Control limits

What Are Control Limits?

Control limits are the boundaries on a control chart that define the expected range of variation for a process that is operating in a state of statistical process control. These limits, typically set at plus or minus three standard deviations from the process mean, help distinguish between common cause variation (natural, inherent fluctuations within a stable process) and special cause variation (unusual, identifiable factors that indicate a process is out of control). Control limits are a core concept within statistical process control, a sub-discipline of quality control, which aims to monitor and maintain the consistency and predictability of a process over time. When data points fall outside these control limits, or exhibit non-random patterns within them, it signals that the process may be experiencing issues that require investigation and corrective action.

History and Origin

The concept of control limits and Shewhart charts was pioneered by Walter A. Shewhart while working at Bell Telephone Laboratories in the 1920s. Shewhart recognized the need for a statistical method to monitor and control manufacturing processes. On May 16, 1924, he created the first "control chart" to differentiate between common cause and special cause variation, laying the groundwork for modern statistical process control. His seminal work, "Economic Control of Quality of Manufactured Product" (1931), formalized these ideas, stressing that bringing a production process into a state of statistical control and keeping it there is essential for predicting future output and managing processes efficiently.¹²,¹¹,

Key Takeaways

Control limits define the expected range of normal variation in a stable process.
They are typically set at three standard deviations above and below the process mean.
Data points falling outside control limits or showing non-random patterns indicate the presence of special cause variation.
Control limits are crucial for identifying when a process needs intervention and distinguishing it from routine process variation.
They are a fundamental tool in statistical process control for continuous process improvement.

Formula and Calculation

For individual observations (X charts) when the process mean (\mu) and standard deviation (\sigma) are known or can be reliably estimated, the general formulas for control limits are:

\text{Upper Control Limit (UCL)} = \mu + 3\sigma \\ \text{Lower Control Limit (LCL)} = \mu - 3\sigma

Where:

(\mu) represents the average or mean of the process being monitored.
(\sigma) represents the standard deviation of the process, indicating the typical dispersion of data points around the mean.
The "3" is a constant that, for a normally distributed process, encompasses approximately 99.73% of data points, assuming the process is stable and only common cause variation is present.

For other types of control charts, such as the X-bar chart (used for sample means) or the R chart (used for sample ranges), specific factors derived from statistical tables are used in place of the direct standard deviation, adjusting for subgroup size.

Interpreting the Control Limits

Interpreting control limits involves more than simply checking if data points fall within the defined upper control limit (UCL) and lower control limit (LCL). A process is considered "in control" if all data points fall between the UCL and LCL and exhibit no discernible non-random patterns. Patterns that indicate an "out-of-control" process, even if within the limits, include:

Points outside the limits: A single point above the UCL or below the LCL signifies a special cause.
Runs: Several consecutive points all above or below the center line.
Trends: A series of points consistently increasing or decreasing.
Cycles: Recurring patterns in the data.

When such a signal occurs, it prompts an investigation to identify and address the special cause responsible, rather than making adjustments based on normal process variation. Effective data analysis of the chart helps process owners determine if intervention is necessary or if the process should be left alone.

Hypothetical Example

Imagine a financial institution that processes an average of 1,000 credit card applications per day. Management wants to monitor the number of errors in these applications using control limits.

Suppose, through historical data analysis, the average number of errors per 1,000 applications is 20, with a standard deviation of 5 errors.

Using the general formula:
(\mu = 20)
(\sigma = 5)

UCL = (\mu + 3\sigma = 20 + (3 \times 5) = 20 + 15 = 35) errors
LCL = (\mu - 3\sigma = 20 - (3 \times 5) = 20 - 15 = 5) errors

So, the upper control limit is 35 errors, and the lower control limit is 5 errors.

If, on a given day, 40 errors are detected, this point falls above the UCL, signaling a special cause variation. The team would then investigate why the error rate was unusually high that day—perhaps a new system update, a temporary staffing shortage, or inadequate training. Conversely, if only 3 errors are found, this point falls below the LCL, also signaling a special cause, which might indicate an unusually effective new procedure or a data reporting error. In either case, it warrants investigation beyond normal fluctuations.

Practical Applications

Control limits are widely applied across various industries to maintain and improve the consistency and quality of processes. In manufacturing, they are essential for monitoring product dimensions, defect rates, and manufacturing efficiency. For example, they might track the weight of packaged goods to ensure consistent output.

In the service sector, particularly within financial institutions, control limits are used to monitor operational processes and financial metrics. They can be applied to track call center wait times, loan approval processing times, or transaction error rates to ensure compliance and efficiency., ¹⁰F⁹or instance, a financial firm might use control charts to monitor key financial ratios or error rates in compliance operations, identifying deviations from expected patterns that require investigation. T⁸he International Monetary Fund (IMF), for example, employs a Data Quality Assessment Framework (DQAF) that, while broader than control limits, emphasizes the importance of statistical processes and accurate data in assessing the quality of macroeconomic datasets, which underpins the need for such quality control measures.,,⁷
⁶
⁵Beyond finance and manufacturing, control limits find use in healthcare (e.g., monitoring patient wait times or infection rates), environmental monitoring, and logistics to ensure process capability and consistent performance.

Limitations and Criticisms

While powerful, control limits and control charts have certain limitations. One challenge is the difficulty in obtaining baseline data, particularly in dynamic environments like healthcare, which can make accurate setup resource-intensive and time-consuming., ⁴T³he effectiveness of control charts relies heavily on understanding the process and proper data analysis to distinguish between common and special causes. Misinterpretation can lead to over-adjustment of a stable process (tampering) or, conversely, a failure to act when a true problem exists.

Another criticism is that traditional control limits, often based on a three-sigma rule, might not be sensitive enough to detect small but persistent shifts in a process. M²ore advanced statistical methods, such as Cumulative Sum (CUSUM) or Exponentially Weighted Moving Average (EWMA) charts, are sometimes employed for greater sensitivity to minor deviations. Additionally, in increasingly complex and high-dimensional data environments, traditional statistical process control methods, including control limits, may face challenges in providing comprehensive insights, sometimes requiring integration with more advanced data science techniques. A¹ fundamental aspect of risk management is recognizing that simply plotting data does not automatically lead to process improvement; strong management commitment is necessary to implement changes based on signals from the control charts.

Control Limits vs. Tolerance Limits

While both control limits and tolerance limits define boundaries, they serve fundamentally different purposes in quality and process management.

Control Limits: These are derived statistically from the actual performance of a process. They represent the "voice of the process," defining the natural, inherent variability of a system operating in statistical control. Their purpose is to signal when a process has changed or is out of control, prompting an investigation into special causes of variation. They are used for process monitoring and improvement.
Tolerance Limits: These are engineering or customer-specified boundaries that define the acceptable range for a product's characteristic or a service's outcome. They represent the "voice of the customer" or the design specification. Tolerance limits do not reflect how the process is actually performing but rather how it should perform to meet requirements. Failing to meet tolerance limits indicates a quality defect, regardless of whether the process is statistically in control.

In essence, control limits tell you what your process is capable of doing, while tolerance limits tell you what your process needs to do to meet requirements. A process can be in statistical control (within its control limits) but still produce outputs that are outside of tolerance limits if the process's natural variation is too wide to meet customer specifications.

FAQs

How are control limits different from specification limits?

Control limits are calculated from process data and show the expected range of normal process behavior. Tolerance limits, also known as specification limits, are external requirements or targets set by customers, designers, or regulations for a product or service's acceptable range. A process can be in statistical control (within its control limits) but still produce outputs that don't meet the tighter specification limits.

Why are control limits typically set at three standard deviations?

The use of three standard deviations (3-sigma) for control limits is a convention established by Walter Shewhart. For a normally distributed process, approximately 99.73% of all data points will fall within plus or minus three standard deviations from the mean. This interval provides a balance: it is wide enough to avoid frequent "false alarms" (interpreting common cause variation as a special cause) but narrow enough to detect significant shifts in the process.

Can control limits change over time?

Yes, control limits are not static. They are recalculated periodically as a process improves or fundamentally changes. When a special cause is identified and eliminated, and the process reaches a new, more stable state, the control limits should be updated to reflect this improved performance. Conversely, if a process degrades, the limits might also need re-evaluation. They are always based on the actual, observed process variation.