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How Do You Set Control Limits?

Published in Statistical Process Control 4 mins read

Setting control limits involves a statistical process to define the acceptable variation in a process, allowing you to distinguish between common cause variation and special cause variation. Here's a breakdown of how to set them:

  1. Collect Data: Gather data from your process over a period of time when the process is stable and in control. This will be your baseline data. You'll want a sufficient number of data points (typically 20-30) to accurately represent the process.

  2. Calculate the Average (Mean): Calculate the average of your collected data. This is often denoted as "x̄" (x-bar).

    • Sum all the individual data points.
    • Divide the sum by the total number of data points.

  3. Calculate the Range or Standard Deviation: Choose either the range method or the standard deviation method, depending on your data and the level of accuracy required. The standard deviation method is generally preferred for its statistical rigor.

    • Range Method (Simpler but less accurate):
      • Determine the range for each subgroup (e.g., each day's data). The range is the difference between the largest and smallest value in the subgroup.
      • Calculate the average range (R̄).
    • Standard Deviation Method (More accurate):
      • Calculate the variance: For each data point, subtract the mean (x̄), square the result, and sum all these squared differences. Then divide this sum by (n-1), where n is the number of data points. This gives you the sample variance (s2).
      • Calculate the standard deviation (s): Find the square root of the variance (s2).

  4. Determine the Control Limit Factors: Use control chart constants that are based on subgroup size (n). These constants are used to calculate the control limits based on either the average range or the standard deviation. These factors can be found in statistical tables or software packages. Common factors are A2, D3, D4 when using range, and A3, B3, B4 when using standard deviation (many different notations exist, so ensure you know what each constant does).

  5. Calculate the Control Limits: Calculate the Upper Control Limit (UCL) and Lower Control Limit (LCL).

    • Using the Range Method (Example for X-bar chart):
      • UCL = x̄ + (A2 * R̄)
      • LCL = x̄ - (A2 * R̄)
    • Using the Standard Deviation Method (Example for X-bar chart):
      • UCL = x̄ + (A3 * s) (Note: A3 may be named something different, but represents a constant multiplied by standard deviation and added to the mean.)
      • LCL = x̄ - (A3 * s) (Note: A3 may be named something different, but represents a constant multiplied by standard deviation and subtracted from the mean.)

  6. Common Standard Deviation Multiples: Often, control limits are set at +/- 3 standard deviations from the mean. This represents a 99.73% confidence interval, meaning that 99.73% of the data points are expected to fall within these limits if the process is in control. The control chart constants are derived based on this common 3-sigma rule.

  7. Create the Control Chart: Plot the data on a control chart with the mean (center line), UCL, and LCL.

  8. Monitor the Process: Continuously monitor the process and plot new data points on the control chart. If a data point falls outside the control limits, or if there is a pattern of data points approaching the limits, this indicates that the process may be out of control and requires investigation.

Example:

Let's say you're monitoring the weight of a product. After collecting 25 samples, you find:

  • Average weight (x̄) = 10 grams
  • Standard deviation (s) = 0.5 grams
  • Assuming an A3 of 1.88 for sample sizes of 1 (individual measurements)

Then:

  • UCL = 10 + (1.88 * 0.5) = 10.94 grams
  • LCL = 10 - (1.88 * 0.5) = 9.06 grams

You would then plot your data on a control chart with a center line at 10 grams, an upper control limit at 10.94 grams, and a lower control limit at 9.06 grams.

In summary, setting control limits involves collecting data, calculating the mean and either the range or standard deviation, and then using these values along with appropriate control chart constants to establish the upper and lower control limits. These limits are then used to monitor the process and identify potential problems.

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