Variable control charts

Title: Variable Control Charts

Authors: Ryan McKee, Philip Rose, Kathryn Siuniak

Introduction
If you have a brand new process, traditional statistics will not help control the system because you will not have any information about the averages and standard deviations for that system. Therefore, you have to determine what values are acceptable for the variables in your system to take. Since you can choose a sample size when taking measurements of the data in your system, you can use simple averages and ranges to construct control charts to determine whether your process is statistically in control. The following analytical methods will describe how to create upper and lower control limits for $$\overline{x}$$ and R charts that will tell you whether or not your system is in control.

Average $$\overline{x}$$
When analyzing the data about a process, a set of samples will be studied. Therefore, samples $$a, b, c...$$ will be studied and each will contain $$a_1, a_2, a_3...$$ components. Because of this matrix of values, a single average cannot be found. Rather, there will be an average for each sample ($$a_1, a_2, a_3...$$) and of those averages $$\overline{x}_a, \overline{x}_b, \overline{x}_c$$ a grand average can be calculated $$\overline{\overline{x}}$$.



Control Limits
The control limits are used for the boundaries of the $$\overline{x}$$ charts. The upper and lower control limits are the upper and lower bounds of the data. The center line is the grand average, $$\overline{\overline{x}}$$. The data should be scattered around the center line but also lie between the lower and upper control limits. Note that the values for $$A_2$$ depend on the number of measurements in each sample. This value can usually be found from a table in any statistical control reference.

$$UCL = \overline{\overline{x}}+A_2 \overline{R}$$

$$LCL = \overline{\overline{x}}-A_2 \overline{R}$$

$$Centerline = \overline{\overline{x}}$$

Range $$\overline{R}$$
The range, R, gives information on the minimum and maximum of each data set. The average of all of these ranges is the average range, $$\overline{R}$$. This value is used to find the upper and lower control limits of both the $$\overline{x}$$ chart and $$R$$ chart. Again, the values for $$D_3$$ and $$D_4$$ depend upon the number of measurements taken in each sample, and can be found in any statistical control reference.

$$R=x_{max} -x_{min}$$



Control Limits
The following equations are used for computing the control limits on the R control charts where $$D_4$$ and $$D_3$$ are tabulated for varies sample sizes.

$$UCL = D_4 \overline{R}$$

$$LCL = D_3 \overline{R}$$

$$Centerline = \overline{R}$$

Phase I
There are two phases involved when constructing control charts for monitoring a new process. Phase I is to determine the control limits. They can be calculated using the upper and lower control limit equations described in earlier sections.

After the limits have been calculated, the data must be compared to them. No data points should fall outside of the upper and lower control limits.

If points do lie outside of the control limits, then those specific points must be studied. If the point in question lies outside of the control limits for a known reason (such as a data logging malfunction) then that point must be thrown out. If the point in question lies outside of the control limits for an unknown reason, then it is either ignored without justification or it is kept and assumed that the point is out of the specified control limits.

Phase II
Phase II deals with monitoring data points for future production after the control limit has been set. If a data point is located outside of the control limit, then there might be a problem in the system. If your data points continue to go above or below the control limit then you might have an indicative shift of the process mean. Control charts can be used to visualize a trend in the process mean. These charts can help you observe if a problem has occurred in a single case or if there has been a shift in the process.

Example Problem
Chemical engineers at the University of Michigan have developed a new process that will that turns bars of lead into golden nuggets. We want to statistically control the diameter of the golden nuggets as they exit the process reactor. We are going to take thirty samples of golden nuggets (where each sample is composed of five nuggets) during a time in which we think the process is in control. The samples of five nuggets are taken once every hour. The data is summarized below.



When you are making $$\overline{x}$$ and R charts for statistical control, you are better off starting with the R chart. The center line for the R chart is $$\overline{R}$$ = $$\sum$$R/N = 14.9788/30 = 0.4993. Since your sample size is 5, the value for $$D_3$$ is 0 and the value for $$D_4$$ is 2.114. Therefore, the upper control limit for the R chart is $$\overline{R}$$$$D_4$$ = 0.4993(2.114) = 1.0555 and the lower control limit for the R chart is $$\overline{R}$$$$D_3$$ = 0.4993(0) = 0.



According to the R chart, there are no out of control conditions. The R chart therefore indicates that the process variability is in control. Now it is okay to produce the $$\overline{x}$$ chart. The center line for the $$\overline{x}$$ chart is $$\overline{\overline{x}}$$ = $$\sum$$$$\overline{x}$$/N = 44.5096/30 = 1.4837. Since your sample size is five, $$A_2$$ = 0.577. The upper control limit for the $$\overline{x}$$ chart is $$\overline{\overline{x}}$$ + $$A_2$$$$\overline{R}$$ = 1.4837 + 0.577(0.4993) = 1.7718. The lower control limit for the $$\overline{x}$$ chart is $$\overline{\overline{x}}$$ - $$A_2$$$$\overline{R}$$ = 1.4837 – 0.577(0.4993) = 1.1956.



The $$\overline{x}$$ chart shows that the process was out of control when the 14th sample was taken. You would need to analyze the process to assign a cause for this out of control point. This could be anything from a faulty pump, contaminated catalyst, broken agitator, etc. If you can find a cause for this sample, then you would throw out the data point and recalculate your control limits using only the remaining 29 sample points. If you cannot find a cause, then you have two options: pretend you found a cause and throw out the point anyway, or keep the point and assume the control limits are appropriate for the control you want. However, if you keep the point, your control limits are going to be wider than they would be otherwise.