R chart

What Is an R chart?

An R chart, or Range chart, is a type of control chart used in statistical process control (SPC) to monitor the variability or dispersion of a process over time. It falls under the broader financial category of Quality Control. The R chart specifically tracks the range of data within small subgroups, providing a visual representation of process variation. By observing the pattern of subgroup ranges, practitioners can determine if the process variability is consistent and predictable, indicating a state of process stability.

History and Origin

The concept of the R chart, along with other fundamental control charts, was pioneered by Walter A. Shewhart while working at Bell Laboratories in the 1920s. Shewhart recognized the need for a statistical method to distinguish between common cause variation, which is inherent to a process, and special cause variation, which arises from identifiable factors. On May 16, 1924, Shewhart introduced the control chart as a critical tool for this distinction, laying the groundwork for modern statistical process control. His work revolutionized quality management, particularly in manufacturing, by providing a scientific basis for process improvement.⁶

Key Takeaways

• An R chart monitors the consistency of process variability over time.
• It tracks the range (difference between maximum and minimum values) within small subgroups of data.
• The chart helps distinguish between common cause and special cause variation in a process.
• An R chart is typically used in conjunction with an X-bar chart to assess both process average and variability.
• Points outside the control limits on an R chart signal that the process variability may be out of control.

Formula and Calculation

The R chart plots the range of each subgroup against time. The key components of an R chart are the center line (CL) and the upper and lower control limits (UCL and LCL).

The formulas are:

Center Line (CL): The center line for an R chart is the average range ($\bar{R}$) of all subgroups.

$CL_R = \bar{R}$

Where:

• (\bar{R}) = The sum of all subgroup ranges divided by the number of subgroups.

Upper Control Limit (UCL):
$UCL_R = D_4 \bar{R}$

Lower Control Limit (LCL):
$LCL_R = D_3 \bar{R}$

Where:

• (D_3) and (D_4) are control chart constants that depend on the subgroup size (n). These constants are statistically derived to ensure that, for a process in statistical control, approximately 99.73% of sample ranges will fall within these limits.⁵

These calculations are based on the principles of statistical distributions and are readily available in statistical quality control handbooks for various subgroup sizes.

Interpreting the R chart

Interpreting the R chart involves analyzing the pattern of plotted points relative to the center line and control limits. If all points fall within the upper and lower control limits and show no discernible patterns (such as trends, cycles, or shifts), it indicates that the process variability is stable and predictable. This means that any observed variation is due to common causes inherent in the process.⁴

Conversely, points falling outside the control limits on the R chart, or specific non-random patterns within the limits, signal the presence of special cause variation. Such signals warrant investigation to identify and address the root cause of the unusual process variation. For instance, a point above the UCL might indicate a sudden increase in variability, possibly due to a faulty machine, a new operator, or inconsistent raw materials. A point below the LCL (which for small subgroup sizes is often zero) could suggest an error in data collection or a temporary reduction in variability that might be unsustainable.

Hypothetical Example

Imagine a company that manufactures precision ball bearings. To ensure consistent quality, they routinely measure the diameter of bearings in subgroups of five, every hour. They want to monitor the consistency of the manufacturing process using an R chart.

Over a day, they collect 20 subgroups of five bearing diameters each. For each subgroup, they calculate the range (maximum diameter - minimum diameter).

Let's assume the ranges for the first 5 subgroups are:

• Subgroup 1: Range = 0.005 mm
• Subgroup 2: Range = 0.004 mm
• Subgroup 3: Range = 0.006 mm
• Subgroup 4: Range = 0.005 mm
• Subgroup 5: Range = 0.007 mm

After collecting data for all 20 subgroups, they calculate the average range ((\bar{R})). Suppose (\bar{R}) for all 20 subgroups is 0.0055 mm.

For a subgroup size (n=5), the control chart constants (D_3) and (D_4) are typically 0 and 2.114, respectively.

Using the formulas:

• Center Line ((CL_R)) = (\bar{R}) = 0.0055 mm
• Lower Control Limit ((LCL_R)) = (D_3 \times \bar{R}) = (0 \times 0.0055 = 0) mm
• Upper Control Limit ((UCL_R)) = (D_4 \times \bar{R}) = (2.114 \times 0.0055 \approx 0.0116) mm

The company then plots each subgroup's range on the R chart. If a range, say for subgroup 15, is calculated as 0.0130 mm, this point would fall above the UCL (0.0116 mm). This out-of-control point on the R chart signals that the variability of the bearing diameters at that time was unusually high, prompting an investigation into the manufacturing process to identify and correct the cause of increased variation. This constant process monitoring allows for proactive quality management.

Practical Applications

R charts are widely used across various industries where variable data is collected in subgroups to monitor process consistency. Their applications span:

• Manufacturing: In automotive, electronics, or food production, R charts track the variability of critical dimensions, weights, or concentrations of products. For example, ensuring the consistent thickness of a metal sheet or the fill volume of a beverage container.³
• Healthcare: Monitoring the consistency of laboratory test results, patient wait times, or medication dosages. While direct R chart use might be less common for single patient data, aggregated subgroup data can reveal inconsistencies in hospital processes.
• Service Industries: Tracking variability in call handling times, customer service response times, or document processing durations to maintain consistent service delivery.
• Environmental Monitoring: Analyzing the consistency of pollutant levels in samples collected over time or across different locations.
• Financial Operations: While less common for direct financial instrument analysis, R charts could be applied to internal processes, such as monitoring the consistency of data entry times or error rates in processing financial transactions. For broader applications of control charts in general, the American Society for Quality (ASQ) provides extensive resources.²

Limitations and Criticisms

While the R chart is a powerful tool for monitoring process variability, it has certain limitations:

• Subgroup Size: The R chart is most effective for small subgroup sizes, typically (n \le 10). For larger subgroups, the range becomes a less efficient estimator of process variability compared to the standard deviation. In such cases, an S chart (standard deviation chart) is generally preferred.
• Normality Assumption: While the R chart is robust to minor deviations from normality, its control limits are derived assuming that the underlying process data is approximately normally distributed. Significant non-normality can lead to misinterpretation of signals.
• Interpretation Requires Expertise: Simply plotting data on an R chart is insufficient. Effective interpretation requires a deep understanding of the process being monitored. Without intimate knowledge of the system, process, or community that the data represents, drawing causal conclusions can be challenging.¹
• Focus on Variability Only: The R chart exclusively monitors process variability. It does not provide information about the process average. Therefore, it must almost always be used in conjunction with an X-bar chart (or another chart for the mean) to provide a complete picture of process stability.

R chart vs. X-bar chart

The R chart and the X-bar chart are often used together as a pair of control charts in statistical process control. While both are designed to monitor processes, they track different aspects of process behavior.

The critical distinction is that the R chart assesses the "consistency" of the process, while the X-bar chart assesses its "level." Before interpreting the X-bar chart, it is essential to ensure that the R chart indicates a stable process variability. If the variability is not in control, then the control limits calculated for the X-bar chart may not be accurate, making any conclusions about the process average unreliable.

FAQs

What does an out-of-control point on an R chart indicate?

An out-of-control point on an R chart indicates that the process variation has changed significantly and is no longer stable. This suggests the presence of a "special cause" of variation that needs to be investigated and addressed.

Can an R chart be used by itself?

While an R chart provides valuable insight into process variability, it is generally not recommended to use it by itself. It should almost always be paired with an X-bar chart (or another chart for the mean) to get a complete understanding of process stability, covering both the consistency of the process and its average level.

What is the ideal subgroup size for an R chart?

The R chart is most effective for small subgroups, typically ranging from 2 to 10 observations. For larger subgroup sizes, the sample standard deviation provides a more efficient measure of variation, and an S chart (Standard Deviation chart) is a more appropriate control chart to use.

How often should data be collected for an R chart?

The frequency of data collection for an R chart depends on the nature of the process being monitored. Data should be collected frequently enough to detect important changes in process variability, but not so frequently that it introduces unnecessary noise or burden. Practical considerations often dictate collecting data at regular intervals, such as hourly, daily, or per batch.