CHAPTER Statistical Quality Control Before studying this chapter you should know or, if necessary, review 1. 2. Quality as a competitive priority, Chapter 2, page 00. Total quality management (TQM) concepts, Chapter 5, pages 00 – 00. 6 LEARNING OBJECTIVES After studying this chapter you should be able to 1 2 3 4 5 6 7 8 9 Describe categories of statistical quality control (SQC). Explain the use of descriptive statistics in measuring quality characteristics. Identify and describe causes of variation. Describe the use of control charts. Identify the differences between x-bar, R-, p-, and c-charts.
Explain the meaning of process capability and the process capability index. Explain the term Six Sigma. Explain the process of acceptance sampling and describe the use of operating characteristic (OC) curves. Describe the challenges inherent in measuring quality in service organizations. CHAPTER OUTLINE What Is Statistical Quality Control? 172 Links to Practice: Intel Corporation 173 Sources of Variation: Common and Assignable Causes 174 Descriptive Statistics 174 Statistical Process Control Methods 176 Control Charts for Variables 178 Control Charts for Attributes 184 C-Charts 188 Process Capability 190 Links to Practice: Motorola, Inc. 196
Acceptance Sampling 196 Implications for Managers 203 Statistical Quality Control in Services 204 Links to Practice: The Ritz-Carlton Hotel Company, L. L. C. ; Nordstrom, Inc. 205 Links to Practice: Marriott International, Inc. 205 OM Across the Organization 206 Inside OM 206 Case: Scharadin Hotels 216 Case: Delta Plastics, Inc. (B) 217 000 171 172 • CHAPTER 6 STATISTICAL QUALITY CONTROL e have all had the experience of purchasing a product only to discover that it is defective in some way or does not function the way it was designed to. This could be a new backpack with a broken zipper or an “out of the box” malfunctioning computer printer.
Many of us have struggled to assemble a product the manufacturer has indicated would need only “minor” assembly, only to ? nd that a piece of the product is missing or defective. As consumers, we expect the products we purchase to function as intended. However, producers of products know that it is not always possible to inspect every product and every aspect of the production process at all times. The challenge is to design ways to maximize the ability to monitor the quality of products being produced and eliminate defects. One way to ensure a quality product is to build quality into the process.
Consider Steinway & Sons, the premier maker of pianos used in concert halls all over the world. Steinway has been making pianos since the 1880s. Since that time the company’s manufacturing process has not changed signi? cantly. It takes the company nine months to a year to produce a piano by fashioning some 12,000-hand crafted parts, carefully measuring and monitoring every part of the process. While many of Steinway’s competitors have moved to mass production, where pianos can be assembled in 20 days, Steinway has maintained a strategy of quality de? ned by skill and craftsmanship.
Steinway’s production process is focused on meticulous process precision and extremely high product consistency. This has contributed to making its name synonymous with top quality. W WHAT IS STATISTICAL QUALITY CONTROL? In Chapter 5 we learned that total quality management (TQM) addresses organizational quality from managerial and philosophical viewpoints. TQM focuses on customer-driven quality standards, managerial leadership, continuous improvement, quality built into product and process design, quality identi? ed problems at the source, and quality made everyone’s responsibility.
However, talking about solving quality problems is not enough. We need speci? c tools that can help us make the right quality decisions. These tools come from the area of statistics and are used to help identify quality problems in the production process as well as in the product itself. Statistical quality control is the subject of this chapter. Statistica1 quality control (SQC) is the term used to describe the set of statistical tools used by quality professionals. Statistical quality control can be divided into three broad categories: 1. Descriptive statistics are used to describe quality characteristics and relationships.
Included are statistics such as the mean, standard deviation, the range, and a measure of the distribution of data. Marketing, Management, Engineering Statistica1 quality control (SQC) The general category of statistical tools used to evaluate organizational quality. Descriptive statistics Statistics used to describe quality characteristics and relationships. WHAT IS STATISTICAL QUALITY CONTROL? • 173 2. Statistical process control (SPC) involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range.
SPC answers the question of whether the process is functioning properly or not. 3. Acceptance sampling is the process of randomly inspecting a sample of goods and deciding whether to accept the entire lot based on the results. Acceptance sampling determines whether a batch of goods should be accepted or rejected. The tools in each of these categories provide different types of information for use in analyzing quality. Descriptive statistics are used to describe certain quality characteristics, such as the central tendency and variability of observed data.
Although descriptions of certain characteristics are helpful, they are not enough to help us evaluate whether there is a problem with quality. Acceptance sampling can help us do this. Acceptance sampling helps us decide whether desirable quality has been achieved for a batch of products, and whether to accept or reject the items produced. Although this information is helpful in making the quality acceptance decision after the product has been produced, it does not help us identify and catch a quality problem during the production process. For this we need tools in the statistical process control (SPC) category.
All three of these statistical quality control categories are helpful in measuring and evaluating the quality of products or services. However, statistical process control (SPC) tools are used most frequently because they identify quality problems during the production process. For this reason, we will devote most of the chapter to this category of tools. The quality control tools we will be learning about do not only measure the value of a quality characteristic. They also help us identify a change or variation in some quality characteristic of the product or process. We will ? st see what types of variation we can observe when measuring quality. Then we will be able to identify speci? c tools used for measuring this variation. Variation in the production process leads to quality defects and lack of product consistency. The Intel Corporation, the world’s largest and most pro? table manufacturer of microprocessors, understands this. Therefore, Intel has implemented a program it calls “copy-exactly” at all its manufacturing facilities. The idea is that regardless of whether the chips are made in Arizona, New Mexico, Ireland, or any of its other plants, they are made in exactly the same way.
This means using the same equipment, the same exact materials, and workers performing the same tasks in the exact same order. The level of detail to which the “copy-exactly” concept goes is meticulous. For example, when a chipmaking machine was found to be a few feet longer at one facility than another, Intel made them match. When water quality was found to be different at one facility, Intel instituted a puri? cation system to eliminate any differences. Even when a worker was found polishing equipment in one direction, he was asked to do it in the approved circular pattern.
Why such attention to exactness of detail? The reason is to minimize all variation. Now let’s look at the different types of variation that exist. Statistical process control (SPC) A statistical tool that involves inspecting a random sample of the output from a process and deciding whether the process is producing products with characteristics that fall within a predetermined range. Acceptance sampling The process of randomly inspecting a sample of goods and deciding whether to accept the entire lot based on the results. LINKS TO PRACTICE Intel Corporation www. intel. com 174 • CHAPTER 6
STATISTICAL QUALITY CONTROL SOURCES OF VARIATION: COMMON AND ASSIGNABLE CAUSES If you look at bottles of a soft drink in a grocery store, you will notice that no two bottles are ? lled to exactly the same level. Some are ? lled slightly higher and some slightly lower. Similarly, if you look at blueberry muf? ns in a bakery, you will notice that some are slightly larger than others and some have more blueberries than others. These types of differences are completely normal. No two products are exactly alike because of slight differences in materials, workers, machines, tools, and other factors.
These are called common, or random, causes of variation. Common causes of variation are based on random causes that we cannot identify. These types of variation are unavoidable and are due to slight differences in processing. An important task in quality control is to ? nd out the range of natural random variation in a process. For example, if the average bottle of a soft drink called Cocoa Fizz contains 16 ounces of liquid, we may determine that the amount of natural variation is between 15. 8 and 16. 2 ounces. If this were the case, we would monitor the production process to make sure that the amount stays within this range.
If production goes out of this range — bottles are found to contain on average 15. 6 ounces — this would lead us to believe that there is a problem with the process because the variation is greater than the natural random variation. The second type of variation that can be observed involves variations where the causes can be precisely identi? ed and eliminated. These are called assignable causes of variation. Examples of this type of variation are poor quality in raw materials, an employee who needs more training, or a machine in need of repair. In each of these examples the problem can be identi? d and corrected. Also, if the problem is allowed to persist, it will continue to create a problem in the quality of the product. In the example of the soft drink bottling operation, bottles ? lled with 15. 6 ounces of liquid would signal a problem. The machine may need to be readjusted. This would be an assignable cause of variation. We can assign the variation to a particular cause (machine needs to be readjusted) and we can correct the problem (readjust the machine). Common causes of variation Random causes that cannot be identi? ed. Assignable causes of variation Causes that can be identi? d and eliminated. DESCRIPTIVE STATISTICS Descriptive statistics can be helpful in describing certain characteristics of a product and a process. The most important descriptive statistics are measures of central tendency such as the mean, measures of variability such as the standard deviation and range, and measures of the distribution of data. We ? rst review these descriptive statistics and then see how we can measure their changes. The Mean Mean (average) A statistic that measures the central tendency of a set of data. In the soft drink bottling example, we stated that the average bottle is ? led with 16 ounces of liquid. The arithmetic average, or the mean, is a statistic that measures the central tendency of a set of data. Knowing the central point of a set of data is highly important. Just think how important that number is when you receive test scores! To compute the mean we simply sum all the observations and divide by the total number of observations. The equation for computing the mean is n xi x i 1 n DESCRIPTIVE STATISTICS • 175 where x xi n the mean observation i, i 1, . . . , n number of observations The Range and Standard Deviation
In the bottling example we also stated that the amount of natural variation in the bottling process is between 15. 8 and 16. 2 ounces. This information provides us with the amount of variability of the data. It tells us how spread out the data is around the mean. There are two measures that can be used to determine the amount of variation in the data. The ? rst measure is the range, which is the difference between the largest and smallest observations. In our example, the range for natural variation is 0. 4 ounces. Another measure of variation is the standard deviation.
The equation for computing the standard deviation is Range The difference between the largest and smallest observations in a set of data. Standard deviation A statistic that measures the amount of data dispersion around the mean. where x xi n standard deviation of a sample the mean observation i, i 1, . . . , n the number of observations in the sample v n (x i i 1 x)2 1 n Small values of the range and standard deviation mean that the observations are closely clustered around the mean. Large values of the range and standard deviation mean that the observations are spread out around the mean.
Figure 6-1 illustrates the differences between a small and a large standard deviation for our bottling operation. You can see that the ? gure shows two distributions, both with a mean of 16 ounces. However, in the ? rst distribution the standard deviation is large and the data are spread out far around the mean. In the second distribution the standard deviation is small and the data are clustered close to the mean. FIGURE 6-1 Normal distributions with varying standard deviations FIGURE 6-2 Differences between symmetric and skewed distributions Skewed distribution Small standard deviation Large standard deviation
Symmetric distribution 15. 7 15. 8 15. 9 16. 0 Mean 16. 1 16. 2 16. 3 15. 7 15. 8 15. 9 16. 0 16. 1 Mean 16. 2 16. 3 176 • CHAPTER 6 STATISTICAL QUALITY CONTROL Distribution of Data A third descriptive statistic used to measure quality characteristics is the shape of the distribution of the observed data. When a distribution is symmetric, there are the same number of observations below and above the mean. This is what we commonly ? nd when only normal variation is present in the data. When a disproportionate number of observations are either above or below the mean, we say that the data has a skewed distribution.
Figure 6-2 shows symmetric and skewed distributions for the bottling operation. STATISTICAL PROCESS CONTROL METHODS Statistical process control methods extend the use of descriptive statistics to monitor the quality of the product and process. As we have learned so far, there are common and assignable causes of variation in the production of every product. Using statistical process control we want to determine the amount of variation that is common or normal. Then we monitor the production process to make sure production stays within this normal range.
That is, we want to make sure the process is in a state of control. The most commonly used tool for monitoring the production process is a control chart. Different types of control charts are used to monitor different aspects of the production process. In this section we will learn how to develop and use control charts. Developing Control Charts Control chart A graph that shows whether a sample of data falls within the common or normal range of variation. Out of control The situation in which a plot of data falls outside preset control limits.
A control chart (also called process chart or quality control chart) is a graph that shows whether a sample of data falls within the common or normal range of variation. A control chart has upper and lower control limits that separate common from assignable causes of variation. The common range of variation is de? ned by the use of control chart limits. We say that a process is out of control when a plot of data reveals that one or more samples fall outside the control limits. Figure 6-3 shows a control chart for the Cocoa Fizz bottling operation. The x axis represents samples (#1, #2, #3, etc. taken from the process over time. The y axis represents the quality characteristic that is being monitored (ounces of liquid). The center line (CL) of the control chart is the mean, or average, of the quality characteristic that is being measured. In Figure 6-3 the mean is 16 ounces. The upper control limit (UCL) is the maximum acceptable variation from the mean for a process that is in a state of control. Similarly, the lower control limit (LCL) is the minimum acceptable variation from the mean for a process that is in a state of control. In our example, the Observation ut of control Variation due to assignable causes UCL = (16. 2) Variation due to normal causes FIGURE 6-3 Volume in ounces Quality control chart for Cocoa Fizz CL = (16. 0) LCL = (15. 8) #1 #2 #3 #4 #5 Sample Number #6 Variation due to assignable causes STATISTICAL PROCESS CONTROL METHODS • 177 upper and lower control limits are 16. 2 and 15. 8 ounces, respectively. You can see that if a sample of observations falls outside the control limits we need to look for assignable causes. The upper and lower control limits on a control chart are usually set at 3 standard deviations from the mean.
If we assume that the data exhibit a normal distribution, these control limits will capture 99. 74 percent of the normal variation. Control limits can be set at 2 standard deviations from the mean. In that case, control limits would capture 95. 44 percent of the values. Figure 6-4 shows the percentage of values that fall within a particular range of standard deviation. Looking at Figure 6-4, we can conclude that observations that fall outside the set range represent assignable causes of variation. However, there is a small probability that a value that falls outside the limits is still due to normal variation.
This is called Type I error, with the error being the chance of concluding that there are assignable causes of variation when only normal variation exists. Another name for this is alpha risk ( ), where alpha refers to the sum of the probabilities in both tails of the distribution that falls outside the con? dence limits. The chance of this happening is given by the percentage or probability represented by the shaded areas of Figure 6-5. For limits of 3 standard deviations from the mean, the probability of a Type I error is . 26% (100% 99. 74%), whereas for limits of 2 standard deviations it is 4. 6% (100% 95. 44%). Types of Control Charts Control charts are one of the most commonly used tools in statistical process control. They can be used to measure any characteristic of a product, such as the weight of a cereal box, the number of chocolates in a box, or the volume of bottled water. The different characteristics that can be measured by control charts can be divided into two groups: variables and attributes. A control chart for variables is used to monitor characteristics that can be measured and have a continuum of values, such as height, weight, or volume.
A soft drink bottling operation is an example of a variable measure, since the amount of liquid in the bottles is measured and can take on a number of different values. Other examples are the weight of a bag of sugar, the temperature of a baking oven, or the diameter of plastic tubing. Variable A product characteristic that can be measured and has a continuum of values (e. g. , height, weight, or volume). Attribute A product characteristic that has a discrete value and can be counted. FIGURE 6-4 Percentage of values captured by different ranges of standard deviation FIGURE 6-5
Chance of Type I error for 3 (sigma-standard deviations) Type 1 error is . 26% –3? –2? Mean 95. 44% 99. 74% +2? +3? –3? –2? Mean 99. 74% +2? +3? 178 • CHAPTER 6 STATISTICAL QUALITY CONTROL A control chart for attributes, on the other hand, is used to monitor characteristics that have discrete values and can be counted. Often they can be evaluated with a simple yes or no decision. Examples include color, taste, or smell. The monitoring of attributes usually takes less time than that of variables because a variable needs to be measured (e. g. , the bottle of soft drink contains 15. 9 ounces of liquid).
An attribute requires only a single decision, such as yes or no, good or bad, acceptable or unacceptable (e. g. , the apple is good or rotten, the meat is good or stale, the shoes have a defect or do not have a defect, the lightbulb works or it does not work) or counting the number of defects (e. g. , the number of broken cookies in the box, the number of dents in the car, the number of barnacles on the bottom of a boat). Statistical process control is used to monitor many different types of variables and attributes. In the next two sections we look at how to develop control charts for variables and control charts for attributes.
CONTROL CHARTS FOR VARIABLES Control charts for variables monitor characteristics that can be measured and have a continuous scale, such as height, weight, volume, or width. When an item is inspected, the variable being monitored is measured and recorded. For example, if we were producing candles, height might be an important variable. We could take samples of candles and measure their heights. Two of the most commonly used control charts for variables monitor both the central tendency of the data (the mean) and the variability of the data (either the standard deviation or the range).
Note that each chart monitors a different type of information. When observed values go outside the control limits, the process is assumed not to be in control. Production is stopped, and employees attempt to identify the cause of the problem and correct it. Next we look at how these charts are developed. Mean (x-Bar) Charts x-bar chart A control chart used to monitor changes in the mean value of a process. A mean control chart is often referred to as an x-bar chart. It is used to monitor changes in the mean of a process. To construct a mean chart we ? rst need to construct the center line of the chart.
To do this we take multiple samples and compute their means. Usually these samples are small, with about four or ? ve observations. Each sample has its own mean, x. The center line of the chart is then computed as the mean of all sample means, where is the number of samples: x x1 x2 x To construct the upper and lower control limits of the chart, we use the following formulas: Upper control limit (UCL) Lower control limit (LCL) x x z z x x where x z x n the average of the sample means standard normal variable (2 for 95. 44% con? dence, 3 for 99. 74% con? ence) standard deviation of the distribution of sample means, computed as /vn population (process) standard deviation sample size (number of observations per sample) Example 6. 1 shows the construction of a mean (x-bar) chart. CONTROL CHARTS FOR VARIABLES • 179 A quality control inspector at the Cocoa Fizz soft drink company has taken twenty-? ve samples with four observations each of the volume of bottles ? lled. The data and the computed means are shown in the table. If the standard deviation of the bottling operation is 0. 14 ounces, use this information to develop control limits of three standard deviations for the bottling operation.
Observations (bottle volume in ounces) 1 2 3 4 15. 85 16. 02 15. 83 15. 93 16. 12 16. 00 15. 85 16. 01 16. 00 15. 91 15. 94 15. 83 16. 20 15. 85 15. 74 15. 93 15. 74 15. 86 16. 21 16. 10 15. 94 16. 01 16. 14 16. 03 15. 75 16. 21 16. 01 15. 86 15. 82 15. 94 16. 02 15. 94 16. 04 15. 98 15. 83 15. 98 15. 64 15. 86 15. 94 15. 89 16. 11 16. 00 16. 01 15. 82 15. 72 15. 85 16. 12 16. 15 15. 85 15. 76 15. 74 15. 98 15. 73 15. 84 15. 96 16. 10 16. 20 16. 01 16. 10 15. 89 16. 12 16. 08 15. 83 15. 94 16. 01 15. 93 15. 81 15. 68 15. 78 16. 04 16. 11 16. 12 15. 84 15. 92 16. 05 16. 12 15. 92 16. 09 16. 12 15. 93 16. 11 16. 02 16. 0 15. 88 15. 98 15. 82 15. 89 15. 89 16. 05 15. 73 15. 73 15. 93 16. 01 16. 01 15. 89 15. 86 16. 08 15. 78 15. 92 15. 98 EXAMPLE 6. 1 Constructing a Mean (x-Bar) Chart Sample Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Total Average x 15. 91 15. 99 15. 92 15. 93 15. 98 16. 03 15. 96 15. 93 15. 96 15. 83 15. 99 15. 96 15. 83 15. 91 16. 05 15. 99 15. 86 16. 01 15. 98 16. 02 16. 00 15. 90 15. 86 15. 94 15. 94 398. 75 Range R 0. 19 0. 27 0. 17 0. 46 0. 47 0. 20 0. 46 0. 20 0. 21 0. 30 0. 29 0. 43 0. 24 0. 37 0. 31 0. 29 0. 33 0. 34 0. 28 0. 20 0. 23 0. 16 0. 32 0. 15 0. 30 7. 17 • Solution
The center line of the control data is the average of the samples: x x The control limits are UCL LCL x x z z 15. 95 15. 95 3 3 . 14 v4 . 14 v4 16. 16 15. 74 398. 75 25 15. 95 x x 180 • CHAPTER 6 STATISTICAL QUALITY CONTROL The resulting control chart is: 16. 20 16. 10 16. 00 Ounces 15. 90 15. 80 15. 70 15. 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 LCL CL UCL Sample Mean This can also be computed using a spreadsheet as shown. A 1 2 X-Bar Chart: Cocoa Fizz 3 G7: =MAX(B7:E7)-MIN(B7:E7) F7: =AVERAGE(B7:E7) 4 5 Bottle Volume in Ounces Sample Num Obs 1 Obs 2 Obs 3 Obs 4 Average Range 6 1 15. 85 16. 02 15. 83 15. 3 15. 91 0. 19 7 2 16. 12 16. 00 15. 85 16. 01 16. 00 0. 27 8 3 16. 00 15. 91 15. 94 15. 83 15. 92 0. 17 9 4 16. 20 15. 85 15. 74 15. 93 15. 93 0. 46 10 5 15. 74 15. 86 16. 21 16. 10 15. 98 0. 47 11 6 15. 94 16. 01 16. 14 16. 03 16. 03 0. 20 12 7 15. 75 16. 21 16. 01 15. 86 15. 96 0. 46 13 8 15. 82 15. 94 16. 02 15. 94 15. 93 0. 20 14 9 16. 04 15. 98 15. 83 15. 98 15. 96 0. 21 15 10 15. 64 15. 86 15. 94 15. 89 15. 83 0. 30 16 11 16. 11 16. 00 16. 01 15. 82 15. 99 0. 29 17 12 15. 72 15. 85 16. 12 16. 15 15. 96 0. 43 18 13 15. 85 15. 76 15. 74 15. 98 15. 83 0. 24 19 14 15. 73 15. 84 15. 96 16. 10 15. 91 0. 37 20 15 16. 20 16. 01 16. 10 15. 9 16. 05 0. 31 21 16 16. 12 16. 08 15. 83 15. 94 15. 99 0. 29 22 17 16. 01 15. 93 15. 81 15. 68 15. 86 0. 33 23 18 15. 78 16. 04 16. 11 16. 12 16. 01 0. 34 24 19 15. 84 15. 92 16. 05 16. 12 15. 98 0. 28 25 20 15. 92 16. 09 16. 12 15. 93 16. 02 0. 20 26 21 16. 11 16. 02 16. 00 15. 88 16. 00 0. 23 27 22 15. 98 15. 82 15. 89 15. 89 15. 90 0. 16 28 23 16. 05 15. 73 15. 73 15. 93 15. 86 0. 32 29 24 16. 01 16. 01 15. 89 15. 86 15. 94 0. 15 30 25 16. 08 15. 78 15. 92 15. 98 15. 94 0. 30 31 15. 95 0. 29 32 Number of Samples 25 Xbar-bar R-bar 33 Number of Observations per Sample 4 34 G32: =AVERAGE(G7:G31) 35 F32: =AVERAGE(F7:F31) 36 B C D E F G
CONTROL CHARTS FOR VARIABLES • 181 A B C 39 Computations for X-Bar Chart 40 Overall Mean (Xbar-bar) = 41 Sigma for Process = 42 Standard Error of the Mean = 43 Z-value for control charts = 44 45 CL: Center Line = 46 LCL: Lower Control Limit = 47 UCL: Upper Control Limit = D 15. 95 0. 14 0. 07 3 15. 95 15. 74 16. 16 E D40: =F32 F G ounces D42: =D41/SQRT(D34) D45: =D40 D46: =D40-D43*D42 D47: =D40+D43*D42 Another way to construct the control limits is to use the sample range as an estimate of the variability of the process. Remember that the range is simply the difference between the largest and smallest values in the sample.
The spread of the range can tell us about the variability of the data. In this case control limits would be constructed as follows: Upper control limit (UCL) Lower control limit (LCL) x x A2 R A2 R where x R A2 average of the sample means average range of the samples factor obtained from Table 6-1. Notice that A2 is a factor that includes three standard deviations of ranges and is dependent on the sample size being considered. A quality control inspector at Cocoa Fizz is using the data from Example 6. 1 to develop control limits. If the average range (R) for the twenty-? ve samples is . 9 ounces (computed as 7. 17 ) and the 25 average mean (x) of the observations is 15. 95 ounces, develop three-sigma control limits for the bottling operation. EXAMPLE 6. 2 • Solution x 15. 95 ounces R . 29 . 73. This leads to the following Constructing a Mean (x-Bar) Chart from the Sample Range The value of A2 is obtained from Table 6. 1. For n limits: The center of the control chart UCL LCL x x A2 R A2 R 15. 95 15. 95 4, A2 CL 15. 95 ounces 16. 16 15. 74 (. 73)(. 29) (. 73)(. 29) 182 • CHAPTER 6 TABLE 6-1 STATISTICAL QUALITY CONTROL Factors for three-sigma control limits of x and R-charts
Source: Factors adapted from the ASTM Manual on Quality Control of Materials. Sample Size n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Factor for x-Chart A2 1. 88 1. 02 0. 73 0. 58 0. 48 0. 42 0. 37 0. 34 0. 31 0. 29 0. 27 0. 25 0. 24 0. 22 0. 21 0. 20 0. 19 0. 19 0. 18 0. 17 0. 17 0. 16 0. 16 0. 15 Factors for R-Chart D3 D4 0 0 0 0 0 0. 08 0. 14 0. 18 0. 22 0. 26 0. 28 0. 31 0. 33 0. 35 0. 36 0. 38 0. 39 0. 40 0. 41 0. 43 0. 43 0. 44 0. 45 0. 46 3. 27 2. 57 2. 28 2. 11 2. 00 1. 92 1. 86 1. 82 1. 78 1. 74 1. 72 1. 69 1. 67 1. 65 1. 64 1. 62 1. 61 1. 60 1. 59 1. 58 1. 57 1. 56 1. 55 1. 54 Range (R) Charts
Range (R) chart A control chart that monitors changes in the dispersion or variability of process. Range (R) charts are another type of control chart for variables. Whereas x-bar charts measure shift in the central tendency of the process, range charts monitor the dispersion or variability of the process. The method for developing and using R-charts is the same as that for x-bar charts. The center line of the control chart is the average range, and the upper and lower control limits are computed as follows: CL UCL LCL R D4 R D3 R where values for D4 and D3 are obtained from Table 6-1. CONTROL CHARTS FOR VARIABLES • 183
The quality control inspector at Cocoa Fizz would like to develop a range (R) chart in order to monitor volume dispersion in the bottling process. Use the data from Example 6. 1 to develop control limits for the sample range. EXAMPLE 6. 3 Constructing a Range (R) Chart • Solution From the data in Example 6. 1 you can see that the average sample range is: 7. 17 25 0. 29 4 R R n From Table 6-1 for n 4: D4 D3 UCL LCL The resulting control chart is: 2. 28 0 D4 R D3 R 2. 28 (0. 29) 0 (0. 29) 0 0. 6612 0. 70 0. 60 0. 50 0. 40 0. 30 0. 20 0. 10 Ounces 0. 00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
LCL CL UCL Sample Mean Using Mean and Range Charts Together You can see that mean and range charts are used to monitor different variables. The mean or x-bar chart measures the central tendency of the process, whereas the range chart measures the dispersion or variance of the process. Since both variables are important, it makes sense to monitor a process using both mean and 184 • CHAPTER 6 FIGURE 6-6 STATISTICAL QUALITY CONTROL Process shifts captured by x-charts and R-charts 15. 8 15. 9 16. 0 16. 1 16. 2 Mean 15. 8 15. 9 UCL R-chart LCL LCL – (a) Shift in mean detected by x-chart but not by R-chart 16. 0 16. 16. 2 Mean – x -chart UCL 15. 8 15. 9 UCL 16. 0 16. 1 16. 2 Mean 15. 8 15. 9 UCL 16. 0 16. 1 16. 2 Mean – x -chart R-chart LCL LCL – (b) Shift in dispersion detected by R-chart but not by x-chart range charts. It is possible to have a shift in the mean of the product but not a change in the dispersion. For example, at the Cocoa Fizz bottling plant the machine setting can shift so that the average bottle ? lled contains not 16. 0 ounces, but 15. 9 ounces of liquid. The dispersion could be the same, and this shift would be detected by an x-bar chart but not by a range chart. This is shown in part (a) of Figure 6-6.
On the other hand, there could be a shift in the dispersion of the product without a change in the mean. Cocoa Fizz may still be producing bottles with an average ? ll of 16. 0 ounces. However, the dispersion of the product may have increased, as shown in part (b) of Figure 6-6. This condition would be detected by a range chart but not by an x-bar chart. Because a shift in either the mean or the range means that the process is out of control, it is important to use both charts to monitor the process. CONTROL CHARTS FOR ATTRIBUTES Control charts for attributes are used to measure quality characteristics that are counted rather than measured.
Attributes are discrete in nature and entail simple yes-or-no decisions. For example, this could be the number of nonfunctioning lightbulbs, the proportion of broken eggs in a carton, the number of rotten apples, the number of scratches on a tile, or the number of complaints issued. Two CONTROL CHARTS FOR ATTRIBUTES • 185 of the most common types of control charts for attributes are p-charts and c-charts. P-charts are used to measure the proportion of items in a sample that are defective. Examples are the proportion of broken cookies in a batch and the proportion of cars produced with a misaligned fender.
P-charts are appropriate when both the number of defectives measured and the size of the total sample can be counted. A proportion can then be computed and used as the statistic of measurement. C-charts count the actual number of defects. For example, we can count the number of complaints from customers in a month, the number of bacteria on a petri dish, or the number of barnacles on the bottom of a boat. However, we cannot compute the proportion of complaints from customers, the proportion of bacteria on a petri dish, or the proportion of barnacles on the bottom of a boat.
Problem-Solving Tip: The primary difference between using a p-chart and a c-chart is as follows. A p-chart is used when both the total sample size and the number of defects can be computed. A c-chart is used when we can compute only the number of defects but cannot compute the proportion that is defective. P-Charts P-charts are used to measure the proportion that is defective in a sample. The computation of the center line as well as the upper and lower control limits is similar to the computation for the other kinds of control charts. The center line is computed as the average proportion defective in the population, p.
This is obtained by taking a number of samples of observations at random and computing the average value of p across all samples. To construct the upper and lower control limits for a p-chart, we use the following formulas: UCL LCL p p z z p p P-chart A control chart that monitors the proportion of defects in a sample. where z p p standard normal variable the sample proportion defective the standard deviation of the average proportion defective As with the other charts, z is selected to be either 2 or 3 standard deviations, depending on the amount of data we wish to capture in our control limits. Usually, however, they are set at 3.
The sample standard deviation is computed as follows: p v p(1 n p) where n is the sample size. 186 • CHAPTER 6 STATISTICAL QUALITY CONTROL EXAMPLE 6. 4 Constructing a p-Chart A production manager at a tire manufacturing plant has inspected the number of defective tires in twenty random samples with twenty observations each. Following are the number of defective tires found in each sample: Sample Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total Number of Defective Tires 3 2 1 2 1 3 3 2 1 2 3 2 2 1 1 2 4 3 1 1 40 Number of Observations Sampled 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 400 Fraction Defective . 5 . 10 . 05 . 10 . 05 . 15 . 15 . 10 . 05 . 10 . 15 . 10 . 10 . 05 .05 . 10 . 20 . 15 . 05 . 05 Construct a three-sigma control chart (z 3) with this information. • Solution The center line of the chart is total number of defective tires total number of observations 40 400 . 067 . 301 . 101 9: 0 CL p .10 p v p p p(1 n p) v .10 . 10 (. 10)(. 90) 20 3(. 067) 3(. 067) UCL LCL z ( p) z ( p) In this example the lower control limit is negative, which sometimes occurs because the computation is an approximation of the binomial distribution. When this occurs, the LCL is rounded up to zero because we cannot have a negative control limit.
CONTROL CHARTS FOR ATTRIBUTES • 187 The resulting control chart is as follows: 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 1 2 3 4 5 6 7 8 9 10 11 12 Sample Number 13 14 15 16 17 18 19 20 Fraction Defective (p) LCL CL UCL p This can also be computed using a spreadsheet as shown below. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 B C D Constructing a p-Chart Size of Each Sample Number Samples # Defective Tires 3 2 1 2 1 3 3 2 1 2 3 2 2 1 1 2 4 3 1 1 20 20 Fraction Defective 0. 15 0. 10 0. 05 0. 10 0. 05 0. 15 0. 15 0. 10 0. 05 0. 10 0. 15 0. 10 0. 10 0. 05 0. 05 0. 10 0. 20 0. 5 0. 05 0. 05 C8: =B8/C$4 Sample # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A B 29 Computations for p-Chart p bar = 30 Sigma_p = 31 32 Z-value for control charts = 33 CL: Center Line = 34 35 LCL: Lower Control Limit = 36 UCL: Upper Control Limit = C 0. 100 0. 067 3 0. 100 0. 000 0. 301 D E F C29: =SUM(B8:B27)/(C4*C5) C30: =SQRT((C29*(1-C29))/C4) C33: =C29 C34: =MAX(C$29-C$31*C$30,0) C35: =C$29+C$31*C$30 188 • CHAPTER 6 STATISTICAL QUALITY CONTROL C-CHARTS C-chart A control chart used to monitor the number of defects per unit. C-charts are used to monitor the number of defects per unit.
Examples are the number of returned meals in a restaurant, the number of trucks that exceed their weight limit in a month, the number of discolorations on a square foot of carpet, and the number of bacteria in a milliliter of water. Note that the types of units of measurement we are considering are a period of time, a surface area, or a volume of liquid. The average number of defects, c, is the center line of the control chart. The upper and lower control limits are computed as follows: UCL LCL c c z vc z vc EXAMPLE 6. 5 Computing a C-Chart The number of weekly customer complaints are monitored at a large hotel using a c-chart.
Complaints have been recorded over the past twenty weeks. Develop three-sigma control limits using the following data: Tota 20 3 44 Week 1 2 3 4 5 6 7 8 No. of Complaints 3 2 3 1 3 3 2 1 9 3 10 11 12 13 14 15 16 17 18 19 1 3 4 2 1 1 1 3 2 2 • Solution The average number of complaints per week is 44 20 UCL LCL c c z vc z vc 2. 2 2. 2 2. 2. Therefore, c 3v2. 2 3v2. 2 6. 65 2. 25 9: 0 2. 2. As in the previous example, the LCL is negative and should be rounded up to zero. Following is the control chart for this example: 7 6 5 4 Complaints Per Week 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Week 11 12 13 14 15 16 17 18 19 20
LCL CL UCL p C-CHARTS • 189 This can also be computed using a spreadsheet as shown below. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 B Computing a C-Chart Number of Complaints 3 2 3 1 3 3 2 1 3 1 3 4 2 1 1 1 3 2 2 3 Week 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A B 26 Computations for a C-Chart 27 c bar = 28 Z-value for control charts = 29 30 Sigma_c = 31 32 CL: Center Line = 33 LCL: Lower Control Limit = 34 UCL: Upper Control Limit = C 2. 2 3 1. 4832397 2. 20 0. 00 6. 65 D E F G C27: =AVERAGE(B5:B24) C30: =SQRT(C27) C31: =C26 C32: =MAX(C$26-C$27*C$29,0) C33: =C$26+C$27*C$29
Before You Go On We have discussed several types of statistical quality control (SQC) techniques. One category of SQC techniques consists of descriptive statistics tools such as the mean, range, and standard deviation. These tools are used to describe quality characteristics and relationships. Another category of SQC techniques consists of statistical process control (SPC) methods that are used to monitor changes in the production process. To understand SPC methods you must understand the differences between common and assignable causes of variation. Common 190 • CHAPTER 6 STATISTICAL QUALITY CONTROL auses of variation are based on random causes that cannot be identi? ed. A certain amount of common or normal variation occurs in every process due to differences in materials, workers, machines, and other factors. Assignable causes of variation, on the other hand, are variations that can be identi? ed and eliminated. An important part of statistical process control (SPC) is monitoring the production process to make sure that the only variations in the process are those due to common or normal causes. Under these conditions we say that a production process is in a state of control.
You should also understand the different types of quality control charts that are used to monitor the production process: x-bar charts, R-range charts, p-charts, and c-charts. PROCESS CAPABILITY So far we have discussed ways of monitoring the production process to ensure that it is in a state of control and that there are no assignable causes of variation. A critical aspect of statistical quality control is evaluating the ability of a production process to meet or exceed preset speci? cations. This is called process capability. To understand exactly what this means, let’s look more closely at the term speci? ation. Product speci? cations, often called tolerances, are preset ranges of acceptable quality characteristics, such as product dimensions. For a product to be considered acceptable, its characteristics must fall within this preset range. Otherwise, the product is not acceptable. Product speci? cations, or tolerance limits, are usually established by design engineers or product design specialists. For example, the speci? cations for the width of a machine part may be speci? ed as 15 inches . 3. This means that the width of the part should be 15 inches, though it is acceptable if it falls within the limits of 14. inches and 15. 3 inches. Similarly, for Cocoa Fizz, the average bottle ? ll may be 16 ounces with tolerances of . 2 ounces. Although the bottles should be ? lled with 16 ounces of liquid, the amount can be as low as 15. 8 or as high as 16. 2 ounces. Speci? cations for a product are preset on the basis of how the product is going to be used or what customer expectations are. As we have learned, any production process has a certain amount of natural variation associated with it. To be capable of producing an acceptable product, the process variation cannot exceed the preset speci? ations. Process capability thus involves evaluating process variability relative to preset product speci? cations in order to determine whether the process is capable of producing an acceptable product. In this section we will learn how to measure process capability. Process capability The ability of a production process to meet or exceed preset speci? cations. Product speci? cations Preset ranges of acceptable quality characteristics. Measuring Process Capability Simply setting up control charts to monitor whether a process is in control does not guarantee process capability.
To produce an acceptable product, the process must be capable and in control before production begins. Let’s look at three examples of process variation relative to design speci? cations for the Cocoa Fizz soft drink company. Let’s say that the speci? cation for the acceptable volume of liquid is preset at 16 ounces . 2 ounces, which is 15. 8 and 16. 2 ounces. In part (a) of Figure 6-7 the process produces 99. 74 percent (three sigma) of the product with volumes between 15. 8 and 16. 2 ounces. You can see that the process variability closely matches the preset speci? cations.
Almost all the output falls within the preset speci? cation range. PROCESS CAPABILITY • 191 In part (b) of Figure 6-7, however, the process produces 99. 74 percent (three sigma) of the product with volumes between 15. 7 and 16. 3 ounces. The process variability is outside the preset speci? cations. A large percentage of the product will fall outside the speci? ed limits. This means that the process is not capable of producing the product within the preset speci? cations. Part (c) of Figure 6-7 shows that the production process produces 99. 74 percent (three sigma) of the product with volumes between 15. and 16. 1 ounces. In this case the process variability is within speci? cations and the process exceeds the minimum capability. Process capability is measured by the process capability index, Cp, which is computed as the ratio of the speci? cation width to the width of the process variability: Cp speci? cation width process width USL 6 LSL Process capability index An index used to measure process capability. where the speci? cation width is the difference between the upper speci? cation limit (USL) and the lower speci? cation limit (LSL) of the process. The process width is Specification Width LSL USL
Specification Width LSL USL 15. 7 15. 8 16. 0 16. 1 16. 2 Mean Process Variability ±3? 15. 9 16. 3 15. 7 15. 8 15. 9 16. 0 16. 1 16. 2 16. 3 Process Variability ±3? (b) Process variability outside specification width (a) Process variability meets specification width Specification Width LSL USL 15. 7 15. 8 15. 9 16. 0 Mean Process Variability ±3? 16. 1 16. 2 16. 3 FIGURE 6-7 Relationship between process variability and speci? cation width (c) Process variability within specification width 192 • CHAPTER 6 STATISTICAL QUALITY CONTROL computed as 6 standard deviations (6 ) of the process being monitored.
The reason we use 6 is that most of the process measurement (99. 74 percent) falls within 3 standard deviations, which is a total of 6 standard deviations. There are three possible ranges of values for Cp that also help us interpret its value: 1: A value of Cp equal to 1 means that the process variability just meets speci? cations, as in Figure 6-7(a). We would then say that the process is minimally capable. 1: A value of Cp below 1 means that the process variability is outside the Cp range of speci? cation, as in Figure 6-7(b). This means that the process is not capable of producing within speci? ation and the process must be improved. Cp 1: A value of Cp above 1 means that the process variability is tighter than specifications and the process exceeds minimal capability, as in Figure 6-7(c). Cp A Cp value of 1 means that 99. 74 percent of the products produced will fall within the speci? cation limits. This also means that . 26 percent (100% 99. 74%) of the products will not be acceptable. Although this percentage sounds very small, when we think of it in terms of parts per million (ppm) we can see that it can still result in a lot of defects. The number . 6 percent corresponds to 2600 parts per million (ppm) defective (0. 0026 1,000,000). That number can seem very high if we think of it in terms of 2600 wrong prescriptions out of a million, or 2600 incorrect medical procedures out of a million, or even 2600 malfunctioning aircraft out of a million. You can see that this number of defects is still high. The way to reduce the ppm defective is to increase process capability. EXAMPLE 6. 6 Three bottling machines at Cocoa Fizz are being evaluated for their capability: Bottling Machine A B C Standard Deviation . 05 . 1 . 2 Computing the CP Value at Cocoa Fizz
If speci? cations are set between 15. 8 and 16. 2 ounces, determine which of the machines are capable of producing within speci? cations. • Solution To determine the capability of each machine we need to divide the speci? cation width (USL LSL 16. 2 15. 8 . 4) by 6 for each machine: Bottling Machine A B C USL LSL . 4 . 4 . 4 6 . 3 . 6 1. 2 Cp USL 6 1. 33 . 67 . 33 LSL .05 . 1 . 2 Looking at the Cp values, only machine A is capable of ? lling bottles within speci? cations, because it is the only machine that has a Cp value at or above 1. PROCESS CAPABILITY • 193 Cp is valuable in measuring process capability.
However, it has one shortcoming: it assumes that process variability is centered on the speci? cation range. Unfortunately, this is not always the case. Figure 6-8 shows data from the Cocoa Fizz example. In the ? gure the speci? cation limits are set between 15. 8 and 16. 2 ounces, with a mean of 16. 0 ounces. However, the process variation is not centered; it has a mean of 15. 9 ounces. Because of this, a certain proportion of products will fall outside the speci? cation range. The problem illustrated in Figure 6-8 is not uncommon, but it can lead to mistakes in the computation of the Cp measure.
Because of this, another measure for process capability is used more frequently: Cpk min USL 3 , LSL 3 where the mean of the process the standard deviation of the process This measure of process capability helps us address a possible lack of centering of the process over the speci? cation range. To use this measure, the process capability of each half of the normal distribution is computed and the minimum of the two is used. Looking at Figure 6-8, we can see that the computed Cp is 1: Process mean: LSL USL Cp 15. 8 16. 2 0. 4 6(0. 067) 1 15. 9 0. 067 Process standard deviation The Cp value of 1. 0 leads us to conclude that the process is capable. However, from the graph you can see that the process is not centered on the speci? cation range Specification Width LSL USL FIGURE 6-8 Process variability not centered across speci? cation width 15. 7 15. 8 15. 9 Mean 16. 0 16. 1 16. 2 16. 3 Process Variability ±3 194 • CHAPTER 6 STATISTICAL QUALITY CONTROL and is producing out-of-spec products. Using only the Cp measure would lead to an incorrect conclusion in this case. Computing Cpk gives us a different answer and leads us to a different conclusion: Cpk Cpk Cpk Cpk min min USL 3 , LSL 3 6. 2 15. 9 15. 9 15. 8 , 3(. 1) 3(. 1) min (1. 00, 0. 33) . 1 . 3 . 33 The computed Cpk value is less than 1, revealing that the process is not capable. EXAMPLE 6. 7 Compute the Cpk measure of process capability for the following machine and interpret the ? ndings. What value would you have obtained with the Cp measure? Machine Data: USL LSL Process Process 110 50 10 70 Computing the Cpk Value • Solution To compute the Cpk measure of process capability: USL 3 LSL 3 Cpk min min , 110 60 60 50 , 3(10) 3(10) min (1. 67, 0. 33) 0. 33 This means that the process is not capable.
The Cp measure of process capability gives us the following measure, 60 6(10) Cp 1 leading us to believe that the process is capable. The reason for the difference in the measures is that the process is not centered on the speci? cation range, as shown in Figure 6-9. PROCESS CAPABILITY • 195 FIGURE 6-9 Process variability not centered across speci? cation width for Example 6. 7 LSL Specification Width USL Process capability of machines is a critical element of statistical process control. 30 50 60 75 90 110 Process Variability Six Sigma Quality
The term Six Sigma® was coined by the Motorola Corporation in the 1980s to describe the high level of quality the company was striving to achieve. Sigma ( ) stands for the number of standard deviations of the process. Recall that 3 sigma ( ) means that 2600 ppm are defective. The level of defects associated with Six Sigma is approximately 3. 4 ppm. Figure 6-10 shows a process distribution with quality levels of 3 sigma ( ) and 6 sigma ( ). You can see the difference in the number of defects produced. Six sigma quality A high level of quality associated with approximately 3. 4 defective parts per million.
LSL Number of defects USL FIGURE 6-10 PPM defective for 3 versus quality (not to scale) 6 2600 ppm 3. 4 ppm Mean ±3 ±6 196 • CHAPTER 6 STATISTICAL QUALITY CONTROL LINKS TO PRACTICE Motorola, Inc. www. motorola. com To achieve the goal of Six Sigma, Motorola has instituted a quality focus in every aspect of its organization. Before a product is designed, marketing ensures that product characteristics are exactly what customers want. Operations ensures that exact product characteristics can be achieved through product design, the manufacturing process, and the materials used.
The Six Sigma concept is an integral part of other functions as well. It is used in the ? nance and accounting departments to reduce costing errors and the time required to close the books at the end of the month. Numerous other companies, such as General Electric and Texas Instruments, have followed Motorola’s leadership and have also instituted the Six Sigma concept. In fact, the Six Sigma quality standard has become a benchmark in many industries. There are two aspects to implementing the Six Sigma concept. The ? rst is the use of technical tools to identify and eliminate causes of quality problems.
These technical tools include the statistical quality control tools discussed in this chapter. They also include the problem-solving tools discussed in Chapter 5, such as cause-and-effect diagrams, ? ow charts, and Pareto analysis. In Six Sigma programs the use of these technical tools is integrated throughout the entire organizational system. The second aspect of Six Sigma implementation is people involvement. In Six Sigma all employees have the training to use technical tools and are responsible for rooting out quality problems. Employees are given martial arts titles that re? ect their skills in the Six Sigma process.
Black belts and master black belts are individuals who have extensive training in the use of technical tools and are responsible for carrying out the implementation of Six Sigma. They are experienced individuals who oversee the measuring, analyzing, process controlling, and improving. They achieve this by acting as coaches, team leaders, and facilitators of the process of continuous improvement. Green belts are individuals who have suf? cient training in technical tools to serve on teams or on small individual projects. Successful Six Sigma implementation requires commitment from top company leaders.
These individuals must promote the process, eliminate barriers to implementation, and ensure that proper resources are available. A key individual is a champion of Six Sigma. This is a person who comes from the top ranks of the organization and is responsible for providing direction and overseeing all aspects of the process. ACCEPTANCE SAMPLING Acceptance sampling, the third branch of statistical quality control, refers to the process of randomly inspecting a certain number of items from a lot or batch in order to decide whether to accept or reject the entire batch. What makes acceptance
ACCEPTANCE SAMPLING • 197 sampling different from statistical process control is that acceptance sampling is performed either before or after the process, rather than during the process. Acceptance sampling before the process involves sampling materials received from a supplier, such as randomly inspecting crates of fruit that will be used in a restaurant, boxes of glass dishes that will be sold in a department store, or metal castings that will be used in a machine shop. Sampling after the process involves sampling ? nished items that are to be shipped either to a customer or to a distribution center.
Examples include randomly testing a certain number of computers from a batch to make sure they meet operational requirements, and randomly inspecting snowboards to make sure that they are not defective. You may be wondering why we would only inspect some items in the lot and not the entire lot. Acceptance sampling is used when inspecting every item is not physically possible or would be overly expensive, or when inspecting a large number of items would lead to errors due to worker fatigue. This last concern is especially important when a large number of items are processed in a short period of time.
Another example of when acceptance sampling would be used is in destructive testing, such as testing eggs for salmonella or vehicles for crash testing. Obviously, in these cases it would not be helpful to test every item! However, 100 percent inspection does make sense if the cost of inspecting an item is less than the cost of passing on a defective item. As you will see in this section, the goal of acceptance sampling is to determine the criteria for acceptance or rejection based on the size of the lot, the size of the sample, and the level of con? ence we wish to attain. Acceptance sampling can be used for both attribute and variable measures, though it is most commonly used for attributes. In this section we will look at the different types of sampling plans and at ways to evaluate how well sampling plans discriminate between good and bad lots. Sampling involves randomly inspecting items from a lot. Sampling Plans A sampling plan is a plan for acceptance sampling that precisely speci? es the parameters of the sampling process and the acceptance/rejection criteria. The variables to be speci? d include the size of the lot (N), the size of the sample inspected from the lot (n), the number of defects above which a lot is rejected (c), and the number of samples that will be taken. There are different types of sampling plans. Some call for single sampling, in which a random sample is drawn from every lot. Each item in the sample is examined and is labeled as either “good” or “bad. ” Depending on the number of defects or “bad” items found, the entire lot is either accepted or rejected. For example, a lot size of 50 cookies is evaluated for acceptance by randomly inspecting 10 cookies from the lot.
The cookies may be inspected to make sure they are not broken or burned. If 4 or more of the 10 cookies inspected are bad, the entire lot is rejected. In this example, the lot size N 50, the sample size n 10, and the maximum number of defects at which a lot is accepted is c 4. These parameters de? ne the acceptance sampling plan. Another type of acceptance sampling is called double sampling. This provides an opportunity to sample the lot a second time if the results of the ? rst sample are inconclusive. In double sampling we ? rst sample a lot of goods according to preset criteria for de? nite acceptance or rejection.
However, if the results fall in the middle range, Sampling plan A plan for acceptance sampling that precisely speci? es the parameters of the sampling process and the acceptance/rejection criteria. 198 • CHAPTER 6 STATISTICAL QUALITY CONTROL they are considered inconclusive and a second sample is taken. For example, a water treatment plant may sample the quality of the water ten times in random intervals throughout the day. Criteria may be set for acceptable or unacceptable water quality, such as . 05 percent chlorine and . 1 percent chlorine. However, a sample of water containing between . 05 percent and . percent chlorine is inconclusive and calls for a second sample of water. In addition to single and double-sampling plans, there are multiple sampling plans. Multiple sampling plans are similar to double sampling plans except that criteria are set for more than two samples. The decision as to which sampling plan to select has a great deal to do with the cost involved in sampling, the time consumed by sampling, and the cost of passing on a defective item. In general, if the cost of collecting a sample is relatively high, single sampling is preferred. An extreme example is collecting a biopsy from a hospital patient.
Because the actual cost of getting the sample is high, we want to get a large sample and sample only once. The opposite is true when the cost of collecting the sample is low but the actual cost of testing is high. This may be the case with a water treatment plant, where collecting the water is inexpensive but the chemical analysis is costly. In this section we focus primarily on single sampling plans. Operating Characteristic (OC) Curves As we have seen, different sampling plans have different capabilities for discriminating between good and bad lots. At one extreme is 100 percent inspection, which has perfect discriminating power.
However, as the size of the sample inspected decreases, so does the chance of accepting a defective lot. We can show the discriminating power of a sampling plan on a graph by means of an operating characteristic (OC) curve. This curve shows the probability or chance of accepting a lot given various proportions of defects in the lot. Figure 6-11 shows a typical OC curve. The x axis shows the percentage of items that are defective in a lot. This is called “lot quality. ” The y axis shows the probability or chance of accepting a lot. You can see that if we use 100 percent inspection we are certain of accepting only lots with zero defects.
However, as the proportion of defects in the lot increases, our chance of accepting the lot decreases. For example, we have a 90 percent probability of accepting a lot with 5 percent defects and an 80 percent probability of accepting a lot with 8 percent defects. Regardless of which sampling plan we have selected, the plan is not perfect. That is, there is still a chance of accepting lots that are “bad” and rejecting “good” lots. The steeper the OC curve, the better our sampling plan is for discriminating between “good” and “bad. ” Figure 6-12 shows three different OC curves, A, B, and C.
Curve A is the most discriminating and curve C the least. You can see that the steeper the slope of the curve, the more discriminating is the sampling plan. When 100 percent inspection is not possible, there is a certain amount of risk for consumers in accepting defective lots and a certain amount of risk for producers in rejecting good lots. There is a small percentage of defects that consumers are willing to accept. This is called the acceptable quality level (AQL) and is generally in the order of 1 – 2 percent. However, sometimes the percentage of defects that passes through is higher than the AQL.
Consumers will usually tolerate a few more defects, but at some point the number of defects reaches a threshold level beyond which consumers will not tolerate them. This threshold level is called the lot tolerance percent defective (LTPD). The Operating characteristic (OC) curve A graph that shows the probability or chance of accepting a lot given various proportions of defects in the lot. Acceptable quality level (AQL) The small percentage of defects that consumers are willing to accept. Lot tolerance percent defective (LTPD) The upper limit of the percentage of defective items consumers are willing to tolerate.
ACCEPTANCE SAMPLING • 199 FIGURE 6-11 Example of an operating characteristic (OC) curve FIGURE 6-12 OC curves with different steepness levels and different levels of discrimination Probability or Chance of Accepting a Lot 1. 00 . 90 90% probability of accepting a lot with 5% defective items . 80 . 70 . 60 . 50 . 40 10% probability of accepting a lot with 24% defective items . 30 . 20 . 10 Curve B: Less Discrimination between “Good” and “Bad” Lots Curve C: Least Discrimination between “Good” and “Bad” Lots Curve A: Highest Discrimination between “Good” and “Bad” Lots Probability or Chance of Accepting a Lot . 00 . 90 . 80 . 70 . 60 . 50 . 40 . 30 . 20 . 10 . 05 . 10 . 15 . 20 . 25 . 30 Proportion of Defective Items in Lot (Lot Quality) . 35 .05 .10 . 15 . 20 . 25 . 30 Proportion of Defective Items in Lot (Lot Quality) .35 LTPD is the upper limit of the percentage of defective items consumers are willing to tolerate. Consumer’s risk is the chance or probability that a lot will be accepted that contains a greater number of defects than the LTPD limit. This is the probability of making a Type II error — that is, accepting a lot that is truly “bad. ” Consumer’s risk or Type II error is enerally denoted by beta ( ). The relationships among AQL, LTPD, and are shown in Figure 6-13. Producer’s risk is the chance or probability that a lot containing an acceptable quality level will be rejected. This is the probability of making a Type I error — that is, rejecting a lot that is “good. ” It is generally denoted by alpha ( ). Producer’s risk is also shown in Figure 6-13. We can determine from an OC curve what the consumer’s and producer’s risks are. However, these values should not be left to chance. Rather, sampling plans are usually designed to meet speci? levels of consumer’s and producer’s risk. For example, one common combination is to have a consumer’s risk ( ) of 10 percent and a producer’s risk ( ) of 5 percent, though many other combinations are possible. Consumer’s risk The chance of accepting a lot that contains a greater number of defects than the LTPD limit. Producer’s risk The chance that a lot containing an acceptable quality level will be rejected. Developing OC Curves An OC curve graphically depicts the discriminating power of a sampling plan. To draw an OC curve, we typically use a cumulative binomial distribution to obtain 00 • CHAPTER 6 FIGURE 6-13 STATISTICAL QUALITY CONTROL An OC curve showing producer’s risk ( ) and consumer’s risk ( ) Probability or Chance of Accepting a Lot 1. 00 . 90 . 80 . 70 . 60 . 50 . 40 . 30 . 20 . 10 . 05 AQL Good Lots . 10 . 15 Probability of accepting a “bad” lot (consumer’s risk ? ) Probability of rejecting a “good” lot (producer’s risk ? ) .20 LTPD .25 .30 .35 Poor Quality Tolerated Lot Quality Bad Quality Not Tolerated