# Calculating an XBAR-S control chart

## 2007-03-02

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

• H. C. Batson, M. Brown, M. Oberstein. An adaptation of quality control chart methods to bacterial vaccine potency testing. J Bacteriol 1951: 61(4); 407-19. Available in pdf format.

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.

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
Week13 1.301 1.204
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