In the realm of Lean Six Sigma, quality improvement and process control are paramount. Among the various tools available for monitoring and controlling processes, the Individual-Moving Range (I-MR) Chart stands out for its simplicity and effectiveness, especially when dealing with continuous data from individual measurements. This article delves into the theory behind Control Charts, with a focus on the construction and application of the I-MR Chart.

Introduction to Control Charts

Control Charts are a cornerstone of statistical process control (SPC), a methodology developed by Walter A. Shewhart in the early 20th century. These charts are used to determine whether a manufacturing or business process is in a state of statistical control. By plotting the performance of a process over time, they help identify trends, shifts, or any out-of-control conditions. Among the various types of control charts, the I-MR Chart is particularly useful for small datasets or when data is collected in single measurements rather than subgroups.

The I-MR Chart: Theory and Purpose

The I-MR Chart consists of two separate but related charts: the Individual (I) Chart and the Moving Range (MR) Chart. The I Chart monitors the process level over time, plotting individual measurements in the order they were collected. The MR Chart, on the other hand, tracks the variability between consecutive measurements, providing insights into the process's stability.

Individual (I) Chart: This chart plots each individual measurement against the order of observation. It helps in detecting shifts or trends in the process average. The central line on the I Chart represents the average (mean) of the collected data, while the upper and lower control limits (UCL and LCL, respectively) are calculated based on the standard deviation of the data, reflecting the expected range of variation in a stable process.

Moving Range (MR) Chart: The MR Chart plots the absolute difference between consecutive measurements, highlighting the process's variability or dispersion. Like the I Chart, it has a central line representing the average moving range, with control limits that help identify when the process variability is out of control.

Construction of the I-MR Chart

To construct an I-MR Chart, follow these steps:

1. Collect Data: Gather continuous data from the process, ensuring measurements are taken in the order of production or occurrence.

2. Calculate the Moving Range: For each pair of consecutive measurements, calculate the moving range (MR) by finding the absolute difference between them.

3. Compute Averages: Calculate the average of the individual measurements for the I Chart (X-bar) and the average moving range for the MR Chart (MR-bar).

4. Determine Control Limits: For the I Chart, calculate the UCL and LCL based on the average ± 3 standard deviations (estimated from the MR data). For the MR Chart, use the average moving range to calculate the control limits, also ± 3 standard deviations, where standard deviation is approximated using a d2 factor (a statistical constant).

5. Plot the Charts: On the I Chart, plot each individual measurement and draw the central line (average) and control limits. On the MR Chart, plot the moving ranges, the central line (average moving range), and its control limits.

6. Interpret the Charts: Analyze the charts for signs of control or out-of-control conditions, looking for patterns such as points outside the control limits, runs, or trends that indicate non-random variation.
   

Application and Benefits

The I-MR Chart is particularly useful in settings where it's impractical to collect data in subgroups or when each measurement is of significant interest. Its application spans across industries, from manufacturing to service processes, enabling practitioners to monitor both the shift in process level and its variability. This dual perspective helps in identifying a wider range of issues affecting process performance and stability, facilitating timely interventions and continuous improvement efforts.

In summary, the Individual-Moving Range Chart is a powerful tool in Lean Six Sigma projects for monitoring process behavior. By understanding its construction and interpretation, practitioners can effectively manage process variability, drive quality improvement, and ensure process stability, aligning with the core objectives of Lean Six Sigma methodologies.

Example

For our example, let's consider a Lean Six Sigma project aimed at improving the quality of coffee produced by a coffee shop. The project team decides to focus on the consistency of coffee temperature as a critical quality attribute. Customers have complained that sometimes the coffee is too hot, and other times it's not hot enough. The team aims to ensure that every cup of coffee served is within the ideal temperature range, enhancing customer satisfaction.

Step 1: Data Collection

The team decides to measure the temperature of the first coffee made every hour over a 10-hour period. Here's the hypothetical data collected (in °C):

1. 85

2. 90

3. 88

4. 84

5. 87

6. 89

7. 90

8. 86

9. 88

10. 85

    
    

Step 2: Creating the I-MR Chart

An I-MR Chart consists of two charts:

• I Chart (Individuals Chart): Shows the individual measurements over time, and it's useful for detecting shifts in the process mean.
  

• MR Chart (Moving Range Chart): Shows the moving range of the process, which is the difference between consecutive measurements. It's useful for detecting changes in the process variability. (see below table)

Average moving range is ( 5 + 2 + 4 + 3 + 2 + 1 + 4 + 2 + 3 ) / 9 = 2.889

Calculations

For the I Chart:

• Mean (μ): The average of all individual measurements, in that case 87.2°C

• Upper Control Limit (UCL): μ+3σ

• Lower Control Limit (LCL): μ-3σ

  
  

  Since we're dealing with individual measurements, we use the moving range to estimate the process standard deviation (σ). The standard deviation estimation (σˉ) is the average moving range (Rˉ) divided by the d2 factor (which is 1.128 for individuals control charts). σˉ=Rˉ/d2 = 2.889 / 1.128 = 2.56

• Upper Control Limit (UCL): μ+3σ = 87.2 + 3 * 2.56 = 94.88

• Lower Control Limit (LCL): μ-3σ = 87.2 - 3 * 2.56 = 79.52

With the data and calculated values let's create the I chart.

The I Chart shows each coffee's temperature across the observed period. The green dashed line represents the average temperature, while the red dashed lines represent the control limits. Any point outside these limits would indicate a potential out-of-control process, suggesting an unusual variation not attributable to random process variability.

For the MR Chart:

• Rˉ = Average moving range = ( 5 + 2 + 4 + 3 + 2 + 1 + 4 + 2 + 3 ) / 9 = 2.889

• Mean of Moving Ranges (MR): The average of the absolute differences between consecutive measurements. For the UCL:

• For the Lower Control Limit (LCL) Calculation

  The LCL for the MR chart is typically calculated with a similar approach but subtracting rather than adding the 3σMR​ term:

However, because moving ranges cannot be negative (it doesn't make sense to have a negative distance between two points), the LCL is often set to 0 if the calculation yields a negative number.

With the data and calculated values let's create the MR chart.

The MR Chart displays the variability in temperature differences between consecutive coffees. The control limits here help identify changes in process dispersion. Since all points fall within the control limits, it suggests that the process variability is stable.

Conclusion

From the charts, we can conclude that the coffee temperature process is in control, with no individual measurements or moving ranges exceeding the calculated control limits. However, if the goal is to narrow down the temperature variability or shift the process mean closer to a specific target (e.g., 85°C to 90°C), the team may need to investigate and potentially adjust the brewing process, taking into account factors like machine calibration, room temperature, or the time coffee is left out before serving.