Control limits for an R chart, which is used in Statistical Quality Control (SQC) to monitor the variability or range within each sample taken from a process, are typically calculated using statistical formulas based on the process data. The control limits for the R chart are essential for identifying patterns, trends, or deviations in the variability of the process and distinguishing between common cause and special cause variation. Here’s how to calculate the control limits for an R chart:

1. Calculate the Average Range (¯R):

• Compute the range (R) for each sample by subtracting the smallest value from the largest value within the sample.
• Calculate the average range (¯R) by taking the mean of all the individual sample ranges.

2. Determine Control Limits:

• Control limits for the R chart are typically calculated using the average range (¯R) and a constant factor (typically denoted as D3).
• The upper control limit (UCL) and lower control limit (LCL) for the R chart can be calculated as follows:
  • UCL = D3 * ¯R
  • LCL = 0 (since range values cannot be negative)

3. Determine the Constant Factor (D3):

• The constant factor (D3) depends on the sample size (n) used in the R chart.
• D3 values are typically found in statistical tables or calculated using statistical software.
• For example, for sample sizes between 2 and 25, D3 values are commonly found in statistical tables.

4. Interpretation:

• Data points falling within the control limits (UCL and LCL) indicate that the process variation is consistent and under control due to common cause variation.
• Data points falling outside the control limits or displaying non-random patterns may indicate special cause variation, requiring investigation and corrective action.

Example:
Suppose we have collected sample ranges (R) from a process over time and calculated the average range (¯R) to be 4.5 units. Using a sample size of 5, we find the constant factor (D3) from a statistical table to be 2.114. We can then calculate the control limits for the R chart as follows:

• UCL = D3 * ¯R = 2.114 * 4.5 = 9.531
• LCL = 0 (since range values cannot be negative)

Therefore, the control limits for the R chart are UCL = 9.531 and LCL = 0.

These control limits help practitioners monitor the variability of the process and identify any deviations or special causes of variation that may affect product quality or process performance.