Shewhart chart

What Is a Shewhart Chart?

A Shewhart chart is a fundamental graphical tool used in statistical process control (SPC) to monitor and analyze process variation over time. This chart, a core component of quality control methodologies, helps distinguish between two types of variation in a process: common cause variation, which is inherent to the process and stable, and special cause variation, which indicates an external, identifiable problem. By plotting data points sequentially, Shewhart charts provide a visual representation that enables practitioners to determine if a process is "in control" and predictable, or "out of control" and requiring intervention.

History and Origin

The Shewhart chart was developed by Walter A. Shewhart while he was working at Bell Telephone Laboratories in the 1920s. On May 16, 1924, Shewhart created the first control chart, a breakthrough that laid the foundation for modern statistical process control. His objective was to introduce statistical methods to monitor and control industrial processes, distinguishing between predictable natural variation and unusual, assignable causes⁴, ⁵. Shewhart's seminal work, "Economic Control of Quality of Manufactured Product," published in 1931, provided a comprehensive exposition of the principles behind quality control and the use of these charts. His theories significantly influenced quality management practices, notably through the work of W. Edwards Deming, who later popularized these concepts in Japan and the United States.

Key Takeaways

A Shewhart chart graphically monitors process performance over time to detect unusual variations.
It distinguishes between common cause (inherent) and special cause (assignable) variation.
The chart includes a central line, an upper control limit (UCL), and a lower control limit (LCL) to define acceptable process boundaries.
Points falling outside the control limits or exhibiting non-random patterns indicate a process is out of statistical control.
Shewhart charts are a cornerstone of continuous improvement efforts across various industries.

Formula and Calculation

The specific formula for a Shewhart chart varies depending on the type of data being monitored (e.g., individual values, averages, ranges, proportions). However, the general structure involves a central line (CL) representing the process average, and control limits (UCL and LCL) typically set at three standard deviations from the central line.

For an X-bar chart (monitoring the average of subgroups), the formulas are:

Central Line (CL):

CL = \bar{\bar{X}}

Upper Control Limit (UCL):

UCL = \bar{\bar{X}} + A_2 \bar{R}

Lower Control Limit (LCL):

LCL = \bar{\bar{X}} - A_2 \bar{R}

Where:

(\bar{\bar{X}}) = The grand average of all subgroup averages (overall central tendency of the process).
(\bar{R}) = The average of the subgroup ranges.
(A_2) = A constant factor that depends on the subgroup size, found in statistical tables for control charts. This factor ensures the control limits are set at the appropriate multiple of the estimated process standard deviation.

These formulas establish boundaries that statistically define the expected range of variation when the process is operating under common causes.

Interpreting the Shewhart Chart

Interpreting a Shewhart chart involves examining the plotted data points in relation to the central line and control limits. A process is considered "in statistical control" if all data points fall within the upper control limit and lower control limit, and there are no discernible non-random patterns (such as trends, shifts, or cycles). When points fall outside these limits, it signals the presence of special cause variation, indicating that the process is out of control and requires investigation to identify and eliminate the root cause.

Beyond single points outside limits, other "out-of-control" conditions include a series of consecutive points on one side of the central line, or a consistent trend upward or downward. These patterns suggest that the process has shifted or is experiencing systematic problems that need to be addressed. The primary goal of using a Shewhart chart is to achieve and maintain a state of statistical control, allowing for predictable process performance.

Hypothetical Example

Consider a company that manufactures precision components. They use a Shewhart chart to monitor the diameter of a critical part, sampling five components every hour. Over a day, they collect 24 subgroups of data.

Collect Data: For each hourly subgroup, they measure the diameter of the five components and calculate their average ((\bar{X})) and range ((R)).
Calculate Overall Averages: After a sufficient number of subgroups (e.g., 20-25), they calculate the grand average of all subgroup averages ((\bar{\bar{X}})) and the average of all subgroup ranges ((\bar{R})).
Determine Control Limits: Using a standard table for control chart constants, they find the (A_2) value corresponding to a subgroup size of 5. They then calculate the Central Line, UCL, and LCL.
- Assume (\bar{\bar{X}}) = 10.00 mm, (\bar{R}) = 0.05 mm, and (A_2) for n=5 is 0.577.
- CL = 10.00 mm
- UCL = 10.00 + (0.577 * 0.05) = 10.02885 mm
- LCL = 10.00 - (0.577 * 0.05) = 9.97115 mm
Plot Data: Each hourly subgroup average ((\bar{X})) is plotted on the chart.
Analyze: If the 3:00 PM subgroup average falls at 10.035 mm, it would plot above the UCL (10.02885 mm). This signals an out-of-control condition, prompting the production team to investigate what specific event or change occurred around 3:00 PM that might have caused the increase in diameter. This could range from a machine malfunction to a change in raw material.

Practical Applications

Shewhart charts are widely applied across various sectors to monitor and improve processes. In manufacturing, they are used to control product dimensions, weights, or defect rates, ensuring consistency and reducing waste. For example, in the automotive industry, these charts are used to assess the quality characteristics of plastic moldings to identify random deviations from quality limits³.

In the service industry, Shewhart charts can track call center wait times, customer satisfaction scores, or transaction processing times. Healthcare organizations utilize them to monitor patient wait times, infection rates, or medication errors, supporting efforts to enhance patient safety and operational efficiency. The National Institute of Standards and Technology (NIST) provides extensive guidance on the application of statistical methods, including control charts, for process improvement across many industries.

Limitations and Criticisms

While powerful, Shewhart charts have certain limitations. One common criticism is their relative insensitivity to small, sustained shifts in the process mean². Because the charts primarily rely on individual points exceeding the three-sigma control limits, a subtle but consistent drift in the process average might not trigger an out-of-control signal quickly. This can lead to a delayed detection of minor process degradations.

Furthermore, the effectiveness of a Shewhart chart depends heavily on accurately calculating the central line and control limits from historical data, known as Phase I. If the initial data used for these calculations are not truly representative of a stable, in-control process, the limits may be misleading, leading to false alarms or missed signals¹. Over-adjusting a stable process based on common cause variation (often called "tampering") can actually increase, rather than decrease, overall process variation. This misuse can undermine the very purpose of the chart, potentially increasing costs and reducing quality.

Shewhart Chart vs. Pareto Chart

While both are valuable quality control tools, a Shewhart chart and a Pareto chart serve different purposes in process improvement.

A Shewhart chart (a type of control chart) is a time-series graph used to monitor a process over time. Its primary function is to determine if a process is stable and predictable (in statistical control) or if it is being influenced by special cause variation. It helps identify when a problem occurred and whether a process is stable.

In contrast, a Pareto chart is a bar graph that displays categories of data in descending order of frequency, along with a cumulative percentage line. It is based on the Pareto principle (the 80/20 rule), which suggests that roughly 80% of problems come from 20% of causes. The primary purpose of a Pareto chart is to identify and prioritize the most significant problems or causes within a process, helping teams focus their improvement efforts on the areas that will yield the greatest impact. It answers the question what are the biggest problems.

Essentially, a Shewhart chart tells you if a process is behaving as expected over time, while a Pareto chart helps identify which specific issues are most prevalent or impactful within that process.

FAQs

What is the purpose of a Shewhart chart?

The purpose of a Shewhart chart is to monitor a process over time to determine if it is stable and predictable, or if it is experiencing unusual variations that require investigation and corrective action. It helps distinguish between inherent process variability and assignable causes of variation.

How do you determine the control limits on a Shewhart chart?

The control limits on a Shewhart chart are typically set at three standard deviations above and below the central line (process average). These limits are calculated using historical data from the process itself, not external specifications.

Can a Shewhart chart be used for any type of data?

Shewhart charts are versatile and can be used for various types of data, including variable data (measurements like length, temperature, time) and attribute data (counts of defects or defective items, proportions). Different types of Shewhart charts (e.g., X-bar and R charts for variables, P charts for proportions) are designed for specific data types.

What does it mean if a point falls outside the control limits?

If a data point falls outside the upper control limit or lower control limit on a Shewhart chart, it indicates that a "special cause" of variation is likely present. This means something unusual has affected the process, and an investigation should be conducted to identify and eliminate the root cause.

What is the difference between a Shewhart chart and a run chart?

Both Shewhart charts and run charts display data over time. However, a Shewhart chart adds statistically calculated control limits (UCL and LCL) and a central line, which allow for a more precise determination of whether a process is in statistical control. A run chart only plots data points and a central line (often the median or average), making it useful for observing trends but less precise for detecting out-of-control conditions compared to a Shewhart chart.