**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

An adaptation of quality control chart methods to bacterial vaccine potency testing.H. C. Batson, M. Brown, M. Oberstein. J Bacteriol 1951: 61(4); 407-19. [Medline] [PDF]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 1.518Each 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.

- Compute a mean and standard deviation for each group,
- Plot the means/standard deviations in sequence (i.e., a run chart),
- 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 0.191Here 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 1.435This 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.

On your own.Compute a control chart for the following data set (see below). This data set represents vaccine potencies, like the above data set, but it uses fixed standards to assess potency. These numbers appear in the same paper, but again some simplifying assumptions have been made. Don't peek until you've done the work, but the answers are available on a separate web page.

`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

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Control charts.