Learning Control Charts

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While attending The 35th South Taiwan Statistics Conference, I first learned about control charts from an experienced professor who had spent many years working in manufacturing environments. Intrigued by the concept, I decided to spend some time organizing my notes and gaining a foundational understanding of what control charts are.

Control charts, introduced by Walter Andrew Shewhart at Bell Laboratories in the 1920s, are one of the fundamental tools of Statistical Process Control (SPC). By plotting process data over time and incorporating a Center Line (CL), an Upper Control Limit (UCL), and a Lower Control Limit (LCL), control charts are used to determine whether a process remains in a state of statistical control.

The underlying principle of a control chart is closely related to hypothesis testing in statistics. Specifically, the null hypothesis ($H_0$) assumes that the process is operating in a state of statistical control. When all sample points fall within the control limits, there is insufficient evidence to reject $H_0$. Conversely, if one or more sample points fall outside the control limits, $H_0$ is rejected, indicating that the process may have been affected by one or more assignable causes. In such cases, further investigation is required to identify and eliminate the source of the abnormal variation.

Shewhart Control Charts

The earliest type of control chart was the Shewhart Control Chart, proposed by Walter A. Shewhart. It is based on the assumption that when a process operates under stable conditions, the statistics computed from repeated samples fluctuate randomly around a fixed central value. By establishing statistically derived control limits, Shewhart control charts provide a practical method for determining whether a process remains in a state of statistical control.

Suppose that a quality characteristic $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$. The standard error of the sample mean is then given by

$$ SE(\bar{X}) = \frac{\sigma}{\sqrt{n}}, $$ where $n$ denotes the sample size.

Taking the process mean $\mu$ as the Center Line (CL), the control limits are defined as

• UCL : $\mu + k \cdot \frac{\sigma}{\sqrt{n}}$
• LCL : $\mu - k \cdot \frac{\sigma}{\sqrt{n}}$

where $k$ is the control limit coefficient, representing the distance between the control limits and the center line. In practice, $k$ is most commonly set to $3$.

Although any positive value can theoretically be chosen for $k$, selecting a value that is too small causes normal process variation to exceed the control limits more frequently, resulting in excessive false alarms. Conversely, if $k$ is too large, genuine process abnormalities may go undetected. Therefore, Shewhart proposed using three standard errors as the control limits, providing a practical balance between the false alarm rate and the ability to detect process shifts.

According to the properties of the normal distribution, approximately $99.73\%$ of observations lie within three standard deviations of the mean. Consequently,

$$ P\left( \mu-3\frac{\sigma}{\sqrt{n}} \le \bar X \le \mu+3\frac{\sigma}{\sqrt{n}} \right) \approx0.9973. $$

When the process remains in a state of statistical control, the probability that a sample point exceeds the control limits due solely to random variation is approximately $1 - 0.9973 = 0.0027$, or about $0.27\%$.

Conclusion

Control charts are statistical tools for monitoring processes, allowing practitioners to determine whether a process remains in a state of statistical control. By continuously monitoring process variation, they enable abnormal changes to be detected at an early stage, helping identify assignable causes and reducing the impact of persistent process abnormalities.

Over nearly a century of development, control charts have evolved from the original Shewhart control chart into a wide variety of techniques designed for different data types and application scenarios. More advanced methods have also been developed to detect subtle process changes, making Statistical Process Control (SPC) more comprehensive and widely applicable.

References

• 【規格上下限】從 LSL、USL 到LCL、UCL：簡單又全面掌握規格上下限的知識. (2024, December 10). 品質塗鴉 Quality Graffiti. Retrieved July 6, 2026, from https://www.qualitygraffiti.com/2024/12/LSL-USL-LCL-UCL.html
• 管制圖(Control Chart). (n.d.). 清華大學品質研究中心 Quality Research Center. Retrieved July 6, 2026, from https://mx.nthu.edu.tw/~ctsu/QRC/studyQ/control_chart/index.htm
• 管制圖. (2023, September 18). Wikipedia, The Free Encyclopedia. Retrieved July 6, 2026, from https://zh.wikipedia.org/zh-tw/管制圖
• 68–95–99.7法則. (2026, May 29). Wikipedia, The Free Encyclopedia. Retrieved July 6, 2026, from https://zh.wikipedia.org/zh-tw/68–95–99.7法則