Overview of statistical methods 283. Figure 9.5. Linearity illustrated.

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1 Overview of statistical methods 283 Figure 9.5. Linearity illustrated. OVERVIEW OF STATISTICAL METHODS Enumerative versus analytic statistical methods How would you respond to the following question? A sample of 100 bottles taken from a lling process has an average of ounces and a standard deviation of 0.1 ounce. The speci cations are 11.9^ 12.1 ounces. Based on these results, should you a. Do nothing? b. Adjust the average to precisely 12 ounces?

2 284 BASIC PRINCIPLES of MEASUREMENT c. Compute a confidence interval about the mean and adjust the process if the nominal fill level is not within the confidence interval? d. None of the above? The correct answer is d, none of the above. The other choices all make the mistake of applying enumerative statistical concepts to an analytic statistical situation. In short, the questions themselves are wrong! For example, based on the data, there is no way to determine if doing nothing is appropriate. Doing something implies that there is a known cause and effect mechanism which can be employed to reach a known objective. There is nothing to suggest that this situation exists. Thus, we can t simply adjust the process average to the nominal value of 12 ounces, even though the process appears to be 5 standard errors below this value. This might have happened because the first 50 were 10 standard errors below the nominal and the last 50 were exactly equal to the nominal (or any of a nearly infinite number of other possible scenarios). The confidence interval calculation fails for the same reason. Figure 9.6 illustrates some processes that could produce the statistics provided above. Some appropriate analytic statistics questions might be:. Is the process central tendency stable over time?. Is the process dispersion stable over time?. Does the process distribution change over time? If any of the above are answered no, then what is the cause of the instability? To help answer this question, ask what is the nature of the variation as revealed by the patterns? when plotted in time-sequence and stratified in various ways. If none of the above are answered no, then, and only then, we can ask such questions as. Is the process meeting the requirements?. Can the process meet the requirements?. Can the process be improved by recentering it?. How can we reduce variation in the process? WHAT ARE ENUMERATIVE AND ANALYTIC STUDIES? Deming (1975) defines enumerative and analytic studies as follows: Enumerative studyöa study in which action will be taken on the universe. Analytic studyöa study in which action will be taken on a process to improve performance in the future.

3 Overview of statistical methods 285 Figure 9.6. Possible processes with similar means and sigmas. The term universe is defined in the usual way: the entire group of interest, e.g., people, material, units of product, which possess certain properties of interest. An example of an enumerative study would be sampling an isolated lot to determine the quality of the lot. In an analytic study the focus is on a process and how to improve it. The focus is the future. Thus, unlike enumerative studies which make inferences about the universe actually studied, analytic studies are interested in a universe which has yet to be produced. Table 9.2 compares analytic studies with enumerative studies (Provost, 1988). Deming (1986) points out that Analysis of variance, t-tests, confidence intervals, and other statistical techniques taught in the books, however interesting, are inappropriate because they provide no basis for prediction and because they bury the information contained in the order of production.

4 286 BASIC PRINCIPLES of MEASUREMENT Table 9.2. Important aspects of analytic studies. ITEM ENUMERATIVE STUDY ANALYTIC STUDY Aim Parameter estimation Prediction Focus Universe Process Method of access Major source of uncertainty Uncertainty quanti able? Environment for the study Counts, statistics Sampling variation Yes Static Models of the process (e.g., ow charts, cause and e ects, mathematical models) Extrapolation into the future No Dynamic These traditional statistical methods have their place, but they are widely abused in the real world. When this is the case, statistics do more to cloud the issue than to enlighten. Analytic study methods provide information for inductive thinking, rather than the largely deductive approach of enumerative statistics. Analytic methods are primarily graphical devices such as run charts, control charts, histograms, interrelationship digraphs, etc. Analytic statistics provide operational guidelines, rather than precise calculations of probability. Thus, such statements as There is a 0.13% probability of a Type I error when acting on a point outside a three-sigma control limit are false (the author admits to having made this error in the past). The future cannot be predicted with a known level of confidence. Instead, based on knowledge obtained from every source, including analytic studies, one can state that one has a certain degree of belief (e.g., high, low) that such and such will result from such and such action on a process. Another difference between the two types of studies is that enumerative statistics proceed from predetermined hypotheses while analytic studies try

5 Overview of statistical methods 287 to help the analyst generate new hypotheses. In the past, this extremely worthwhile approach has been criticized by some statisticians as fishing or rationalizing. However, this author believes that using data to develop plausible explanations retrospectively is a perfectly legitimate way of creating new theories to be tested. To refuse to explore possibilities suggested by data is to take a very limited view of the scope of statistics in quality improvement and control. Enumerative statistical methods This section discusses the basic concept of statistical inference. The reader should also consult the Glossary in the Appendix for additional information. Inferential statistics belong to the enumerative class of statistical methods. The term inference is defined as 1) the act or process of deriving logical conclusions from premises known or assumed to be true, or 2) the act of reasoning from factual knowledge or evidence. Inferential statistics provide information that is used in the process of inference. As can be seen from the definitions, inference involves two domains: the premises and the evidence or factual knowledge. Additionally, there are two conceptual frameworks for addressing premises questions in inference: the design-based approach and the model-based approach. As discussed by Koch and Gillings (1983), a statistical analysis whose only assumptions are random selection of units or random allocation of units to experimental conditions results in design-based inferences; or, equivalently, randomization-based inferences. The objective is to structure sampling such that the sampled population has the same characteristics as the target population. If this is accomplished then inferences from the sample are said to have internal validity. A limitation on design-based inferences for experimental studies is that formal conclusions are restricted to the finite population of subjects that actually received treatment, that is, they lack external validity. However, if sites and subjects are selected at random from larger eligible sets, then models with random effects provide one possible way of addressing both internal and external validity considerations. One important consideration for external validity is that the sample coverage includes all relevant subpopulations; another is that treatment differences be homogeneous across subpopulations. A common application of design-based inference is the survey. Alternatively, if assumptions external to the study design are required to extend inferences to the target population, then statistical analyses based on postulated probability distributional forms (e.g., binomial, normal, etc.) or other stochastic processes yield model-based inferences. A focus of distinction

6 288 BASIC PRINCIPLES of MEASUREMENT between design-based and model-based studies is the population to which the results are generalized rather than the nature of the statistical methods applied. When using a model-based approach, external validity requires substantive justification for the model s assumptions, as well as statistical evaluation of the assumptions. Statistical inference is used to provide probabilistic statements regarding a scientific inference. Science attempts to provide answers to basic questions, such as can this machine meet our requirements? Is the quality of this lot within the terms of our contract? Does the new method of processing produce better results than the old? These questions are answered by conducting an experiment, which produces data. If the data vary, then statistical inference is necessary to interpret the answers to the questions posed. A statistical model is developed to describe the probabilistic structure relating the observed data to the quantity of interest (the parameters), i.e., a scientific hypothesis is formulated. Rules are applied to the data and the scientific hypothesis is either rejected or not. In formal tests of a hypothesis, there are usually two mutually exclusive and exhaustive hypotheses formulated: a null hypothesis and an alternate hypothesis. Formal hypothesis testing is discussed later in this chapter. DISCRETE AND CONTINUOUS DATA Data are said to be discrete when they take on only a finite number of points that can be represented by the non-negative integers. An example of discrete data is the number of defects in a sample. Data are said to be continuous when they exist on an interval, or on several intervals. An example of continuous data is the measurement of ph. Quality methods exist based on probability functions for both discrete and continuous data. METHODS OF ENUMERATION Enumeration involves counting techniques for very large numbers of possible outcomes. This occurs for even surprisingly small sample sizes. In Six Sigma, these methods are commonly used in a wide variety of statistical procedures. The basis for all of the enumerative methods described here is the multiplication principle. The multiplication principle states that the number of possible outcomes of a series of experiments is equal to the product of the number of outcomes of each experiment. For example, consider flipping a coin twice. On the first flip there are two possible outcomes (heads/tails) and on the second

7 348 MEASUREMENT SYSTEMS ANALYSIS Measurement Concept Interpretation for Attribute Data Table 10.7 (cont.) Suggested Metrics and Comments Stability The variability between attribute R&R studies at di erent times. Metric Repeatability Reproducibility Stability Measure for Metric Standard deviation of repeatabilities Standard deviation of reproducibilities Accuracy Standard deviation of accuracies Bias Average bias Linearity When an inspector evaluates items covering the full set of categories, her classi cations are consistent across the categories. Range of inaccuracy and bias across all categories. Requires knowledge of the true value. Note: Because there is no natural ordering for nominal data, the concept of linearity doesn t really have a precise analog for attribute data on this scale. However, the suggested metrics will highlight interactions between inspectors and speci c categories. Operational definitions An operational definition is defined as a requirement that includes a means of measurement. High quality solder is a requirement that must be operationalized by a clear definition of what high quality solder means. This might include verbal descriptions, magnification power, photographs, physical comparison specimens, and many more criteria. EXAMPLES OF OPERATIONAL DEFINITIONS 1. Operational de nition of the Ozone Transport Assessment Group s (OTAG) goal Goal: To identify reductions and recommend transported ozone and its precursors which, in combination with other measures, will enable attainment and maintenance of the ozone standard in the OTAG region.

8 Attribute measurement error analysis 349 Suggested operational de nition of the goal: 1. A general modeled reduction in ozone and ozone precursors aloft throughout the OTAG region; and 2. A reduction of ozone and ozone precursors both aloft and at ground level at the boundaries of non-attainment area modeling domains in the OTAG region; and 3. A minimization of increases in peak ground level ozone concentrations in the OTAG region. (This component of the operationalde nitionisinreview.) 2. Wellesley College Child Care Policy Research Partnership operational de nition of unmet need 1. Standard of comparison to judge the adequacy of neighborhood services: the median availability of services in the larger region (Hampden County). 2. Thus, our de nition of unmet need: The di erence between the care available in the neighborhood and the median level of care in the surrounding region (stated in terms of child care slots indexed to the age-appropriate child populationö slots-per-tots ). 3. Operational de nitions of acids and bases 1. An acid is any substance that increases the concentration of the H + ion when it dissolves in water. 2. A base is any substance that increases the concentration of the OH^ ion when it dissolves in water. 4. Operational de nition of intelligence 1. Administer the Stanford-Binet IQ test to a person and score the result. The person s intelligence is the score on the test. 5. Operational de nition of dark blue carpet A carpet will be deemed to be dark blue if 1. Judged by an inspector medically certi ed as having passed the U.S. Air Force test for color-blindness 1.1. It matches the PANTONEcolor card 7462 Cwhen both carpet and card are illuminated by GE cool white uorescent tubes; 1.2. Card and carpet are viewed at a distance between 16 inches and 24 inches. HOW TO CONDUCT ATTRIBUTE INSPECTION STUDIES Some commonly used approaches to attribute inspection analysis are shown in Table 10.8.

9 318 BASIC PRINCIPLES of MEASUREMENT assumptions, and it is simple. Resampling doesn t impose as much baggage between the engineering problem and the statistical result as conventional methods. It can also be used for more advanced problems, such as modeling, design of experiments, etc. For a discussion of the theory behind resampling, see Efron (1982). For a presentation of numerous examples using a resampling computer program see Simon (1992). PRINCIPLES OF STATISTICAL PROCESS CONTROL Terms and concepts DISTRIBUTIONS A central concept in statistical process control (SPC) is that every measurable phenomenon is a statistical distribution. In other words, an observed set of data constitutes a sample of the effects of unknown common causes. It follows that, after we have done everything to eliminate special causes of variations, there will still remain a certain amount of variability exhibiting the state of control. Figure 9.25 illustrates the relationships between common causes, special causes, and distributions. Figure Distributions. From Continuing Process Control and Process Capability Improvement, p. 4a. Copyright # 1983 by Ford Motor Company. Used by permission of the publisher. There are three basic properties of a distribution: location, spread, and shape. The location refers to the typical value of the distribution, such as the mean. The spread of the distribution is the amount by which smaller values differ from larger ones. The standard deviation and variance are measures of distribution spread. The shape of a distribution is its patternöpeakedness, symmetry, etc. A given phenomenon may have any one of a number of distribution shapes, e.g., the distribution may be bell-shaped, rectangular-shaped, etc.

10 CENTRAL LIMIT THEOREM The central limit theorem can be stated as follows: Principles of statistical process control 319 Irrespective of the shape of the distribution of the population or universe, the distribution of average values of samples drawn from that universe will tend toward a normal distribution as the sample size grows without bound. It can also be shown that the average of sample averages will equal the average of the universe and that the standard deviation of the averages equals the standard deviation of the universe divided by the square root of the sample size. Shewhart performed experiments that showed that small sample sizes were needed to get approximately normal distributions from even wildly non-normal universes. Figure 9.26 was created by Shewhart using samples of four measurements. Figure Illustration of the central limit theorem. From Economic Control of Quality of Manufactured Product, gure 59. Copyright # 1931, 1980 by ASQC Quality Press. Used by permission of the publisher.

11 320 BASIC PRINCIPLES of MEASUREMENT The practical implications of the central limit theorem are immense. Consider that without the central limit theorem effects, we would have to develop a separate statistical model for every non-normal distribution encountered in practice. This would be the only way to determine if the system were exhibiting chance variation. Because of the central limit theorem we can use averages of small samples to evaluate any process using the normal distribution. The central limit theorem is the basis for the most powerful of statistical process control tools, Shewhart control charts. Objectives and benefits Without SPC, the bases for decisions regarding quality improvement are based on intuition, after-the-fact product inspection, or seat-of-the-pants data analysis. SPC provides a scientific basis for decisions regarding process improvement. PREVENTION VERSUS DETECTION A process control system is essentially a feedback system that links process outcomes with process inputs. There are four main elements involved, the process itself, information about the process, action taken on the process, and action taken on the output from the process. The way these elements fit together is shown in Figure Figure A process control system.

12 Principles of statistical process control 321 By the process, we mean the whole combination of people, equipment, input materials, methods, and environment that work together to produce output. The performance information is obtained, in part, from evaluation of the process output. The output of a process includes more than product, it also includes information about the operating state of the process such as temperature, cycle times, etc. Action taken on a process is future-oriented in the sense that it will affect output yet to come. Action on the output is pastoriented because it involves detecting out-of-specification output that has already been produced. There has been a tendency in the past to concentrate attention on the detection-oriented strategy of product inspection. With this approach, we wait until an output has been produced, then the output is inspected and either accepted or rejected. SPC takes you in a completely different direction: improvement in the future. A key concept is the smaller the variation around the target, the better. Thus, under this school of thought, it is not enough to merely meet the requirements; continuous improvement is called for even if the requirements are already being met. The concept of never-ending, continuous improvement is at the heart of SPC and Six Sigma. Common and special causes of variation Shewhart (1931, 1980) defined control as follows: A phenomenon will be said to be controlled when, through the use of past experience, we can predict, at least within limits, how the phenomenon may be expected to vary in the future. Here it is understood that prediction within limits means that we can state, at least approximately, the probability that the observed phenomenon will fall within the given limits. The critical point in this definition is that control is not defined as the complete absence of variation. Control is simply a state where all variation is predictable variation. A controlled process isn t necessarily a sign of good management, nor is an out-of-control process necessarily producing non-conforming product. In all forms of prediction there is an element of risk. For our purposes, we will call any unknown random cause of variation a chance cause or a common cause, the terms are synonymous and will be used as such. If the influence of any particular chance cause is very small, and if the number of chance causes of variation are very large and relatively constant, we have a situation where the variation is predictable within limits. You can see from the definition above, that a system such as this qualifies as a controlled system. Where Dr. Shewhart used the term chance cause, Dr. W. Edwards Deming coined the

13 Figure Should these variations be left to chance? From Economic Control of Quality of Manufactured Product,p.13.Copyright# 1931, 1980 by ASQC Quality Press. Used by permission of the publisher. Figure Types of variation. 322

14 Principles of statistical process control 323 Figure Charts from Figure 9.28 with control limits shown. From Economic Control of Quality of Manufactured Product, p. 13. Copyright # 1931, 1980 by ASQC Quality Press. Used by permission of the publisher. term common cause to describe the same phenomenon. Both terms are encountered in practice. Needless to say, not all phenomena arise from constant systems of common causes. At times, the variation is caused by a source of variation that is not part of the constant system. These sources of variation were called assignable causes by Shewhart, special causes of variation by Deming. Experience indicates that

15 324 BASIC PRINCIPLES of MEASUREMENT special causes of variation can usually be found without undue difficulty, leading to a process that is less variable. Statistical tools are needed to help us effectively separate the effects of special causes of variation from chance cause variation. This leads us to another definition: Statistical process controlöthe use of valid analytical statistical methods to identify the existence of special causes of variation in a process. The basic rule of statistical process control is: Variation from common-cause systems should be left to chance, but special causes of variation should be identi ed and eliminated. This is Shewhart s original rule. However, the rule should not be misinterpreted as meaning that variation from common causes should be ignored. Rather, common-cause variation is explored off-line. That is, we look for long-term process improvements to address common-cause variation. Figure 9.28 illustrates the need for statistical methods to determine the category of variation. The answer to the question should these variations be left to chance? can only be obtained through the use of statistical methods. Figure 9.29 illustrates the basic concept. In short, variation between the two control limits designated by the dashed lines will be deemed as variation from the common-cause system. Any variability beyond these fixed limits will be assumed to have come from special causes of variation. We will call any system exhibiting only common-cause variation, statistically controlled. It must be noted that the control limits are not simply pulled out of the air, they are calculated from actual process data using valid statistical methods. Figure 9.28 is shown below as Figure 9.30, only with the control limits drawn on it; notice that process (a) is exhibiting variations from special causes, while process (b) is not. This implies that the type of action needed to reduce the variability in each case is of a different nature. Without statistical guidance there could be endless debate over whether special or common causes were to blame for variability.

16 Statistical process control (SPC) 429 TAMPERING EFFECTS AND DIAGNOSIS Tampering occurs when adjustments are made to a process that is in statistical control. Adjusting a controlled process will always increase process variability, an obviously undesirable result. The best means of diagnosing tampering is to conduct a process capability study (see Chapter 13) and to use a control chart to provide guidelines for adjusting the process. Perhaps the best analysis of the effects of tampering is from Deming (1986). Deming describes four common types of tampering by drawing the analogy of aiming a funnel to hit a desired target. These funnel rules are described by Deming (1986, p. 328): 1. Leave the funnel xed, aimed at the target, no adjustment. 2. At drop k(k¼ 1, 2, 3,...) the marble will come to rest at point z k, measured from the target. (In other words, z k is the error at drop k.) Move the funnel the distance z k from the last position. Memory Set the funnel at each drop right over the spot z k, measured from the target. No memory. 4. Setthefunnelateachdroprightoverthespot(z k )whereitlastcameto rest. No memory. Rule #1 is the best rule for stable processes. By following this rule, the process average will remain stable and the variance will be minimized. Rule #2 produces a stable output but one with twice the variance of rule #1. Rule #3 results in a system that explodes, i.e., a symmetrical pattern will appear with a variance that increases without bound. Rule #4 creates a pattern that steadily moves away from the target, without limit (see figure 12.20). At first glance, one might wonder about the relevance of such apparently abstract rules. However, upon more careful consideration, one finds many practical situations where these rules apply. Rule #1 is the ideal situation and it can be approximated by using control charts to guide decision-making. If process adjustments are made only when special causes are indicated and identified, a pattern similar to that produced by rule #1 will result. Rule #2 has intuitive appeal for many people. It is commonly encountered in such activities as gage calibration (check the standard once and adjust the gage accordingly) or in some automated equipment (using an automatic gage, check the size of the last feature produced and make a compensating adjustment). Since the system produces a stable result, this situation can go unnoticed indefinitely. However, as shown by Taguchi (1986), increased variance translates to poorer quality and higher cost. The rationale that leads to rule #3 goes something like this: A measurement was taken and it was found to be 10 units above the desired target. This hap-

17 430 STATISTICAL PROCESS CONTROL TECHNIQUES pened because the process was set 10 units too high. I want the average to equal the target. To accomplish this I must try to get the next unit to be 10 units too low. This might be used, for example, in preparing a chemical solution. While reasonable on its face, the result of this approach is a wildly oscillating system. A common example of rule #4 is the train-the-trainer method. A master spends a short time training a group of experts, who then train others, who train others, etc. An example is on-the-job training. Another is creating a setup by using a piece from the last job. Yet another is a gage calibration system where standards are used to create other standards, which are used to create still others, and so on. Just how far the final result will be from the ideal depends on how many levels deep the scheme has progressed Rule # Rule # Rule # Rule #4 Figure Funnel rule simulation results. SHORT RUN STATISTICAL PROCESS CONTROL TECHNIQUES Short production runs are a way of life with many manufacturing companies. In the future, this will be the case even more often. The trend in manufacturing has been toward smaller production runs with product tailored to the specific needs of individual customers. Henry Ford s days of the customer can have any color, as long as it s black have long since passed. Classical SPC methods, such as X and R charts, were developed in the era of mass production of identical parts. Production runs often lasted for weeks, months, or even years. Many of the SPC rules of thumb currently in use