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**Statistically-Based Quality Improvement for Variables**

Chapter 12 Statistically-Based Quality Improvement for Variables S. Thomas Foster, Jr. Boise State University PowerPoint prepared by Dave Magee University of Kentucky Lexington Community College ©2004 Prentice-Hall

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**Chapter Overview Statistical Fundamentals Process Control Charts**

Some Control Chart Concepts for Variables Process Capability for Variables Other Statistical Techniques in Quality Management

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**Statistical Fundamentals Slide 1 of 11**

Statistical Thinking A decision-making skill demonstrated by the ability to draw to conclusions based on data. Based on three concepts All work occurs in a system of interconnected processes. All processes have variation (the amount of which tends to be underestimated). Understanding and reducing variation are important keys to success. Guides people to make decisions based on data, which needs to be done in business.

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**Statistical Fundamentals Slide 2 of 11**

Why Do Statistics Sometimes Fail in the Workplace? Many times statistical tools do not create the desired result, because many firms fail to implement quality control in a substantive way. They prefer form over substance.

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**Statistical Fundamentals Slide 3 of 11**

Reasons for Failure of Statistical Tools Lack of knowledge about the tools; therefore, tools are misapplied. General disdain for all things mathematical creates a natural barrier to the use of statistics. Cultural barriers in a company make the use of statistics for continual improvement difficult. Statistical specialists have trouble communicating with managerial generalists. Statistics generally are poorly taught, emphasizing mathematical development rather than application.

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**Statistical Fundamentals Slide 4 of 11**

Reasons for Failure of Statistical Tools (continued) People have a poor understanding of the scientific method. Organizations lack patience in collecting data. All decisions have to be made “yesterday.” Statistics are viewed as something to buttress an already-held opinion rather than a method for informing and improving decision making. People fear using statistics because they fear they may violate critical statistical assumptions.

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**Statistical Fundamentals Slide 5 of 11**

Reasons for Failure of Statistical Tools (continued) Most people don’t understand random variation resulting in too much process tampering. Statistical tools are often reactive and focus on effects rather than causes. People make mistakes with statistics, because of Type I error (producer’s risk) and Type II error (consumer’s risk). These erroneous decisions can result in high costs or lost future sales.

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**Statistical Fundamentals Slide 6 of 11**

Understanding Process Variation All processes exhibit variation. Some variation can be controlled and some cannot. Two types of process variation Random Nonrandom Statistical tools presented here are useful for determining whether variation is random.

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**Statistical Fundamentals Slide 7 of 11**

Random variation Random variation is centered around a mean and occurs with a consistent amount of dispersion. This type of variation cannot be controlled. Hence, we refer to it as “uncontrolled variation.” The statistical tools discussed in this chapter are not designed to detect random variation. Nonrandom variation Nonrandom or “special cause” variation results from some event. The event may be a shift in a process mean or some unexpected occurrence.

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**Statistical Fundamentals Slide 8 of 11**

Random Variation Figure 12.1

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**Statistical Fundamentals Slide 9 of 11**

Nonrandom Variation Figure 12.2

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**Statistical Fundamentals Slide 10 of 11**

Process Stability Means that the variation we observe in the process is random variation (common cause) and not nonrandom variation (special or assignable causes). To determine process stability we use process charts. Process charts are graphs designed to signal process workers when nonrandom variation is occurring in a process. Sampling Methods Process control requires that data be gathered. Samples are cheaper, take less time and are less intrusive than 100% inspection.

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**Statistical Fundamentals Slide 11 of 11**

Sampling Methods Random samples Randomization is useful because it ensures independence among observations. To randomize means to sample in such a way that every piece of product has an equal chance of being selected for inspection. Systematic samples Systematic samples have some of the benefits of random samples without the difficulty of randomizing. Sampling by Rational Subgroup A rational subgroup is a group of data that is logically homogenous; variation within the data can provide a yardstick for computing limits on the standard variation between subgroups.

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**Process Control Charts Slide 1 of 20**

Statistical Process Control Charts Tools for monitoring process variation. The figure on the following slide shows a process control chart. It has an upper limit, a center line, and a lower limit. Variables and Attributes To select the proper process chart, we must differentiate between variables and attributes. A variable is a continuous measurement such as weight, height, or volume. An attribute is the result of a binomial process that results in an either-or-situation.

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**Process Control Charts Slide 2 of 20**

Figure 12.3

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**Process Control Charts Slide 3 of 20**

Variables and Attributes Variables Attributes X (process population average) P (proportion defective) X-bar (mean or average) np (number defective) R (range) C (number conforming) MR (moving range) U (number nonconforming) S (standard deviation)

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**Process Control Charts Slide 4 of 20**

Central Requirements for Properly Using Process Charts 1. You must understand the generic process for implementing process charts. You must know how to interpret process charts. You need to know when different process charts are used. You need to know how to compute limits for the different types of process charts. 2. 3. 4.

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**Process Control Charts Slide 5 of 20**

A Generalized Procedure for Developing Process Charts Identify critical operations in the process where inspection might be needed. These are operations in which, if the operation is performed improperly, the product will be negatively affected. Identify critical product characteristics. These are the aspects of the product that will result in either good or poor function of the product. Determine whether the critical product characteristic is a variable or an attribute.

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**Process Control Charts Slide 6 of 20**

A Generalized Procedure for Developing Process Charts (continued) Select the appropriate process control chart from among the many types of control charts. This decision process and types of charts available are discussed later. Establish the control limits and use the chart to continually monitor and improve. Update the limits when changes have been made to the process.

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**Process Control Charts Slide 7 of 20**

Understanding Control Charts A process chart is nothing more than an application of hypothesis testing where the null hypothesis is that the product meets requirements. An X-bar chart is a variables chart that monitors average measurement. Control charts draw a sampling distribution rather than a population distribution. Control charts make use of the central limit theorem, which states that when we plot sample means, the sampling distribution approximates a normal distribution.

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**Process Control Charts Slide 8 of 20**

X-bar and R Charts The X-bar chart is a process chart used to monitor the average of the characteristics being measured. To set up an X-bar chart Select samples from the process for the characteristic being measured. Then form the samples into rational subgroups. Next, find the average value of each sample by dividing the sums of the measurements by the sample size and plot the value on the process control X-bar chart.

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**Process Control Charts Slide 9 of 20**

X-bar and R Charts (continued) The R chart is used to monitor the dispersion of the process. It is used in conjunction with the X-bar chart when the process characteristic is a variable. To develop an R chart Collect samples from the process and organize them into subgroups, usually of three to six items. Next, compute the range, R, by taking the difference of the high value in the subgroup minus the low value. Then plot the R values on the R chart.

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**Process Control Charts Slide 10 of 20**

X-bar and R Charts Figure 12.6

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**Process Control Charts Slide 11 of 20**

Interpreting Control Charts Before introducing other types of process charts, we discuss the interpretation of the charts. The figures in the next several slides show different signals for concern that are sent by a control chart, as in the second and third boxes. When a point is found to be outside of the control limits, we call this an “out of control” situation. When a process is out of control, the variation is probably no longer random.

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**Process Control Charts Slide 12 of 20**

Control Chart Evidence for Investigation Figure 12.10

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**Process Control Charts Slide 13 of 20**

Control Chart Evidence for Investigation Figure 12.10

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**Process Control Charts Slide 14 of 20**

Implications of a Process Out of Control If a process loses control and becomes nonrandom, the process should be stopped immediately. In many modern process industries where just-in-time is used, this will result in the stoppage of several work stations. The team of workers who are to address the problem should use a structured problem solving process.

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**Process Control Charts Slide 15 of 20**

X and Moving Range (MR) Charts for Population Data At times, it may not be possible to draw samples. This may occur because a process is so slow that only one or two units per day are produced. X chart. A chart used to monitor the mean of a process for population values. MR chart. A chart for plotting variables when samples are not possible. If data are not normally distributed, other charts are available.

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**Process Control Charts Slide 16 of 20**

g and h Charts A g chart is used when data are geometrically distributed h charts are useful when data is hypergeometrically distributed. If a histogram of data appears like either of these distributions, you may want to use either an h or a g chart instead of an X chart. Figure 12.13

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**Process Control Charts Slide 17 of 20**

Median Charts Many be used when it is too time consuming or inconvenient to compute subgroup averages or when there is concern about the accuracy of computed means. Small sample sample sizes are generally used like the x-bar chart. Equations for computing the control limits are: = Mean of medians = sum of the medians/number of medians

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**Process Control Charts Slide 18 of 20**

x-bar and s Charts When dispersion of the process is of particular concern the s (standard deviation) chart is used in place of the R chart. Different formulas are used to compute the limits for the x-bar chart.

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**Process Control Charts Slide 19 of 20**

Other Control Charts Moving Average Chart. The moving average chart is an interesting chart that is used for monitoring variables and measurement on a continuous scale. The chart uses past information to predict what the next process outcome will be. Using this chart, we can adjust a process in anticipation of its going out of control. Cusum Chart. The cumulative sum, or cusum, chart is used to identify slight but sustained shifts in a universe where there is no independence between observations.

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**Process Control Charts Slide 20 of 20**

Summary of Variable Chart Formulas Chart LCL CL UCL

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**Some Control Charts Concepts for Variables Slide 1 of 4**

Choosing the Correct Variables Control Chart Obviously, it is key to choose the correct control chart. Figure in the textbook shows a decision tree for the basic control charts. This flow chart helps to show when certain charts should be selected for use. Corrective Action. When a process is out of control, corrective action is needed. Corrective action steps are similar to continuous improvement processes.

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**Some Control Charts Concepts for Variables Slide 2 of 4**

Corrective Action (continued) Correction action steps : Carefully identify the problem. Form the correct team to evaluate and solve the problem. Use structured brainstorming along with fishbone diagrams or affinity diagrams to identify causes of the problem. Brainstorm to identify potential solutions to problems. Eliminate the cause. Restart the process. Document the problem, root causes, and solutions. Communicate the results of the process to all personnel so that this process becomes reinforced and ingrained in the organization.

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**Some Control Charts Concepts for Variables Slide 3 of 4**

How Do We Use Control Charts to Continuously Improve? One of the goals of the control chart user is to reduce variation. Over time, as processes are improved, control limits are recomputed to show improvements in stability. As upper and lower control limits get closer and closer together, the process improving. Two key concepts: The focus of control charts should be on continuous improvement. Control chart limits should be updated only when there is a change to the process. Otherwise any changes are unexpected.

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**Some Control Charts Concepts for Variables Slide 4 of 4**

Tampering With the Process One of the cardinal rules of process charts is that you should never tamper with the process. You might wonder, why don’t we make adjustments to the process any time the process deviates from the target? The reason is that random effects are just that—random. This means that these effects cannot be controlled. If we make adjustments to a random process, we actually inject nonrandom activity into the process.

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**Process Capability Slide 1 of 11**

Process Stability and Capability Once a process is stable, the next emphasis is to ensure that the process is capable. Process capability refers to the ability of a process to produce a product that meets specifications. Six-sigma program such as those pioneered by Motorola Corporation result in highly capable processes.

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**Process Capability Slide 2 of 11**

Six-Sigma Quality Figure 12.20

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**Process Capability Slide 3 of 11**

Process Versus Sampling Distribution To understand process capability we must first understand the differences between population and sampling distributions. Population distributions are distributions with all the items or observations of interest to a decision maker. A population is defined as a collection of all the items or observations of interest to a decision maker. A sample is subset of the population. Sampling distributions are distributions that reflect the distributions of sample means.

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**Process Capability Slide 4 of 11**

Capability Studies Now that we have defined process capability, we can discuss how to determine whether a process is capable. That is, we want to know if individual products meet specifications. There are two purposes for performing process capability studies: 1. To determine whether a process consistently results in products that meet specifications 2. To determine whether a process is in need of monitoring through the use of permanent process charts.

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**Process Capability Slide 5 of 11**

Capability Studies (continued) Process capability studies help process managers understand whether the range over which natural variation of a process occurs is the result of the system of common (or random) causes.

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**Process Capability Slide 6 of 11**

Capability Studies (continued) Five steps in performing process capability studies: 1. Select a critical operation. These may be bottlenecks, costly steps of the process, or places in the process where problems have occurred in the past. 2. Take k samples of size n, where x is an individual observation. Where 19 < k < 26 If x is an attribute n > 50, (as in the case of a binomial) Or if x is a measurement 1 < n < 11 (Note: Small sample sizes can lead to erroneous conclusions.)

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**Process Capability Slide 7 of 11**

Capability Studies (continued) Five steps in performing process capability studies: 3. Use a trial control chart to see whether the process is stable. 4. Compare process natural tolerance limits with specification limits. Note that natural tolerance limits are three standard deviation limits for the population distribution. This can be compared with the specification limits.

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**Process Capability Slide 8 of 11**

Capability Studies (continued) Five steps in performing process capability studies: 5. Compute capability indexes: To compute capability indexes, you compute an upper capability index (Cpu), a lower capability index (Cpl), and a capability index (Cpk). The formulas are:

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**Process Capability Slide 9 of 11**

Capability Studies (continued) Although different firms use different benchmarks, the generally accepted benchmarks for process capability are 1.25, 1.33, and 2.0. We will say that processes that achieve capability indexes (Cpk) of: 1.25 are capable 1.33 are highly capable 2.0 are world-class capable (six sigma)

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**Process Capability Slide 10 of 11**

Ppk Population capability index Used when data is not arranged in subgroups, but is only available as population data. Formulas:

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**Process Capability Slide 11 of 11**

The Difference Between Capability and Stability Once again, a process is capable if individual products consistently meet specifications. A process is stable if only common variation is present in the process. It is possible to have a process that is stable but not capable. This would happen where random variation was very high. It is probably not so common that an incapable process would be stable.

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**Summary Statistical Fundamentals Process Control Charts**

Some Control Chart Concepts for Variables Process Capability for Variables Other Statistical Techniques in Quality Management

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