Reading Assignment:

Download and read Chapter 10 in Essentials of Quality With Cases and Experiential Exercises.  Review the Discussion Questions at the end of the chapter to be sure that you understand what you have read.

Normally a separate control chart must be created for each product and variable to be monitored.  This is usually not a problem for repetitive processes where the same product is produced day-in and day-out.  However, in cases where one process is used produce several products, such as a drilling operation, a traditional x-bar chart would have to be created for each short-run product.  Among other drawbacks, the short-run time on each product would make the observance of trend signals in the data more difficult.  A special form of the x-bar chart called a Delta chart may be used in this case.  In our example, if the same drill is used to bore both 0.50" and 0.75" holes (only the drill bit changes) and the variances of both hole distributions are about the same, one chart may be used for both hole sizes by plotting the difference from nominal rather than the actual dimension of the hole.  The chart used to plot the difference from nominal is called a Delta chart.

Consider a sample of two parts taken from the process when it is producing a 0.50" nominal hole diameter.  The actual measured hole diameter for part 1 is 0.52" and for part 2 is 0.49".  For part one, the delta statistic is
0.52 - 0.50 = 0.02".  The delta statistic for part 2 is 0.49 - 0.50 = -0.01".  The Delta chart uses the delta statistic rather than the actual measured dimension.  So, if the next sample is taken when the process is producing a 0.75" nominal diameter hole, the delta statistic can be plotted on the same Delta chart.  Consider that the second sample also consists of two parts with actual measured hole diameters of 0.77" and 0.74".  The delta statistics would be 0.77 - 0.75 = 0.02" and 0.74 - 0.75 = -0.01" respectively.

Consider a process that produces jelly beans.  There are many ways in which a jelly bean can be defective.  It can be the wrong color, mal-formed, or crushed, among others.  If it is more important to track the production of defective jelly beans rather than some variable (e.g. jelly bean length or weight), then an attribute control chart is needed.  An attribute control chart allows us to lump all the ways a jelly bean can be defective together.  That is, we plot the number of defective jelly beans regardless of the reason why they are defective.  Of course, when you investigate a possible out-of-control signal, you would need to determine why the jelly beans are defective to find the assignable (root) cause and eliminate it.

Attribute control charts are used to evaluate variation in in a process where the measurement is an attribute--i.e. is discrete or count data (e.g. pass/fail, number of defects). There are two main types of attribute control charts. One type, based on the binomial distribution (e.g. p, np-chart), is used for defective units.  This type of chart would be appropriate for the preceding jelly bean example.

Consider the same jelly bean process.  This time we are interested in the number of spots on the jelly beans.  Our customer has agreed to accept jelly beans with no more than 4 small spots--more than 4 small spots and the jelly bean is considered to be defective.  For this example we would need a chart based on on the Poisson distribution (e.g. c, u chart) to plot the number of defects.

Consider the effect of using a p-chart instead of a c-chart in this example.  In the first sample taken during the day, we find the average number of defects (spots) to be 0, and we find no defective jelly beans.  We would plot a 0 on the p-chart.  In the second sample we find 1 spot on average, but still no defective jelly beans.  We again plot a 0 on the p-chart.  In the third, fourth, and fifth samples we find 2, 3, and 4 spots on average respectively but no defective jelly beans.  We plot a 0 on the p-chart for all three samples.  Finally in the sixth sample we find 5 defects on average and a number of rejected jelly beans.  The p-chart would indicate a major change between the fifth and sixth sample when in realitythere was a gradual increase in the average number of defects over the 6 samples.  The c-chart would plot 0, 1, 2, 3, 4, and 5 respectively--clearly showing the rising trend in average number of defects.

As with the variables charts, the patterns in the data plotted on the attribute control charts provide evidence of the process being in-control (only common cause variation present) or out-of-control (common cause and assignable cause variation present).  These out-of-control signals are interpreted in the same basic way as was discussed in the section on variables control charts.

Answer Discussion Questions 8, 9, 12-14 at the end of Chapter 10 of the text.

Do Problems 3-9, 13, 16 at the end of Chapter 10 of the text.

Do experiential Exercise 2 at the end of Chapter 10 of the text.