Shewhart charts

Shewhart charts are a fundamental tool within Statistical Process Control (SPC), a methodology focused on monitoring and controlling processes to ensure consistent quality and output. These charts provide a graphical representation of process data over time, allowing for the identification of unusual trends or deviations that may indicate a process is out of statistical control. By distinguishing between natural, random process variation and specific, assignable causes of variation, Shewhart charts help organizations make data-driven decisions to improve efficiency and reduce defects. They are widely applied in various fields beyond manufacturing, including service industry, healthcare, and even investment performance analysis.

History and Origin

Shewhart charts owe their existence to Walter A. Shewhart, an American physicist and statistician, who developed the concept while working at Bell Telephone Laboratories in the 1920s.¹⁰ Bell Labs sought to enhance the reliability of its telephone transmission systems and to improve the clarity of voice transmissions in carbon microphones.⁹ Shewhart's pioneering work laid the groundwork for modern quality control by recognizing that every process exhibits process variation, but that this variation could be categorized.

On May 16, 1924, Shewhart created the first control chart, proposing it as a tool to differentiate between common cause variation, which is inherent to a process, and special cause variation, which arises from specific, identifiable factors.⁸ This distinction was critical because addressing common cause variation requires changes to the process itself, while special cause variation demands investigation into its unique root cause. His work was foundational for the field of statistical quality control and significantly influenced later quality pioneers like W. Edwards Deming, who further popularized the use of SPC, particularly in post-World War II Japan.⁷

Key Takeaways

• Shewhart charts are graphical tools used in Statistical Process Control (SPC) to monitor and assess process stability over time.
• They help differentiate between routine, expected process variation (common causes) and unusual, unexpected variation (special causes).
• The charts typically include a central limit (process average) and statistically derived upper control limit and lower control limit lines.
• Points falling outside the control limits, or specific patterns within the limits, signal the presence of special cause variation requiring investigation.
• Shewhart charts are widely applicable for data analysis in any process where output can be measured numerically or attributed by count.

Formula and Calculation

The general formula for setting the control limits in many Shewhart charts is based on the process average and a multiple of the process's standard deviation. While specific formulas vary by chart type (e.g., X-bar, R, P, C charts), the core principle remains consistent.

For a typical Shewhart chart, the control limits are calculated as:

Where:

• (\text{UCL}) = Upper Control Limit
• (\text{LCL}) = Lower Control Limit
• (\bar{\bar{X}}) = Grand average of all subgroup averages (the process average)
• (\bar{R}) = Average range of subgroups
• (A_2) = A constant factor that depends on the subgroup size, used to calculate the control limits for the average, derived from statistical tables.

These formulas are for X-bar and R charts, which are commonly used together to monitor both the average and variability of a process using data points collected in subgroups.

Interpreting the Shewhart Chart

Interpreting a Shewhart chart involves more than just looking for points outside the control limits. While a point beyond either the upper or lower control limit is a clear signal of a special cause of variation, other patterns within the control limits can also indicate an unstable process. These patterns, often called "runs" or "trends," suggest that the process behavior has shifted.

For example, a series of consecutive points above or below the center line, or a consistent upward or downward trend, might signal a non-random event influencing the process. The purpose of identifying these signals is to prompt investigation into the root cause of the unusual variation. If a special cause is identified and eliminated, the control limits may be recalculated to reflect the improved process. If no special cause is found, the variation is considered inherent process variability. The goal is to bring a process into a state of statistical control, where only common cause variation remains.

Hypothetical Example

Imagine a small online retail company, "GadgetGo," that aims to ship customer orders within two business days. To monitor their shipping process, GadgetGo decides to implement a Shewhart chart for the average daily shipping time (in hours from order placement to shipment confirmation).

1. Data Collection: For 25 consecutive days, GadgetGo randomly selects 5 orders each day and records the time it took to ship them. They calculate the average shipping time for each subgroup of 5 orders, and the range of times within each subgroup.
2. Calculate Averages:
   • The average of all 25 daily averages ((\bar{\bar{X}})) is calculated, let's say it's 36 hours. This becomes the center line for their chart.
   • The average of all 25 daily ranges ((\bar{R})) is calculated, say 10 hours.
3. Determine Control Limits: Using a standard (A_2) factor for a subgroup size of 5 (which is 0.577), the control limits are calculated:
   • UCL = (36 + (0.577 \times 10) = 36 + 5.77 = 41.77) hours
   • LCL = (36 - (0.577 \times 10) = 36 - 5.77 = 30.23) hours
4. Plot Data: Each day's average shipping time is plotted on the chart.
5. Interpretation: For the first 20 days, all points fall within the UCL and LCL, and there are no suspicious patterns, indicating the shipping process is stable. On day 21, the average shipping time jumps to 43 hours, exceeding the UCL. This signals a "special cause" of variation.
6. Action: GadgetGo investigates and discovers that their primary packing machine broke down on day 21, forcing manual packing, which significantly slowed down the process. They fix the machine, and subsequent days' averages return to within control limits. This incident was a special cause, acted upon and resolved, allowing the process to return to its statistically controlled state.

Practical Applications

Shewhart charts are versatile tools for risk management and process improvement across numerous sectors:

• Manufacturing: They are indispensable in factories for monitoring product dimensions, defect rates, and process parameters like temperature or pressure. This ensures consistent product quality assurance and reduces waste.⁶
• Healthcare: Hospitals use Shewhart charts to track patient wait times, infection rates, or medication errors, aiming to improve patient safety and operational efficiency.
• Finance: Beyond investment performance, financial institutions can apply Shewhart charts to monitor transaction processing times, error rates in data entry, or call center wait times, enhancing operational efficiency.
• Government and Regulation: Regulatory bodies, such as the Food and Drug Administration (FDA), encourage or mandate the use of statistical methods, including control charts, for process validation and continued process verification in industries like pharmaceuticals and medical devices to ensure product quality and safety.⁵,⁴ This application helps ensure that processes remain in a state of control throughout their lifecycle.

Limitations and Criticisms

While powerful, Shewhart charts have certain limitations. One common criticism is that they primarily detect shifts in the process mean or variability after they have occurred. They are less effective at detecting small, gradual shifts in a process. For such subtle changes, other control charts, such as Cumulative Sum (CUSUM) or Exponentially Weighted Moving Average (EWMA) charts, might be more sensitive.³,²

Another limitation is the assumption of normally distributed data for some chart types (e.g., X-bar and R charts) for accurate limit setting, though attributes charts (P, C, U charts) do not have this requirement. Improper selection of chart type or incorrect calculation of control limits can lead to misinterpretations, causing unnecessary process adjustments (tampering) or failing to detect actual problems. Over-adjusting a process that is already in statistical control, based on common cause variation, can actually increase process instability. Furthermore, the effectiveness of Shewhart charts relies heavily on consistent data collection and a clear understanding of the process being monitored.

Shewhart Charts vs. Histograms

Shewhart charts and histograms are both tools for statistical analysis and data visualization, but they serve different primary purposes.

While a histogram can show the spread of data from a process, it does not reveal whether the process is stable over time or if any unusual events occurred during the data collection period. Shewhart charts, by plotting data sequentially and using control limits, specifically address process behavior and stability, guiding interventions when a process deviates from its expected statistical behavior.¹

FAQs

What is the main purpose of a Shewhart chart?

The main purpose of a Shewhart chart is to distinguish between common cause variation (inherent, random variation in a stable process) and special cause variation (unusual, assignable causes that make a process unstable). This distinction helps in deciding whether to adjust the process or investigate a specific problem.

Can Shewhart charts be used for financial data?

Yes, Shewhart charts can be applied to financial data, though their application requires careful consideration of the nature of financial markets and underlying assumptions. They can be used to monitor metrics like trading volume stability, expense rates, or daily transaction counts within an organization, contributing to project management or operational oversight.

How often should data be collected for a Shewhart chart?

The frequency of data collection for a Shewhart chart depends on the process being monitored and the type of variation one wishes to detect. For high-volume processes, data might be collected hourly or daily. For slower processes, weekly or monthly collection might suffice. The key is to collect enough data points to represent the process adequately and allow for meaningful statistical analysis.

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

If a data point falls outside the control limits on a Shewhart chart, it indicates the presence of a "special cause" of variation. This means something unusual has happened in the process that is not part of its normal, common cause variability. It signals that an investigation is needed to identify the root cause of this deviation and take corrective action.

Are Shewhart charts still relevant today?

Yes, Shewhart charts remain highly relevant today. They are foundational tools in disciplines like Statistical Process Control (SPC), Lean, and Six Sigma, continuing to be used globally for process monitoring and improvement in industries ranging from automotive to healthcare and telecommunications. Their simplicity and effectiveness in identifying process instability ensure their continued importance.