# Calculating an XBAR-S control chart

## 2007/03/02

[StATS]: Calculating an XBAR-S control chart (March 2, 2007)

The following data represents a weekly evaluation of vaccine potency. The data is taken from

but I have taken some liberties with the data to simplify the calculations.

Week01 0.716 0.771 0.924 Week02 0.978 1.212 1.176 Week03 0.644 0.903 0.869 Week04 0.869 0.716 0.869 Week05 1.398 1.301 0.934 Week06 1.218 0.924 1.398 Week07 0.876 0.591 0.644 Week08 1.215 1.241 1.021 Week09 1.021 0.954 0.491 Week10 0.690 0.477 0.785 Week11 1.301 1.279 1.220 Week12 1.644 1.176 1.114 Week13 1.146 1.256 1.518

Each week, three lots of vaccine are tested for potency. Calculate a control chart for this data.

While many experts in quality control would use an XBAR-R chart for this data, the XBAR-S chart also works well. There are three steps in calculating an XBAR-S chart.

1. Compute a mean and standard deviation for each group,
2. Plot the means/standard deviations in sequence (i.e., a run chart),
3. Draw reference lines at the overall mean and at the three sigma limits.

The mean and standard deviation for each week are shown below

 Mean Stdev Week01 0.804 0.108 Week02 1.122 0.126 Week03 0.805 0.141 Week04 0.818 0.088 Week05 1.211 0.245 Week06 1.180 0.239 Week07 0.704 0.152 Week08 1.159 0.120 Week09 0.822 0.289 Week10 0.651 0.158 Week11 1.267 0.042 Week12 1.311 0.290 Week13 1.307 0.191

Here is a plot of the means

and of the standard deviations.

I have included a single reference line at the average of all the data points. For these two charts, the data values fluctuate more or less randomly above and below the reference line. If you noticed eight or more consecutive points on the same side of the center line, you would declare the process to be out of control.

The final step is to compute control limits. These limits are placed at three sigma distance from the overall mean and variation inside these limits is considered normal variation. The formula for control limits for the XBAR chart is

where the constant A3 comes from the following table.

 n A3 B3 B4 2 2.659 0 3.267 3 1.954 0 2.568 4 1.628 0 2.266 5 1.427 0 2.089 6 1.287 0.030 1.970 7 1.182 0.118 1.882 8 1.099 0.185 1.815 9 1.032 0.239 1.761 10 0.975 0.284 1.716 11 0.927 0.321 1.679 12 0.886 0.354 1.646 13 0.850 0.382 1.618 14 0.817 0.406 1.594 15 0.789 0.428 1.572 16 0.763 0.448 1.552 17 0.739 0.466 1.534 18 0.718 0.482 1.518 19 0.698 0.497 1.503 20 0.680 0.510 1.490 21 0.663 0.523 1.477 22 0.647 0.534 1.466 23 0.633 0.545 1.455 24 0.619 0.555 1.455 25 0.606 0.565 1.435

This table can be found in many books and on several websites. I selected this table from

In this particular example, n=3, so A3=1.954. This produces lower and upper control limits of 0.68 and 1.34. If a single data point falls outside the control limits, your process is out of control.

It is optional, but you can also compute warning limits at two sigma units away from the mean. If you notice two out of three consecutive points outside the warning limits, then your process is out of control. Here is a control chart that includes warning limits.

Notice that the tenth week is below the lower control limit and that two consecutive weeks (11 and 12) fall above the upper warning limit. You can also compute control limits for the standard deviations using the formula

where the constants B3 and B4 come from the same table. When n is small (5 or less), the value of B3 is zero which places no effective lower control limit on the chart. What this tells you is that an individual standard deviation can be extremely small without raising any concern.

Here is a plot of the standard deviations with control limits.

There are no points outside the control or warning limits.

Week01 1.097 1.204 Week03 1.030 1.362 Week05 0.682 0.978 Week07 0.820 1.080 Week09 1.042 0.858 Week11 1.398 1.146 Week13 1.301 1.204