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Control charts are essential tools in statistical process control (SPC) to monitor process stability. When we talk about "three-sigma control limits," we're referencing how these limits are established by using three standard deviations from the process mean. This is significant because it provides a wide range encompassing 99.73% of the data if the process is statistically stable.

Key points about three-sigma control limits:

• The **Upper Control Limit (UCL)** is calculated by adding three times the process standard deviation (σ) divided by the square root of subgroup size (n) to the process mean (µ).
• The **Lower Control Limit (LCL)**, by contrast, is determined by subtracting the same value from the mean.
• Three sigma limits help identify when a process may be out of control due to irregular variations that deviate significantly from the mean.

In the given exercise, the UCL is 48.75 and the LCL is 42.71. These values are crucial thresholds that define the boundaries of acceptable process variation based on statistical theory.

The subgroup size, noted as "n" in control chart formulas, is a critical component. It determines how many observations are grouped together to calculate the control limits.

Why is subgroup size important?

• The larger the subgroup size, the smaller the variability between the subgroups, leading to narrower control limits. This can help in detecting smaller shifts in the process mean.
• Smaller subgroups might have more variability, resulting in wider control limits, which could miss subtle variations that might indicate a process drifting from its intended target.

In solving the provided exercise, we calculated the subgroup size as 5. This means five consecutive data points were used to compute the \( \bar{X} \) values and establish control limits.
This setup helps balance sensitivity to changes and practicality in data collection.

The process standard deviation, denoted by σ, reflects the amount of variation within a process. Understanding this concept is pivotal when setting control charts, as it influences the positioning of the control limits. A higher process standard deviation indicates greater variability, potentially requiring broader control limits for effective monitoring.

Considerations for process standard deviation:

• Standard deviation is used in computing control limits and essentially defines how variable each subgroup's mean might be from the process mean as shown by \( \frac{\sigma}{\sqrt{n}} \).
• A precise estimation of σ is critical, as inaccuracies can lead to either too many false alarms (if σ is underestimated) or missed signals of process issues (if overestimated).

In the example problem, σ was given as 2.25. This value served as an anchor for calculating the control limits and explained how subgroup means could fluctuate in a controlled process.
An accurate standard deviation ensures we set realistic and meaningful thresholds for process evaluation.