X-bar/R Charts

Overview of process:

Step 1: Collect k subgroups (usually at least 20 subgroups) of size n

Step 2: Calculate R and x-bar for each subgroup

Step 3: Plot the statistics calculated in Step 2 on the corresponding charts. NOTE: The X-bar chart goes at the top of the page and the R chart goes at the bottom of the page--tradition is the only explanation for why!

Step 4: Calculate limits for the R chart

Step 5: Conduct runs tests on the R chart. If there are any signals of special causes of variation within subgroups, search for the cause, take corrective action, and return to Step 1. If there are no signals, go to Step 6.

Step 6:Calculate control limits for the X-bar chart.

Step 7: Conduct runs tests on the X-bar chart. If there are any signals of special causes of variation between subgroups, search for the cause, take corrective action, and return to step 1. Otherwise, go to Step 8.

Step 8: Develop theories about how to improve the process, and try these out. Look for patterns on your control charts to determine if the change you made resulted in improvement.

R Chart

The R chart looks at variation within subgroups. We expect to find some variation from one subgroup to another. The control limits on the R chart, along with the results of the runs tests, determine how much variation we should expect to see if the process is stable with respect to variability. If the R chart fails to show stability, you should be asking, "Why is the amount of spread in one group inconsistent with with amount of spread in another group?" or "What is happening while I am collecting each subgroup--because there seems to be something different happening while different subgroups are being collected."

Use the following formulas for calculating limits for the R chart. Control chart constants can be found at the bottom of this page and depend on n, the subgroup size.

X-Bar Chart

The X-bar Chart looks at variation between subgroups. In effect, the X-bar chart is answering the question, "Are there additional sources of variation between subgroups that were not accounted for within the subgroup?" The following formulas are used to calculate control limits. Again, control chart constants can be found at the bottom of this page.

If there are no signals of instability on the X-bar Chart or on the R Chart, then we conclude that the process appears to be stable (i.e., operating in a predictable manner).

If we believe that conditions in effect while these observations were collected will continue, we can predict what this process will produce in the near future. To do this we estimate the process standard deviation, σ, by calculating R-bar divided by d₂. We use the center line on the X-bar chart to estimate the process mean, μ. Predicted output would then be between μ - 3σ and μ + 3σ.

Control Chart Constants

On to Using Excel to Plot X-Bar/R Charts

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