing enterprises have not yet reached the stage of conducting quality anomaly analysis).

So, what other charts can we use for quality analysis related to SPC, besides the SPC control charts?

3σ Distribution Chart (My own coined term)

The chart below illustrates a 6sigma diagram; actual data rarely shows such an ideal state. When we look at control charts, they typically appear as shown below: 

However, we often want to know the number or proportion of these points located in the upper and lower ABC zones. The horizontal axis divisions of a normal distribution histogram are often automatically generated, and not necessarily corresponding to each sigma. The following chart divides the upper and lower ABC zones into intervals for each sigma. 

Variable Subgroup Control Chart (Another coined term)

Often, the number of samples in each batch is not fixed (not sampled by a fixed subgroup size). For the obtained data, we use the average to plot the chart and display each batch's data points on the chart to intuitively observe the distribution of the points.

Rainbow Chart (Pre-control Chart)

The pre-control chart is a newer, simpler, more user-friendly, and more economical quality control technique. Compared to traditional Shewhart control charts, it is statistically more powerful, allowing quicker identification and response to anomalies by setting predefined limits. It is suitable for real-time production line monitoring and small batch production environments.

Capability Analysis Chart

The capability analysis chart is a comprehensive tool for process capability analysis. It includes a data distribution chart, overall and within-group distribution fitting curves, and indicators such as PPK, CPK, and Ca. These charts and indicators help us evaluate process performance and capability comprehensively, identify potential issues, and provide a basis for process improvement. Through these analyses, we can better understand process stability and consistency, thus more effectively controlling quality. 

Capability Comparison Chart

The capability comparison chart visually compares specification limits, within-group actual deviation (range), and overall actual deviation (range). It allows us to see:

• Whether the data is within the specified range.
• The variation within subgroups.
• The overall data variation.

This chart helps identify issues within the process, such as if within-group deviation is much smaller than the overall deviation, it may indicate significant differences between batches. Conversely, if within-group and overall deviations are close, it indicates a more consistent and stable process. 

Subgroup Distribution Chart

This chart shows the distribution of measurement values for each subgroup. Each subgroup's measurement data is represented by a box plot. The box plot displays the median, interquartile range, and potential outliers for each subgroup. Red dots indicate outliers within each subgroup, significantly deviating from other data points, suggesting further attention and analysis may be needed.

This chart helps us intuitively understand the central tendency and dispersion of each subgroup's data, identify potential outliers, and thus better conduct quality control and data analysis. 

Normality Test Chart

The normality test chart is a tool used to assess whether data follows a normal distribution. By plotting data points on a specific chart, such as a Q-Q plot (Quantile-Quantile Plot) or a P-P plot (Probability-Probability Plot), it compares the actual data distribution to the theoretical normal distribution. If the data points roughly follow a straight line, the data conforms to a normal distribution; significant deviations indicate that the data may not follow a normal distribution. This chart is intuitive and easy to interpret, commonly used in statistical analysis. 

Machine Learning Anomaly Detection Chart

Machine learning for anomaly detection involves classifying data into two categories using regression-based binary classification methods, where the less frequent category is deemed anomalous. Algorithms such as linear regression, support vector machines (SVM), random forests, and K-means can be used.

However, machine learning for anomaly detection has a critical weakness: the judgment process is often a black box, making it difficult to understand. Unlike SPC's standard eight anomaly detection rules, we cannot clearly know why the machine learning algorithm determines a point as anomalous, making it challenging to find the underlying cause. 

Among the analysis charts and tools mentioned above, there are always some that suit your needs.

Statistical Process Control (SPC), as a systematic analytical tool, has been widely implemented in the manufacturing industry. It is not only easy to implement but also demonstrates significant effectiveness in improving product quality, optimizing production processes, and enhancing work performance. This discussion will elaborate on the ease of implementation, significant effectiveness, and work performance enhancement of SPC, while comparing it with other manufacturing analysis systems in terms of cost and employee participation.

Ease of Implementation

Simple and User-friendly Tools:

SPC primarily relies on statistical tools such as control charts and process capability indices. These tools are conceptually simple, easy to understand, and apply. Enterprises can equip employees with these basic tools through simple training.

Modern manufacturing enterprises are usually equipped with data acquisition systems and computer-aided tools, which can automatically generate the control charts and data analysis results required for SPC, further reducing the implementation difficulty.

Wide Range of Applications:

SPC is suitable for manufacturing enterprises of various scales and types, ranging from small businesses to large multinational corporations, from manual operations to automated production lines.

Whether it is a continuous production process or a discrete production process, SPC can be effectively applied to achieve comprehensive quality control.

Low Implementation Cost:

Compared to other complex analytical tools, the implementation cost of SPC is relatively low. Enterprises only need basic statistical knowledge and tools to start implementation, without requiring substantial upfront investment.

The automation of data collection and analysis further reduces labor costs, making SPC an affordable and efficient choice for enterprises.

Significant Effectiveness

Real-time Monitoring and Feedback:

SPC monitors the production process in real-time through control charts, promptly identifying and providing feedback on abnormal situations, preventing defective products from flowing into the next stage or the final market.

This real-time monitoring mechanism not only improves product quality but also reduces rework and scrap costs, directly enhancing production efficiency and economic benefits.

Process Stability and Capability Enhancement:

SPC can identify and control variations in the process, helping enterprises achieve process stability. A stable production process signifies product quality consistency and reliability.

Through continuous process capability analysis (e.g., Cp, Cpk), enterprises can continuously optimize the production process, enhance process capability, and make production more efficient.

Prevention-oriented Quality Management:

SPC emphasizes prevention rather than post-event correction, preventing quality problems from occurring by controlling process variation. This prevention-oriented management philosophy helps enterprises fundamentally solve quality issues and elevate overall quality levels.

Work Performance Enhancement

Data-driven Decision-making:

The data and analysis results provided by SPC offer scientific evidence for enterprise decision-making. Management can make informed decisions based on data, optimize resource allocation, and enhance production efficiency.

With data support, employees can better understand and control the production process, improving individual and team work performance.

Employee Skill Enhancement:

Implementing SPC requires employees to master basic statistical tools and quality management knowledge, which implicitly enhances their skill levels.

Through participation in SPC implementation, employees gain a deeper understanding and identification with quality management, leading to improved work motivation and a sense of responsibility, further promoting work performance enhancement.

Continuous Improvement Culture:

SPC advocates continuous improvement, fostering a corporate culture that pursues excellence by continuously monitoring and optimizing the production process.

Under the guidance of this culture, enterprises and employees continuously seek opportunities for improvement and enhancement, resulting in continuous improvement in work performance and overall competitiveness.

Comparison with Other Manufacturing Analysis Systems

Comparison with BI:

• Implementation Difficulty: BI systems require complex data warehouses and ETL (Extract, Transform, Load) processes; in contrast, SPC implementation is simpler.
• Real-time Capability: BI systems are typically used for strategic analysis in high-level decision-making, with lower data update frequency, while SPC emphasizes real-time process monitoring and feedback.
• Cost: BI system implementation costs are relatively high, involving substantial software and hardware investments and complex implementation processes, whereas SPC costs are lower, mainly involving basic statistical tools and simple training.
• Employee Participation: BI systems are primarily targeted at management and decision-making levels, while SPC requires extensive participation from frontline employees, promoting the enhancement of quality awareness among all staff.

Comparison with MES:

• Functional Focus: MES covers various aspects of production execution, including scheduling, tracking, quality control, etc., while SPC focuses on quality control and process monitoring, with a more focused function.
• Complexity: MES systems are highly complex, with high implementation costs and long cycles, whereas SPC implementation is relatively simple and cost-effective.
• Cost: MES system implementation is expensive, involving hardware, software, and extensive training investments, while SPC is less costly and easy to integrate into existing systems.
• Employee Participation: MES systems are primarily used by production management personnel, while SPC implementation requires the participation of all employees, especially the active cooperation of frontline workers, to enhance the overall quality control level.

Comparison with APS:

• Analysis Scope: APS focuses on production planning and scheduling optimization, while SPC focuses on quality monitoring and improvement in the production process.
• Complexity: APS requires complex algorithms and models for planning and scheduling optimization, with higher implementation and maintenance difficulty, whereas SPC is relatively simple and easy to use.
• Cost: APS system implementation and maintenance costs are relatively high, while SPC costs are lower and suitable for enterprises of all sizes.
• Employee Participation: APS is primarily used by planning personnel and management, while SPC requires the participation of all employees, from operators to management, to collectively enhance production quality.

Comparison with SCADA:

• Focus: SPC focuses on statistical analysis and quality control in the production process, while SCADA focuses on real-time monitoring and data acquisition, covering comprehensive monitoring of production equipment and process parameters.
• Implementation Complexity: SPC implementation is simple, mainly relying on control charts and process capability analysis, with quick results. SCADA implementation is complex, requiring the installation of sensors, configuration of hardware and software, and involving a significant amount of system integration work.
• Cost: SPC costs are lower, mainly involving statistical tools and employee training. SCADA costs are higher, including hardware, sensors, and high-performance software systems.
• Employee Participation: SPC requires the participation of all employees on the production line, especially operators and quality control personnel. SCADA is mainly used by engineers and operators for equipment monitoring and control, with relatively narrow participation.

Comparison with Simulation Analysis:

• Focus: SPC focuses on quality control and variation management in actual production processes, while simulation analysis optimizes production layout, processes, and resource allocation through modeling and simulation.
• Implementation Complexity: SPC implementation and use are relatively simple, mainly relying on statistical analysis tools. Simulation analysis requires complex modeling and simulation techniques, with higher implementation and maintenance difficulty.
• Cost: SPC costs are lower and suitable for enterprises of all sizes. Simulation analysis costs are higher, involving specialized software, hardware, and technical personnel.
• Employee Participation: SPC requires the participation of all employees, especially the cooperation of operators. Simulation analysis is primarily used by engineers and management for strategic decision-making and process optimization.

Comparison with Equipment Analysis Software:

• Focus: SPC focuses on process quality control and statistical analysis, controlling variations in the production process. Equipment analysis software focuses on equipment operating status, maintenance needs, and fault prediction.
• Implementation Complexity: SPC implementation is simple, mainly involving statistical tools and basic training. Equipment analysis software implementation is complex, usually requiring the integration of equipment sensors and advanced analysis software.
• Cost: SPC costs are lower and easy to integrate into existing systems. Equipment analysis software implementation costs are higher, involving hardware, software, and technical personnel.
• Employee Participation: SPC requires the participation of all employees, especially frontline workers and quality management personnel. Equipment analysis software is mainly used by equipment maintenance personnel and engineers to monitor and maintain equipment status.

Comparison with QMS:

• Focus: QMS covers the entire enterprise's quality management system, including document control, audits, supplier management, etc., while SPC focuses on statistical analysis and quality control in the production process.
• Implementation Complexity: QMS system implementation is highly complex, requiring comprehensive quality management system establishment, while SPC can be quickly implemented within the existing production system.
• Cost: QMS systems involve comprehensive quality management system establishment, with higher costs, while SPC only requires basic statistical tools and simple training, resulting in lower costs.
• Employee Participation: QMS systems require the participation of all employees, but the implementation process is complex and time-consuming. SPC implementation is simple and can quickly yield results, enhancing employee participation and a sense of responsibility.

Conclusion

Statistical Process Control (SPC), with its ease of implementation, significant effectiveness, and notable enhancement of work performance, stands out as the most accessible, effective, and performance-demonstrating analytical tool in the manufacturing field.

Compared to other manufacturing analysis systems, SPC possesses unique advantages in real-time monitoring, focus on quality control and prevention, simple implementation, low cost, and employee participation. By widely applying SPC, manufacturing enterprises can achieve continuous improvement in quality management, enhancing overall production efficiency and market competitiveness.

Our point is that the SPC system is easier to implement, promote, cost-effective, and yields obvious benefits. Other manufacturing systems/systems are very important, but they are not at the same level as SPC tools/systems and are even more critical than SPC.

Our suggestion is that a low-cost, high-benefit SPC project can be used to enhance digitalization and improve product quality.