## Tuesday, May 20, 2014

### Basics of a control chart

The best way for a product to meet a customer's fitness for use, it should be produced in a consistent and reliable manner. It must be capable of operating with very little change or variability around the optimal dimensions of the quality characteristics of the product. There are may problem solving tools helpful in achieving stability and reducing variability. One such tool is the control chart. What are the basic principles of the control chart?

A control chart is a graphical display of a quality characteristic that has been computed or measured from a sample versus a sample number or time. Think of the chart as an x-axis and y-axis, focusing only on the first quadrant values (zero and positive values only). The x-axis would have time or sample number, while the y-axis is the sample quality characteristic. Examples could be "average ring diameter", "copper concentration", or "sample fraction nonconforming". The chart contains a horizontal center line which represents the average value of the quality characteristic measure when the system is in the "in-control" state.

Two other horizontal lines, called the "upper control limit" (UCL) and "lower control limit" (LCL), are shown.
Sample values are plotted according to the measure of the quality characteristic. A process is said to be in control when all the points plot inside the control limits, in which case no action is required. If a point plots outside the control limits, the process is out of control and quality control experts will investigate to determine the cause of the out of control process and what course of action to take to get the process back in control.
Even if all the points plot inside the control limit, the process could be out of control if the points are not in a nonrandom manner. For example, if 17 of 20 sample points are below the center line but above the LCL and only 3 are above the center line and below the UCL, we would suspect that something is wrong. The points plotted should represent a random pattern.

Let's examine the statistical basis of the control chart. Suppose we have an x-bar control chart for piston-ring diameter. Note that x-bar is the sample average. Suppose the average ring diameter is 70 mm and the standard deviation of the process is 0.015 mm. If sample of size n = 10 were are taken, the sample standard deviation of x-bar is the standard deviation of the process divided by the square root of n. Therefore, the standard deviation of x-bar is 0.0047. If we use the 3-standard deviation control limits, the UCL would be 70 + 3(.0047) = 70.0141 and the LCL would be 70 - 3(.0047) = 69.9859. Now we can plot the sample values and see if they fall within the control limits and if they display a random pattern.

The most important use of a control chart is to improve the process. In real world applications, most processes do not operate in statistical control. Therefore, we will look for assignable causes for this out of control process. If they can be eliminated, the variability will be reduced and the process will be improved, hopefully to the point of being back in control.

This guide gives the very basics of what control charts are and how they are used. In upcoming articles, I will examine how to choose control limits, how to analyze patterns on control charts, and the other six tools of statistical process control.