X-bar and S Chart: Essential Tools for Quality Control in Lean Six Sigma

In the realm of Lean Six Sigma, maintaining quality control is paramount for the success of any organization. A critical aspect of this is the monitoring and controlling of processes to ensure they remain within predefined limits. Among the most effective tools for this purpose are the X-bar and S charts, both of which play a vital role in the statistical monitoring of process variations. This article delves into the theory behind these charts and guides on their construction, offering insights into how they can be leveraged to uphold quality control in line with Lean Six Sigma principles.

Understanding X-bar and S Charts

The X-bar and S charts are types of control charts used in statistical process control (SPC) to monitor the process mean (average) and process variability (standard deviation) over time, respectively. These charts are particularly useful in environments where sample sizes may vary, which is a common scenario in many manufacturing and service processes.

• X-bar Chart: The X-bar chart is utilized to monitor the process mean. It is constructed using the average of sample means (X-bar) from several samples taken from a process at different times. This chart helps in identifying shifts or trends in the process average over time, signaling when the process may be going out of control.

• S Chart: The S chart, on the other hand, monitors the variability of the process. It is built using the standard deviation (S) of the same samples used in the X-bar chart. The S chart is critical for detecting changes in process dispersion or variability, which can be indicative of issues such as machine wear or changes in raw materials.
  

Constructing X-bar and S Charts

To effectively use X-bar and S charts for quality control, it is important to understand the steps involved in their construction:

1. Data Collection: Collect samples from the process at regular intervals. The size of each sample can vary, but typically, 3 to 5 samples are collected at each interval.

2. Calculate Averages and Standard Deviations: For each set of samples, calculate the average (X-bar) and standard deviation (S). These calculations form the data points for the X-bar and S charts, respectively.

3. Determine Control Limits: Control limits are calculated using the process mean and standard deviation. For the X-bar chart, the Upper Control Limit (UCL) and Lower Control Limit (LCL) are typically set at ±3 standard deviations from the process mean. For the S chart, the control limits are also calculated based on standard deviations, adjusted for sample size.

4. Plot the Charts: Plot the calculated averages (X-bar) and standard deviations (S) against the control limits on two separate charts. Time or sequence of sampling is usually on the horizontal axis, with the statistical measure (mean or standard deviation) on the vertical axis.

5. Analyze the Charts: Regularly review the charts to identify any patterns or trends. Points outside the control limits, or certain patterns within the limits, may indicate that the process is out of control and requires investigation.

Theoretical Foundation

The theoretical foundation of X-bar and S charts is grounded in the principles of statistical quality control. By applying the Central Limit Theorem, these charts assume that regardless of the distribution of the process data, the distribution of the sample means tends to be normal. This underpinning allows for the effective use of control limits to detect shifts in the process mean or variability.

Conclusion

X-bar and S charts are indispensable tools in the Lean Six Sigma toolkit for quality control. Their ability to monitor both the central tendency and dispersion of a process makes them highly effective in identifying variations that could lead to quality issues. By understanding the theory behind these charts and following the steps to construct and analyze them, organizations can significantly enhance their quality control efforts, leading to improved process performance and product quality.

Real-life based example

To create an X-bar and S chart for a Lean Six Sigma project, let's use a practical example from a manufacturing context, specifically in the production of precision components. Our goal is to ensure the diameter of produced components is within specified control limits to maintain quality.

Step 1: Collect Data

Assume we're examining the diameter (in mm) of precision components. We collect measurements from 12 subgroups (batches of components), each subgroup containing 7 measurements.

Here's an example dataset:

Step 2: Calculate Subgroup Means and Standard Deviations

For each subgroup, we calculate the mean (X-bar) and standard deviation (S).

*X-bar for Subgroup 1:

(50.099 + 49.972 + 50.130 + 50.305 + 49.953 + 49.953 + 50.316) / 7 = 50.104

**S for Subgroup 1:

I do not detail more the maths behind the calculation of the subgroup1 here ; indeed it would take to long and it is not the topic of this article.

Step 3: Determine Overall Mean (X-double-bar) and Average Standard Deviation (S-bar)

X-double-bar is the average of all subgroup means:

(50.104 + 49.95 + 49.89 + 49.92 + 50.003 + 49.936 + 49.997 + 50.041 + 50.004 + 49.985 + 50.016 + 50.032) / 12 = 49.979 (mm)

S-bar is the average of all subgroup standard deviations:

(0.146 + 0.166 + 0.204 + 0.125 + 0.186 + 0.182+ 0.161 + 0.152 + 0.116 + 0.133 + 0.115 + 0.172) / 12 = (0.176 mm)

Step 4: Calculate Control Limits

• X-bar Chart Control Limits:

  • Upper Control Limit (UCL) = X-double-bar + A3 * S-bar

  • Lower Control Limit (LCL) = X-double-bar - A3 * S-bar

• S Chart Control Limits:

  1. Upper Control Limit (UCL) = B4 * S-bar

  2. Lower Control Limit (LCL) = B3 * S-bar
     

First let's find A3, B4 and B3. With below table: (n = subgroup size ; n=7 in this case)

So we find: A3=1.182

B4=1.882

B3=0.118 And we know: X-double-bar = 49.979

S-bar = 0.176

Now let's compute:

• X-bar Chart Control Limits:

  • Upper Control Limit (UCL) = X-double-bar + A3 * S-bar

  • Upper Control Limit (UCL) = 49.979 + 1.182 * 0.176 = 50.187

  • Lower Control Limit (LCL) = X-double-bar - A3 * S-bar

  • Lower Control Limit (LCL) = 49.979 + 1.182 * 0.176 = 49.770
    

• S Chart Control Limits:

  • Upper Control Limit (UCL) = B4 * S-bar

  • Upper Control Limit (UCL) = 1.882 * 0.176 = 0.331

  • Lower Control Limit (LCL) = B3 * S-bar

  • Lower Control Limit (LCL) = 0.118 * 0.176 = 0.0207

Step 5: Create the X-bar and S Charts

We'll plot the means and standard deviations of each subgroup on their respective charts, including the control limits.

Step 6: Brief Interpretation of the Updated X-bar and S Charts

X-bar Chart Insights

• The X-bar chart with control limits at UCL = 50.187 mm and LCL = 49.770 mm is designed to monitor the average diameter of produced components. Assuming all plotted mean values fall within these limits, it suggests that the process mean is stable and under control, indicating consistent component sizes across subgroups without significant deviation.
  

S Chart Insights

• The S chart, with UCL = 0.331 mm and LCL = 0.0207 mm, tracks the variability of component diameters within each subgroup. If all standard deviations are within these bounds, it indicates that the process variability is under control, reflecting consistent production quality and a stable process.
  

Combined Interpretation

• Observing both charts, if points remain within their respective control limits without any unusual patterns, the process is considered stable and capable. Any points or patterns outside the expected ranges would necessitate further investigation to identify and correct underlying issues.