Understanding control charts and concepts of variation

Controlled variation vs. uncontrolled variation

Common cause variation

Common cause variability, also known as controlled variation, is the natural, everyday variation inherent in any process. This type of variation is caused by many small, random factors. It is always present and cannot be eliminated without fundamentally changing the process.

A controlled process is stable and predictable over time, which is essential for making reliable decisions about future performance. For example, slight differences in temperature or humidity during production might be responsible for common cause variation. The example below shows the output of a process with typical common cause variation. A stable process changes with constant variation around its mean. It does not drift, shift, or spike.

Special cause variation

Special cause variation, sometimes known as uncontrolled variation or assignable cause variation, happens when something unusual affects a process. This type of variation is not inherent to the process and arises from specific, identifiable sources. Special causes are often sporadic and can be corrected once identified. Uncontrolled variation can lead to defects, inefficiencies, and inferior quality. Examples include a miscalibrated machine, a change in raw material, or a sudden power outage. Identifying and addressing special causes is critical to maintaining process stability. The example below shows the output of a process changing when a component changes to a new batch of raw materials.

Short-term vs. long-term estimators of sigma

To estimate the variability of sample of data, you often use the sample standard deviation. In statistical process control, the focus is on distinguishing between common- and special-cause variability, and the sample standard deviation is not a good estimator of sigma in this case.

In analysis of variance, you use within-group variability to judge the between-group variability. The same thing happens in control charts: within-subgroup variation or short-term variation is used to judge the stability of the process. Using short-term variation in a control chart makes the chart more sensitive to special causes, which means finding (and fixing) more process problems.

Short-term estimator

A short-term estimator of $\sigma$ measures the variability within subgroups of data, capturing the process's inherent consistency over a brief period of time. It reflects common cause variation and is used to set control limits that are sensitive to detecting special causes. For example, $\sigma$ might be estimated using data from consecutive production runs if there is not a basis for subgrouping, or it might be estimated for each group of five units processed together in a specialized fixture. In the image below, the short-term estimate of $\sigma$ is represented by the blue box that can include nearly all the variation for each rolling brief period of time.

Long-term estimator

A longer-term estimator of $\sigma$ measures the overall variability across all data, including both common and special causes, over an extended period of time. The long-term $\sigma$ estimate is measured by the sample standard deviation. For instance, the long-term s estimate might include data from multiple shifts, operators, or batches, capturing the full range of variation in the process. In the image above, the long-term estimate of $\sigma$ is much larger than the short-term estimate of $\sigma$, due to the presence of special causes as the process exhibits noticeable gradual trends and sudden shifts.

Historical development of control charts

Walter Shewhart, known as the father of statistical quality control, invented the control chart in the 1920s while working at Bell Labs. His work laid the foundation for modern statistical process control by introducing the concept of monitoring processes using statistical methods. Shewhart's control charts were initially used to monitor manufacturing processes and ensure product quality.

In the 1950s, L.H.C. Tippett addressed a challenge in estimating short-term variability for subgroup size one. He introduced the use of the average moving range ( ) to estimate variability, which became a standard method for control charts on individual values.

In the 1980s, W. Edwards Deming and Donald Wheeler popularized the use of I-MR (individual and moving range) charts. These charts became widely adopted for processes where data are collected one point at a time. Control charts used today generally include the following elements:

●       Individual points are graphed on a Y axis, sorted by an X axis representing time or production order.

●       A process center line showing the average of the process historically.

●       Control limits, which are calculated based on historic process variation and indicate specific thresholds beyond which a user can conclude that a problem has occurred with the process that needs to be addressed immediately.

●       Often, specific points are highlighted to indicate that other process problems, such as process drifting, have occurred and need to be addressed.

Three-sigma limits

Control charts typically use limits set at ±3 times an estimate of $\sigma$ from the process mean. Different control charts use different short-term estimators of $\sigma$. Shewhart found that using the multiple of 3 balances the trade-off between detecting true process changes and avoiding false alarms. Simply put, if a point falls outside these limits, it is much more likely to be due to a special cause of variation rather than random chance. This approach has become a cornerstone of statistical process control.

Rational subgroups

Rational subgrouping is a method of selecting data to ensure that control limits reflect only common cause variation. The goal is to make the control chart more sensitive to detecting special causes. Subgroups are formed by collecting data points that are close together in time or under similar conditions. This minimizes the chance of including special causes within a subgroup. For example, in a manufacturing process, a rational subgroup might consist of parts produced together in the same fixture used by one operator on one machine.

Why rational subgroups matter

If within-subgroup variability includes both common and special causes, the control limits will be wider than they should be, making the chart less sensitive to detecting special causes. Giving extra thought to rational subgrouping helps eliminate special cause variation from occurring within subgroups, helping to narrow the overall control limits. These narrower limits ensure that the control chart provides the most opportunities for signaling special cause events.

Traditional use and evolution of the control chart

When control charts were first introduced, calculations were made manually. Simpler measures, such as ranges, were preferred over standard deviations because they were easier to compute, thus making control charts accessible to practitioners in the early 20th century.

With the advent of computers and statistical software like JMP, more complex calculations, such as standard deviations and advanced control chart types, are now feasible. These advancements have improved the precision and versatility of control charts.

Over time, control charts have evolved to include other types, such as EWMA (exponentially weighted moving average) and CUSUM (cumulative sum) charts and MDMVCC (model-driven multivariate control chart). These charts are particularly effective at detecting small shifts in processes and shifts in interrelated processes, making them valuable tools for modern quality control and process improvement.