# X-Bar R and X-Bar S control charts

X-Bar charts are designed for situations where you’re creating charts from variable (measured) data. When creating an X-Bar chart, each point on the chart doesn’t represent an individual value, rather each point is determined by taking multiple measurements and averaging them together. The group of points you average together is called a “subgroup”, and the number of points in the subgroup is the subgroup size. By subgrouping, X-Bar charts take advantage of the the Central Limit Theorem to insure that the data being plotted is normally distributed, which gives variable control charts much of their power to identify out-of-control conditions.

SuperEasyStats can easily create two kinds of X-Bar charts:

• X-Bar R – When the subgroup size is constant, and consists of between 2 and 9 values, use the X-Bar R option. SuperEasyStats will generate to charts – one is the chart of the average values (X-Bar) of the subgroups, along with calculated control limits, and the other is the range (hence the ‘R’) of each subgroup along with calculated control limits. The subgroup range is simply the difference between the largest and smallest values in the subgroup.
• X-Bar S – If the subgroup size varies, or it consists of between 10 and 25 values, use the X-Bar S option. This option also creates two charts. The first is the chart of the average values (X-Bar) along with calculated control limits, and the second chart shows the standard deviation (hence the ‘S’) of each subgroup along with calculated control limits.

Let’s look at the following example. A small parts supplier manufactures bolts for its customers. The inner diameter of the bolts must be 0.5 centimeters. The manufacturer wants to use an X-Bar R chart to insure that their manufacturing process is in control. The supplier runs a single 8 hour shift each day. They decide that two times per hour they will pull a sample of 6 bolts from the line to create their chart.

They start by creating an X-Bar data entry template in SuperEasyStats. To do that they click “Control Charts” on the SuperEasyStats menu, and select “X-Bar R or X-Bar S Chart” from the list of options. Once the template is created, they change the “Subgroup size” option to 6, and begin entering their data, as shown at left.

You can see that each row is labeled with the  hour of the shift, and an abbreviation indicating the group of bolts for that hour (e.g., “8am-9am g1” is the first group of bolts measured that hour). The numerical values in each row are the measured inner diameters of the bolts in that sample.

After a full 8-hour shift, 16 data rows have been filled in. The next step is to generate the control chart. They do this by selecting “Control Charts” from the SuperEasyStats menu.

When the chart sheet is created, notice that at the top is an option to select either X-Bar R or X-Bar S. By default X-Bar R is selected, and those are the charts that are displayed. If you want to switch to X-Bar S charts, simply change the selection and the charts will automatically update.

As described above, SuperEasyStats will create two charts. The first will plot the averages of each row of data, and the second chart will plot the range (largest value minus the smallest value) of each row.

The first thing to notice is that the R-chart (on the bottom) appears to be in control, but there are multiple out-of-control conditions in the X-Bar chart. The list next to each chart shows which out-of-control conditions have been detected and allows you to show or hide those conditions. For example, if you hide the “Above mean” and “Below mean” conditions, you can clearly see which points are part of the downward trend. Seven or more consecutive points either increasing or decreasing is unlikely to happen by chance, and probably indicates that something unusual happened during the manufacturing process. Exactly what caused the trend is unknown at this point, but should be investigated.