Chapter 16 Introduction to Quality ©

Some Benefits of Utilizing Statistical Quality Methods Increased Productivity Increased Sales Increased Profits

Causes of Process Variation Common causes Assignable causes Common causes (also called random or uncontrollable causes) of variation are those causes that are random in occurrence and are inherent in all processes. Management, not the workers, are responsible for these causes. Assignable causes (also called special causes) of variation are the result of external sources outside the system. These causes can and must be detected, and corrective action must be taken to remove them from the process. Failing to do so will increase variation and lower quality.

Stable Process stable A process is stable (in-control) if all assignable causes are removed; thus, variation results only from common causes.

Overall Mean, Average Sample Standard Deviation, Process Standard Deviation average of these sample means overall mean A sequence of K samples, each of n observations, is taken over time on a measurable characteristic of the output of a production process. The sample means denoted X i for I = 1, 2,..., K can be graphed on an X – chart. The average of these sample means is the overall mean of the sample observations s-chartaverage sample standard deviation The sample standard deviations denoted s i for 1 = 1, 2,...,K can be graphed on an s-chart. The average sample standard deviation is process standard deviation The process standard deviation, , is the standard deviation of the population from which the samples were drawn, and it must be estimated from sample data.

Estimate of Process Standard Deviation Based on s estimate of process standard deviation An estimate of process standard deviation, is Where s is the average sample standard deviation, and the control chart factor, c 4, which depends on the sample size, n, can be found in Table 16.1 or the Factors for Control Charts table in the appendix. If the population distribution is normal, the estimator is unbiased.

Factors for Control Charts (Table 16.1) nc4c4 A3A3 B3B3 B4B

X - Chart X – Chart center line The X – Chart is a time plot of the sequence of sample means. For convenience in interpretation, three lines are drawn on this chart. The center line is lower control limit In addition, there are three-standard error control limits. The lower control limit is upper control limit And the upper control limit is Where certain values of A 3 are given in Table 16.1 or the Control Chart Constants Table in the Appendix.

s - Chart s-chart center line The s-chart is a time plot of the sequence of sample standard deviations. The center line on the s-chart is lower control limit For three-standard error limits, the lower control limit is upper control limit And the upper control limit is Where values for the control chart constants B 3 and B 4 are shown in Table 16.1.

Out of Control Patterns Certain patterns of data points in a control chart indicate that a process might be out-of-control. Three of these patterns are: 1)A value outside the control limits (One point more than 3 sigmas from center line); 2)Trend in sample statistics (six points in a row, all decreasing or increasing); 3)Too many points on one side of the center line (nine points in a row on same side of center line)

Two Measures of Process Capability Assume that management sets lower (L) and upper (U) tolerance limits for process performance. Process capability is judged by the extent to which lies between these limits. Capability Index (i) Capability Index. This measure is appropriate when the sample data are centered between the tolerance limits, i.e.. The index is A satisfactory value of this index is usually taken to be one that is at least [This implies that the natural rate of tolerance of the process should be no more than 75% of (U – L), the width of the range of acceptable values.]

Two Measures of Process Capability (continued) C pk Index (ii) C pk Index. When the sample data are not centered between the tolerance limits, it is necessary to allow for the fact that the process is operating closer to one tolerance limit than the other. The resulting measure, called the C pk index, is Again, this is taken to be satisfactory if its value is at least 1.33.

Defects and Defectives defect defective “A defect is a single nonconforming quality characteristic of an item. An item may have several defects. The term defective refers to items having one or more defects” (reference 4).

Average of Sample Proportions average of the sample proportionsoverall proportion of nonconforming A sequence of K samples, each of n observations, is taken over time, and the proportion of sample members not conforming to standards is determined. These sample proportions denoted p i for i = 1, 2,..., K can be plotted on a p-chart. If the samples are of the same size, the average of the sample proportions is the overall proportion of nonconforming items. This is

p - Chart p-chart The p-chart is a time plot of the sequence of sample proportions of nonconforming items with center line given by: lower and upper control limits The lower and upper control limits are:

Sample Mean Number of Occurrences sample mean number of occurrences A sequence of K samples is inspected over time. For each item, the number of occurrences of some event, such as an imperfection, is recorded. These numbers of occurrences are denoted c i for i = 1, 2,..., K. The sample mean number of occurrences is then

c - Chart c-chart The c-chart is a time plot of the number of occurrences of an event. The center is: lower control limit For three-standard error limits, the lower control limit is: upper control limit and the upper control limit is

Key Words 4Assignable Causes 4c-chart 4Common Causes 4Control Chart Constants 4C p Index 4C pk Index 4Defect 4Defective 4Deming, W. Edwards 4Estimate of Process Standard Deviation based on s 4 Estimate of Process Standard Deviation based on R 4 Natural Tolerance 4 Out-of-Control Patterns 4 p-chart 4 Process Capability Indices 4 Process Standard Deviation 4 R-chart 4 s-chart

Key Words (continued) 4Specification Limits 4Stable Process 4Taguchi 4X-Chart