Probability & Statistics - Chapter 1 - PDF

Chapter 1 Introduction to Statistics and Data Analysis 1.1 Overview: Statistical Inference, Samples, Populations, and the Role of Probability Beginning in the 1980s and continuing into the 21st century, an inordinate amount of attention has been focused on improvement of quality in American industry. Much has been said and written about the Japanese “industrial miracle,” which began in the middle of the 20th century. The Japanese were able to succeed where we and other countries had failed–namely, to create an atmosphere that allows the production of high-quality products. Much of the success of the Japanese has been attributed to the use of statistical methods and statistical thinking among management personnel. Use of Scientiﬁc Data The use of statistical methods in manufacturing, development of food products, computer software, energy sources, pharmaceuticals, and many other areas involves the gathering of information or scientiﬁc data. Of course, the gathering of data is nothing new. It has been done for well over a thousand years. Data have been collected, summarized, reported, and stored for perusal. However, there is a profound distinction between collection of scientiﬁc information and inferential statistics. It is the latter that has received rightful attention in recent decades. The oﬀspring of inferential statistics has been a large “toolbox” of statistical methods employed by statistical practitioners. These statistical methods are de- signed to contribute to the process of making scientiﬁc judgments in the face of uncertainty and variation. The product density of a particular material from a manufacturing process will not always be the same. Indeed, if the process involved is a batch process rather than continuous, there will be not only variation in ma- terial density among the batches that come oﬀ the line (batch-to-batch variation), but also within-batch variation. Statistical methods are used to analyze data from a process such as this one in order to gain more sense of where in the process changes may be made to improve the quality of the process. In this process, qual- 1 2 Chapter 1 Introduction to Statistics and Data Analysis ity may well be deﬁned in relation to closeness to a target density value in harmony with what portion of the time this closeness criterion is met. An engineer may be concerned with a speciﬁc instrument that is used to measure sulfur monoxide in the air during pollution studies. If the engineer has doubts about the eﬀectiveness of the instrument, there are two sources of variation that must be dealt with. The ﬁrst is the variation in sulfur monoxide values that are found at the same locale on the same day. The second is the variation between values observed and the true amount of sulfur monoxide that is in the air at the time. If either of these two sources of variation is exceedingly large (according to some standard set by the engineer), the instrument may need to be replaced. In a biomedical study of a new drug that reduces hypertension, 85% of patients experienced relief, while it is generally recognized that the current drug, or “old” drug, brings relief to 80% of pa- tients that have chronic hypertension. However, the new drug is more expensive to make and may result in certain side eﬀects. Should the new drug be adopted? This is a problem that is encountered (often with much more complexity) frequently by pharmaceutical ﬁrms in conjunction with the FDA (Federal Drug Administration). Again, the consideration of variation needs to be taken into account. The “85%” value is based on a certain number of patients chosen for the study. Perhaps if the study were repeated with new patients the observed number of “successes” would be 75%! It is the natural variation from study to study that must be taken into account in the decision process. Clearly this variation is important, since variation from patient to patient is endemic to the problem. Variability in Scientiﬁc Data In the problems discussed above the statistical methods used involve dealing with variability, and in each case the variability to be studied is that encountered in scientiﬁc data. If the observed product density in the process were always the same and were always on target, there would be no need for statistical methods. If the device for measuring sulfur monoxide always gives the same value and the value is accurate (i.e., it is correct), no statistical analysis is needed. If there were no patient-to-patient variability inherent in the response to the drug (i.e., it either always brings relief or not), life would be simple for scientists in the pharmaceutical ﬁrms and FDA and no statistician would be needed in the decision process. Statistics researchers have produced an enormous number of analytical methods that allow for analysis of data from systems like those described above. This reﬂects the true nature of the science that we call inferential statistics, namely, using techniques that allow us to go beyond merely reporting data to drawing conclusions (or inferences) about the scientiﬁc system. Statisticians make use of fundamental laws of probability and statistical inference to draw conclusions about scientiﬁc systems. Information is gathered in the form of samples, or collections of observations. The process of sampling is introduced in Chapter 2, and the discussion continues throughout the entire book. Samples are collected from populations, which are collections of all individ- uals or individual items of a particular type. At times a population signiﬁes a scientiﬁc system. For example, a manufacturer of computer boards may wish to eliminate defects. A sampling process may involve collecting information on 50 computer boards sampled randomly from the process. Here, the population is all 1.1 Overview: Statistical Inference, Samples, Populations, and the Role of Probability 3 computer boards manufactured by the ﬁrm over a speciﬁc period of time. If an improvement is made in the computer board process and a second sample of boards is collected, any conclusions drawn regarding the eﬀectiveness of the change in pro- cess should extend to the entire population of computer boards produced under the “improved process.” In a drug experiment, a sample of patients is taken and each is given a speciﬁc drug to reduce blood pressure. The interest is focused on drawing conclusions about the population of those who suﬀer from hypertension. Often, it is very important to collect scientiﬁc data in a systematic way, with planning being high on the agenda. At times the planning is, by necessity, quite limited. We often focus only on certain properties or characteristics of the items or objects in the population. Each characteristic has particular engineering or, say, biological importance to the “customer,” the scientist or engineer who seeks to learn about the population. For example, in one of the illustrations above the quality of the process had to do with the product density of the output of a process. An engineer may need to study the eﬀect of process conditions, temperature, humidity, amount of a particular ingredient, and so on. He or she can systematically move these factors to whatever levels are suggested according to whatever prescription or experimental design is desired. However, a forest scientist who is interested in a study of factors that inﬂuence wood density in a certain kind of tree cannot necessarily design an experiment. This case may require an observational study in which data are collected in the ﬁeld but factor levels can not be preselected. Both of these types of studies lend themselves to methods of statistical inference. In the former, the quality of the inferences will depend on proper planning of the experiment. In the latter, the scientist is at the mercy of what can be gathered. For example, it is sad if an agronomist is interested in studying the eﬀect of rainfall on plant yield and the data are gathered during a drought. The importance of statistical thinking by managers and the use of statistical inference by scientiﬁc personnel is widely acknowledged. Research scientists gain much from scientiﬁc data. Data provide understanding of scientiﬁc phenomena. Product and process engineers learn a great deal in their oﬀ-line eﬀorts to improve the process. They also gain valuable insight by gathering production data (on- line monitoring) on a regular basis. This allows them to determine necessary modiﬁcations in order to keep the process at a desired level of quality. There are times when a scientiﬁc practitioner wishes only to gain some sort of summary of a set of data represented in the sample. In other words, inferential statistics is not required. Rather, a set of single-number statistics or descriptive statistics is helpful. These numbers give a sense of center of the location of the data, variability in the data, and the general nature of the distribution of observations in the sample. Though no speciﬁc statistical methods leading to statistical inference are incorporated, much can be learned. At times, descriptive statistics are accompanied by graphics. Modern statistical software packages allow for computation of means, medians, standard deviations, and other single- number statistics as well as production of graphs that show a “footprint” of the nature of the sample. Deﬁnitions and illustrations of the single-number statistics and graphs, including histograms, stem-and-leaf plots, scatter plots, dot plots, and box plots, will be given in sections that follow. 4 Chapter 1 Introduction to Statistics and Data Analysis The Role of Probability In this book, Chapters 2 to 6 deal with fundamental notions of probability. A thorough grounding in these concepts allows the reader to have a better under- standing of statistical inference. Without some formalism of probability theory, the student cannot appreciate the true interpretation from data analysis through modern statistical methods. It is quite natural to study probability prior to study- ing statistical inference. Elements of probability allow us to quantify the strength or “conﬁdence” in our conclusions. In this sense, concepts in probability form a major component that supplements statistical methods and helps us gauge the strength of the statistical inference. The discipline of probability, then, provides the transition between descriptive statistics and inferential methods. Elements of probability allow the conclusion to be put into the language that the science or engineering practitioners require. An example follows that will enable the reader to understand the notion of a P -value, which often provides the “bottom line” in the interpretation of results from the use of statistical methods. Example 1.1: Suppose that an engineer encounters data from a manufacturing process in which 100 items are sampled and 10 are found to be defective. It is expected and antic- ipated that occasionally there will be defective items. Obviously these 100 items represent the sample. However, it has been determined that in the long run, the company can only tolerate 5% defective in the process. Now, the elements of prob- ability allow the engineer to determine how conclusive the sample information is regarding the nature of the process. In this case, the population conceptually represents all possible items from the process. Suppose we learn that if the process is acceptable, that is, if it does produce items no more than 5% of which are de- fective, there is a probability of 0.0282 of obtaining 10 or more defective items in a random sample of 100 items from the process. This small probability suggests that the process does, indeed, have a long-run rate of defective items that exceeds 5%. In other words, under the condition of an acceptable process, the sample in- formation obtained would rarely occur. However, it did occur! Clearly, though, it would occur with a much higher probability if the process defective rate exceeded 5% by a signiﬁcant amount. From this example it becomes clear that the elements of probability aid in the translation of sample information into something conclusive or inconclusive about the scientiﬁc system. In fact, what was learned likely is alarming information to the engineer or manager. Statistical methods, which we will actually detail in Chapter 10, produced a P -value of 0.0282. The result suggests that the process very likely is not acceptable. The concept of a P-value is dealt with at length in succeeding chapters. The example that follows provides a second illustration. Example 1.2: Often the nature of the scientiﬁc study will dictate the role that probability and deductive reasoning play in statistical inference. Exercise 9.40 on page 294 provides data associated with a study conducted at the Virginia Polytechnic Institute and State University on the development of a relationship between the roots of trees and the action of a fungus. Minerals are transferred from the fungus to the trees and sugars from the trees to the fungus. Two samples of 10 northern red oak seedlings were planted in a greenhouse, one containing seedlings treated with nitrogen and 1.1 Overview: Statistical Inference, Samples, Populations, and the Role of Probability 5 the other containing seedlings with no nitrogen. All other environmental conditions were held constant. All seedlings contained the fungus Pisolithus tinctorus. More details are supplied in Chapter 9. The stem weights in grams were recorded after the end of 140 days. The data are given in Table 1.1. Table 1.1: Data Set for Example 1.2 No Nitrogen Nitrogen 0.32 0.26 0.53 0.43 0.28 0.47 0.37 0.49 0.47 0.52 0.43 0.75 0.36 0.79 0.42 0.86 0.38 0.62 0.43 0.46 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Figure 1.1: A dot plot of stem weight data. In this example there are two samples from two separate populations. The purpose of the experiment is to determine if the use of nitrogen has an inﬂuence on the growth of the roots. The study is a comparative study (i.e., we seek to compare the two populations with regard to a certain important characteristic). It is instructive to plot the data as shown in the dot plot of Figure 1.1. The ◦ values represent the “nitrogen” data and the × values represent the “no-nitrogen” data. Notice that the general appearance of the data might suggest to the reader that, on average, the use of nitrogen increases the stem weight. Four nitrogen ob- servations are considerably larger than any of the no-nitrogen observations. Most of the no-nitrogen observations appear to be below the center of the data. The appearance of the data set would seem to indicate that nitrogen is eﬀective. But how can this be quantiﬁed? How can all of the apparent visual evidence be summa- rized in some sense? As in the preceding example, the fundamentals of probability can be used. The conclusions may be summarized in a probability statement or P-value. We will not show here the statistical inference that produces the summary probability. As in Example 1.1, these methods will be discussed in Chapter 10. The issue revolves around the “probability that data like these could be observed” given that nitrogen has no eﬀect, in other words, given that both samples were generated from the same population. Suppose that this probability is small, say 0.03. That would certainly be strong evidence that the use of nitrogen does indeed inﬂuence (apparently increases) average stem weight of the red oak seedlings. 6 Chapter 1 Introduction to Statistics and Data Analysis How Do Probability and Statistical Inference Work Together? It is important for the reader to understand the clear distinction between the discipline of probability, a science in its own right, and the discipline of inferen- tial statistics. As we have already indicated, the use or application of concepts in probability allows real-life interpretation of the results of statistical inference. As a result, it can be said that statistical inference makes use of concepts in probability. One can glean from the two examples above that the sample information is made available to the analyst and, with the aid of statistical methods and elements of probability, conclusions are drawn about some feature of the population (the pro- cess does not appear to be acceptable in Example 1.1, and nitrogen does appear to inﬂuence average stem weights in Example 1.2). Thus for a statistical problem, the sample along with inferential statistics allows us to draw conclu- sions about the population, with inferential statistics making clear use of elements of probability. This reasoning is inductive in nature. Now as we move into Chapter 2 and beyond, the reader will note that, unlike what we do in our two examples here, we will not focus on solving statistical problems. Many examples will be given in which no sample is involved. There will be a population clearly described with all features of the population known. Then questions of im- portance will focus on the nature of data that might hypothetically be drawn from the population. Thus, one can say that elements in probability allow us to draw conclusions about characteristics of hypothetical data taken from the population, based on known features of the population. This type of reasoning is deductive in nature. Figure 1.2 shows the fundamental relationship between probability and inferential statistics. Probability Population Sample Statistical Inference Figure 1.2: Fundamental relationship between probability and inferential statistics. Now, in the grand scheme of things, which is more important, the ﬁeld of probability or the ﬁeld of statistics? They are both very important and clearly are complementary. The only certainty concerning the pedagogy of the two disciplines lies in the fact that if statistics is to be taught at more than merely a “cookbook” level, then the discipline of probability must be taught ﬁrst. This rule stems from the fact that nothing can be learned about a population from a sample until the analyst learns the rudiments of uncertainty in that sample. For example, consider Example 1.1. The question centers around whether or not the population, deﬁned by the process, is no more than 5% defective. In other words, the conjecture is that on the average 5 out of 100 items are defective. Now, the sample contains 100 items and 10 are defective. Does this support the conjecture or refute it? On the 1.2 Sampling Procedures; Collection of Data 7 surface it would appear to be a refutation of the conjecture because 10 out of 100 seem to be “a bit much.” But without elements of probability, how do we know? Only through the study of material in future chapters will we learn the conditions under which the process is acceptable (5% defective). The probability of obtaining 10 or more defective items in a sample of 100 is 0.0282. We have given two examples where the elements of probability provide a sum- mary that the scientist or engineer can use as evidence on which to build a decision. The bridge between the data and the conclusion is, of course, based on foundations of statistical inference, distribution theory, and sampling distributions discussed in future chapters. 1.2 Sampling Procedures; Collection of Data In Section 1.1 we discussed very brieﬂy the notion of sampling and the sampling process. While sampling appears to be a simple concept, the complexity of the questions that must be answered about the population or populations necessitates that the sampling process be very complex at times. While the notion of sampling is discussed in a technical way in Chapter 8, we shall endeavor here to give some common-sense notions of sampling. This is a natural transition to a discussion of the concept of variability. Simple Random Sampling The importance of proper sampling revolves around the degree of conﬁdence with which the analyst is able to answer the questions being asked. Let us assume that only a single population exists in the problem. Recall that in Example 1.2 two populations were involved. Simple random sampling implies that any particular sample of a speciﬁed sample size has the same chance of being selected as any other sample of the same size. The term sample size simply means the number of elements in the sample. Obviously, a table of random numbers can be utilized in sample selection in many instances. The virtue of simple random sampling is that it aids in the elimination of the problem of having the sample reﬂect a diﬀerent (possibly more conﬁned) population than the one about which inferences need to be made. For example, a sample is to be chosen to answer certain questions regarding political preferences in a certain state in the United States. The sample involves the choice of, say, 1000 families, and a survey is to be conducted. Now, suppose it turns out that random sampling is not used. Rather, all or nearly all of the 1000 families chosen live in an urban setting. It is believed that political preferences in rural areas diﬀer from those in urban areas. In other words, the sample drawn actually conﬁned the population and thus the inferences need to be conﬁned to the “limited population,” and in this case conﬁning may be undesirable. If, indeed, the inferences need to be made about the state as a whole, the sample of size 1000 described here is often referred to as a biased sample. As we hinted earlier, simple random sampling is not always appropriate. Which alternative approach is used depends on the complexity of the problem. Often, for example, the sampling units are not homogeneous and naturally divide themselves into nonoverlapping groups that are homogeneous. These groups are called strata, 8 Chapter 1 Introduction to Statistics and Data Analysis and a procedure called stratiﬁed random sampling involves random selection of a sample within each stratum. The purpose is to be sure that each of the strata is neither over- nor underrepresented. For example, suppose a sample survey is conducted in order to gather preliminary opinions regarding a bond referendum that is being considered in a certain city. The city is subdivided into several ethnic groups which represent natural strata. In order not to disregard or overrepresent any group, separate random samples of families could be chosen from each group. Experimental Design The concept of randomness or random assignment plays a huge role in the area of experimental design, which was introduced very brieﬂy in Section 1.1 and is an important staple in almost any area of engineering or experimental science. This will be discussed at length in Chapters 13 through 15. However, it is instructive to give a brief presentation here in the context of random sampling. A set of so-called treatments or treatment combinations becomes the populations to be studied or compared in some sense. An example is the nitrogen versus no-nitrogen treat- ments in Example 1.2. Another simple example would be “placebo” versus “active drug,” or in a corrosion fatigue study we might have treatment combinations that involve specimens that are coated or uncoated as well as conditions of low or high humidity to which the specimens are exposed. In fact, there are four treatment or factor combinations (i.e., 4 populations), and many scientiﬁc questions may be asked and answered through statistical and inferential methods. Consider ﬁrst the situation in Example 1.2. There are 20 diseased seedlings involved in the exper- iment. It is easy to see from the data themselves that the seedlings are diﬀerent from each other. Within the nitrogen group (or the no-nitrogen group) there is considerable variability in the stem weights. This variability is due to what is generally called the experimental unit. This is a very important concept in in- ferential statistics, in fact one whose description will not end in this chapter. The nature of the variability is very important. If it is too large, stemming from a condition of excessive nonhomogeneity in experimental units, the variability will “wash out” any detectable diﬀerence between the two populations. Recall that in this case that did not occur. The dot plot in Figure 1.1 and P-value indicated a clear distinction between these two conditions. What role do those experimental units play in the data- taking process itself? The common-sense and, indeed, quite standard approach is to assign the 20 seedlings or experimental units randomly to the two treat- ments or conditions. In the drug study, we may decide to use a total of 200 available patients, patients that clearly will be diﬀerent in some sense. They are the experimental units. However, they all may have the same chronic condition for which the drug is a potential treatment. Then in a so-called completely ran- domized design, 100 patients are assigned randomly to the placebo and 100 to the active drug. Again, it is these experimental units within a group or treatment that produce the variability in data results (i.e., variability in the measured result), say blood pressure, or whatever drug eﬃcacy value is important. In the corrosion fatigue study, the experimental units are the specimens that are the subjects of the corrosion. 1.2 Sampling Procedures; Collection of Data 9 Why Assign Experimental Units Randomly? What is the possible negative impact of not randomly assigning experimental units to the treatments or treatment combinations? This is seen most clearly in the case of the drug study. Among the characteristics of the patients that produce variability in the results are age, gender, and weight. Suppose merely by chance the placebo group contains a sample of people that are predominately heavier than those in the treatment group. Perhaps heavier individuals have a tendency to have a higher blood pressure. This clearly biases the result, and indeed, any result obtained through the application of statistical inference may have little to do with the drug and more to do with diﬀerences in weights among the two samples of patients. We should emphasize the attachment of importance to the term variability. Excessive variability among experimental units “camouﬂages” scientiﬁc ﬁndings. In future sections, we attempt to characterize and quantify measures of variability. In sections that follow, we introduce and discuss speciﬁc quantities that can be computed in samples; the quantities give a sense of the nature of the sample with respect to center of location of the data and variability in the data. A discussion of several of these single-number measures serves to provide a preview of what statistical information will be important components of the statistical methods that are used in future chapters. These measures that help characterize the nature of the data set fall into the category of descriptive statistics. This material is a prelude to a brief presentation of pictorial and graphical methods that go even further in characterization of the data set. The reader should understand that the statistical methods illustrated here will be used throughout the text. In order to oﬀer the reader a clearer picture of what is involved in experimental design studies, we oﬀer Example 1.3. Example 1.3: A corrosion study was made in order to determine whether coating an aluminum metal with a corrosion retardation substance reduced the amount of corrosion. The coating is a protectant that is advertised to minimize fatigue damage in this type of material. Also of interest is the inﬂuence of humidity on the amount of corrosion. A corrosion measurement can be expressed in thousands of cycles to failure. Two levels of coating, no coating and chemical corrosion coating, were used. In addition, the two relative humidity levels are 20% relative humidity and 80% relative humidity. The experiment involves four treatment combinations that are listed in the table that follows. There are eight experimental units used, and they are aluminum specimens prepared; two are assigned randomly to each of the four treatment combinations. The data are presented in Table 1.2. The corrosion data are averages of two specimens. A plot of the averages is pictured in Figure 1.3. A relatively large value of cycles to failure represents a small amount of corrosion. As one might expect, an increase in humidity appears to make the corrosion worse. The use of the chemical corrosion coating procedure appears to reduce corrosion. In this experimental design illustration, the engineer has systematically selected the four treatment combinations. In order to connect this situation to concepts with which the reader has been exposed to this point, it should be assumed that the 10 Chapter 1 Introduction to Statistics and Data Analysis Table 1.2: Data for Example 1.3 Average Corrosion in Coating Humidity Thousands of Cycles to Failure 20% 975 Uncoated 80% 350 20% 1750 Chemical Corrosion 80% 1550 2000 Chemical Corrosion Coating Average Corrosion 1000 Uncoated 0 0 20% 80% Humidity Figure 1.3: Corrosion results for Example 1.3. conditions representing the four treatment combinations are four separate popula- tions and that the two corrosion values observed for each population are important pieces of information. The importance of the average in capturing and summariz- ing certain features in the population will be highlighted in Section 1.3. While we might draw conclusions about the role of humidity and the impact of coating the specimens from the ﬁgure, we cannot truly evaluate the results from an analyti- cal point of view without taking into account the variability around the average. Again, as we indicated earlier, if the two corrosion values for each treatment com- bination are close together, the picture in Figure 1.3 may be an accurate depiction. But if each corrosion value in the ﬁgure is an average of two values that are widely dispersed, then this variability may, indeed, truly “wash away” any information that appears to come through when one observes averages only. The foregoing example illustrates these concepts: (1) random assignment of treatment combinations (coating, humidity) to experi- mental units (specimens) (2) the use of sample averages (average corrosion values) in summarizing sample information (3) the need for consideration of measures of variability in the analysis of any sample or sets of samples 1.3 Measures of Location: The Sample Mean and Median 11 This example suggests the need for what follows in Sections 1.3 and 1.4, namely, descriptive statistics that indicate measures of center of location in a set of data, and those that measure variability. 1.3 Measures of Location: The Sample Mean and Median Measures of location are designed to provide the analyst with some quantitative values of where the center, or some other location, of data is located. In Example 1.2, it appears as if the center of the nitrogen sample clearly exceeds that of the no-nitrogen sample. One obvious and very useful measure is the sample mean. The mean is simply a numerical average. Deﬁnition 1.1: Suppose that the observations in a sample are x1 , x2 ,... , xn. The sample mean, denoted by x̄, is n xi x1 + x 2 + · · · + x n x̄ = =. i=1 n n There are other measures of central tendency that are discussed in detail in future chapters. One important measure is the sample median. The purpose of the sample median is to reﬂect the central tendency of the sample in such a way that it is uninﬂuenced by extreme values or outliers. Deﬁnition 1.2: Given that the observations in a sample are x1 , x2 ,... , xn , arranged in increasing order of magnitude, the sample median is x(n+1)/2 , if n is odd, x̃ = 1 2 (xn/2 + xn/2+1 ), if n is even. As an example, suppose the data set is the following: 1.7, 2.2, 3.9, 3.11, and 14.7. The sample mean and median are, respectively, x̄ = 5.12, x̃ = 3.9. Clearly, the mean is inﬂuenced considerably by the presence of the extreme obser- vation, 14.7, whereas the median places emphasis on the true “center” of the data set. In the case of the two-sample data set of Example 1.2, the two measures of central tendency for the individual samples are x̄ (no nitrogen) = 0.399 gram, 0.38 + 0.42 x̃ (no nitrogen) = = 0.400 gram, 2 x̄ (nitrogen) = 0.565 gram, 0.49 + 0.52 x̃ (nitrogen) = = 0.505 gram. 2 Clearly there is a diﬀerence in concept between the mean and median. It may be of interest to the reader with an engineering background that the sample mean 12 Chapter 1 Introduction to Statistics and Data Analysis is the centroid of the data in a sample. In a sense, it is the point at which a fulcrum can be placed to balance a system of “weights” which are the locations of the individual data. This is shown in Figure 1.4 with regard to the with-nitrogen sample. x 0.565 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Figure 1.4: Sample mean as a centroid of the with-nitrogen stem weight. In future chapters, the basis for the computation of x̄ is that of an estimate of the population mean. As we indicated earlier, the purpose of statistical infer- ence is to draw conclusions about population characteristics or parameters and estimation is a very important feature of statistical inference. The median and mean can be quite diﬀerent from each other. Note, however, that in the case of the stem weight data the sample mean value for no-nitrogen is quite similar to the median value. Other Measures of Locations There are several other methods of quantifying the center of location of the data in the sample. We will not deal with them at this point. For the most part, alternatives to the sample mean are designed to produce values that represent compromises between the mean and the median. Rarely do we make use of these other measures. However, it is instructive to discuss one class of estimators, namely the class of trimmed means. A trimmed mean is computed by “trimming away” a certain percent of both the largest and the smallest set of values. For example, the 10% trimmed mean is found by eliminating the largest 10% and smallest 10% and computing the average of the remaining values. For example, in the case of the stem weight data, we would eliminate the largest and smallest since the sample size is 10 for each sample. So for the without-nitrogen group the 10% trimmed mean is given by 0.32 + 0.37 + 0.47 + 0.43 + 0.36 + 0.42 + 0.38 + 0.43 x̄tr(10) = = 0.39750, 8 and for the 10% trimmed mean for the with-nitrogen group we have 0.43 + 0.47 + 0.49 + 0.52 + 0.75 + 0.79 + 0.62 + 0.46 x̄tr(10) = = 0.56625. 8 Note that in this case, as expected, the trimmed means are close to both the mean and the median for the individual samples. The trimmed mean is, of course, more insensitive to outliers than the sample mean but not as insensitive as the median. On the other hand, the trimmed mean approach makes use of more information than the sample median. Note that the sample median is, indeed, a special case of the trimmed mean in which all of the sample data are eliminated apart from the middle one or two observations. / / Exercises 13 Exercises 1.1 The following measurements were recorded for polymer? Explain. the drying time, in hours, of a certain brand of latex (c) Calculate the sample mean tensile strength of the paint. two samples. 3.4 2.5 4.8 2.9 3.6 (d) Calculate the median for both. Discuss the simi- 2.8 3.3 5.6 3.7 2.8 larity or lack of similarity between the mean and 4.4 4.0 5.2 3.0 4.8 median of each group. Assume that the measurements are a simple random sample. 1.4 In a study conducted by the Department of Me- (a) What is the sample size for the above sample? chanical Engineering at Virginia Tech, the steel rods (b) Calculate the sample mean for these data. supplied by two diﬀerent companies were compared. (c) Calculate the sample median. Ten sample springs were made out of the steel rods supplied by each company, and a measure of ﬂexibility (d) Plot the data by way of a dot plot. was recorded for each. The data are as follows: (e) Compute the 20% trimmed mean for the above Company A: 9.3 8.8 6.8 8.7 8.5 data set. 6.7 8.0 6.5 9.2 7.0 (f) Is the sample mean for these data more or less de- Company B: 11.0 9.8 9.9 10.2 10.1 scriptive as a center of location than the trimmed 9.7 11.0 11.1 10.2 9.6 mean? (a) Calculate the sample mean and median for the data for the two companies. 1.2 According to the journal Chemical Engineering, (b) Plot the data for the two companies on the same an important property of a ﬁber is its water ab- line and give your impression regarding any appar- sorbency. A random sample of 20 pieces of cotton ﬁber ent diﬀerences between the two companies. was taken and the absorbency on each piece was mea- sured. The following are the absorbency values: 1.5 Twenty adult males between the ages of 30 and 18.71 21.41 20.72 21.81 19.29 22.43 20.17 40 participated in a study to evaluate the eﬀect of a 23.71 19.44 20.50 18.92 20.33 23.00 22.85 speciﬁc health regimen involving diet and exercise on 19.25 21.77 22.11 19.77 18.04 21.12 the blood cholesterol. Ten were randomly selected to (a) Calculate the sample mean and median for the be a control group, and ten others were assigned to above sample values. take part in the regimen as the treatment group for a (b) Compute the 10% trimmed mean. period of 6 months. The following data show the re- (c) Do a dot plot of the absorbency data. duction in cholesterol experienced for the time period for the 20 subjects: (d) Using only the values of the mean, median, and trimmed mean, do you have evidence of outliers in Control group: 7 3 −4 14 2 the data? 5 22 −7 9 5 Treatment group: −6 5 9 4 4 1.3 A certain polymer is used for evacuation systems 12 37 5 3 3 for aircraft. It is important that the polymer be re- (a) Do a dot plot of the data for both groups on the sistant to the aging process. Twenty specimens of the same graph. polymer were used in an experiment. Ten were as- (b) Compute the mean, median, and 10% trimmed signed randomly to be exposed to an accelerated batch mean for both groups. aging process that involved exposure to high tempera- (c) Explain why the diﬀerence in means suggests one tures for 10 days. Measurements of tensile strength of conclusion about the eﬀect of the regimen, while the specimens were made, and the following data were the diﬀerence in medians or trimmed means sug- recorded on tensile strength in psi: gests a diﬀerent conclusion. No aging: 227 222 218 217 225 218 216 229 228 221 1.6 The tensile strength of silicone rubber is thought Aging: 219 214 215 211 209 to be a function of curing temperature. A study was 218 203 204 201 205 carried out in which samples of 12 specimens of the rub- (a) Do a dot plot of the data. ber were prepared using curing temperatures of 20◦ C (b) From your plot, does it appear as if the aging pro- and 45◦ C. The data below show the tensile strength cess has had an eﬀect on the tensile strength of this values in megapascals. 14 Chapter 1 Introduction to Statistics and Data Analysis 20◦ C: 2.07 2.14 2.22 2.03 2.21 2.03 (b) Compute sample mean tensile strength for both 2.05 2.18 2.09 2.14 2.11 2.02 samples. ◦ 45 C: 2.52 2.15 2.49 2.03 2.37 2.05 1.99 2.42 2.08 2.42 2.29 2.01 (c) Does it appear as if curing temperature has an inﬂuence on tensile strength, based on the plot? Comment further. (a) Show a dot plot of the data with both low and high (d) Does anything else appear to be inﬂuenced by an temperature tensile strength values. increase in curing temperature? Explain. 1.4 Measures of Variability Sample variability plays an important role in data analysis. Process and product variability is a fact of life in engineering and scientiﬁc systems: The control or reduction of process variability is often a source of major diﬃculty. More and more process engineers and managers are learning that product quality and, as a result, proﬁts derived from manufactured products are very much a function of process variability. As a result, much of Chapters 9 through 15 deals with data analysis and modeling procedures in which sample variability plays a major role. Even in small data analysis problems, the success of a particular statistical method may depend on the magnitude of the variability among the observations in the sample. Measures of location in a sample do not provide a proper summary of the nature of a data set. For instance, in Example 1.2 we cannot conclude that the use of nitrogen enhances growth without taking sample variability into account. While the details of the analysis of this type of data set are deferred to Chap- ter 9, it should be clear from Figure 1.1 that variability among the no-nitrogen observations and variability among the nitrogen observations are certainly of some consequence. In fact, it appears that the variability within the nitrogen sample is larger than that of the no-nitrogen sample. Perhaps there is something about the inclusion of nitrogen that not only increases the stem height (x̄ of 0.565 gram compared to an x̄ of 0.399 gram for the no-nitrogen sample) but also increases the variability in stem height (i.e., renders the stem height more inconsistent). As another example, contrast the two data sets below. Each contains two samples and the diﬀerence in the means is roughly the same for the two samples, but data set B seems to provide a much sharper contrast between the two populations from which the samples were taken. If the purpose of such an experiment is to detect diﬀerences between the two populations, the task is accomplished in the case of data set B. However, in data set A the large variability within the two samples creates diﬃculty. In fact, it is not clear that there is a distinction between the two populations. Data set A: X X X X X X 0 X X 0 0 X X X 0 0 0 0 0 0 0 0 xX x0 Data set B: X X X X X X X X X X X 0 0 0 0 0 0 0 0 0 0 0 xX x0 1.4 Measures of Variability 15 Sample Range and Sample Standard Deviation Just as there are many measures of central tendency or location, there are many measures of spread or variability. Perhaps the simplest one is the sample range Xmax − Xmin. The range can be very useful and is discussed at length in Chapter 17 on statistical quality control. The sample measure of spread that is used most often is the sample standard deviation. We again let x1 , x2 ,... , xn denote sample values. Deﬁnition 1.3: The sample variance, denoted by s2 , is given by n (xi − x̄)2 s2 =. i=1 n−1 The sample standard deviation, denoted by s, is the positive square root of s2 , that is, √ s = s2. It should be clear to the reader that the sample standard deviation is, in fact, a measure of variability. Large variability in a data set produces relatively large values of (x − x̄)2 and thus a large sample variance. The quantity n − 1 is often called the degrees of freedom associated with the variance estimate. In this simple example, the degrees of freedom depict the number of independent pieces of information available for computing variability. For example, suppose that we wish to compute the sample variance and standard deviation of the data set (5, 17, 6, 4). The sample average is x̄ = 8. The computation of the variance involves (5 − 8)2 + (17 − 8)2 + (6 − 8)2 + (4 − 8)2 = (−3)2 + 92 + (−2)2 + (−4)2. n The quantities inside parentheses sum to zero. In general, (xi − x̄) = 0 (see i=1 Exercise 1.16 on page 31). Then the computation of a sample variance does not involve n independent squared deviations from the mean x̄. In fact, since the last value of x − x̄ is determined by the initial n − 1 of them, we say that these are n − 1 “pieces of information” that produce s2. Thus, there are n − 1 degrees of freedom rather than n degrees of freedom for computing a sample variance. Example 1.4: In an example discussed extensively in Chapter 10, an engineer is interested in testing the “bias” in a pH meter. Data are collected on the meter by measuring the pH of a neutral substance (pH = 7.0). A sample of size 10 is taken, with results given by 7.07 7.00 7.10 6.97 7.00 7.03 7.01 7.01 6.98 7.08. The sample mean x̄ is given by 7.07 + 7.00 + 7.10 + · · · + 7.08 x̄ = = 7.0250. 10 16 Chapter 1 Introduction to Statistics and Data Analysis The sample variance s2 is given by 1 s2 = [(7.07 − 7.025)2 + (7.00 − 7.025)2 + (7.10 − 7.025)2 9 + · · · + (7.08 − 7.025)2 ] = 0.001939. As a result, the sample standard deviation is given by √ s = 0.001939 = 0.044. So the sample standard deviation is 0.0440 with n − 1 = 9 degrees of freedom. Units for Standard Deviation and Variance It should be apparent from Deﬁnition 1.3 that the variance is a measure of the average squared deviation from the mean x̄. We use the term average squared deviation even though the deﬁnition makes use of a division by degrees of freedom n − 1 rather than n. Of course, if n is large, the diﬀerence in the denominator is inconsequential. As a result, the sample variance possesses units that are the square of the units in the observed data whereas the sample standard deviation is found in linear units. As an example, consider the data of Example 1.2. The stem weights are measured in grams. As a result, the sample standard deviations are in grams and the variances are measured in grams2. In fact, the individual standard deviations are 0.0728 gram for the no-nitrogen case and 0.1867 gram for the nitrogen group. Note that the standard deviation does indicate considerably larger variability in the nitrogen sample. This condition was displayed in Figure 1.1. Which Variability Measure Is More Important? As we indicated earlier, the sample range has applications in the area of statistical quality control. It may appear to the reader that the use of both the sample variance and the sample standard deviation is redundant. Both measures reﬂect the same concept in measuring variability, but the sample standard deviation measures variability in linear units whereas the sample variance is measured in squared units. Both play huge roles in the use of statistical methods. Much of what is accomplished in the context of statistical inference involves drawing conclusions about characteristics of populations. Among these characteristics are constants which are called population parameters. Two important parameters are the population mean and the population variance. The sample variance plays an explicit role in the statistical methods used to draw inferences about the population variance. The sample standard deviation has an important role along with the sample mean in inferences that are made about the population mean. In general, the variance is considered more in inferential theory, while the standard deviation is used more in applications. 1.5 Discrete and Continuous Data 17 Exercises 1.7 Consider the drying time data for Exercise 1.1 1.10 For the data of Exercise 1.4 on page 13, com- on page 13. Compute the sample variance and sample pute both the mean and the variance in “ﬂexibility” standard deviation. for both company A and company B. Does there ap- pear to be a diﬀerence in ﬂexibility between company 1.8 Compute the sample variance and standard devi- A and company B? ation for the water absorbency data of Exercise 1.2 on page 13. 1.11 Consider the data in Exercise 1.5 on page 13. Compute the sample variance and the sample standard 1.9 Exercise 1.3 on page 13 showed tensile strength deviation for both control and treatment groups. data for two samples, one in which specimens were ex- posed to an aging process and one in which there was 1.12 For Exercise 1.6 on page 13, compute the sample no aging of the specimens. standard deviation in tensile strength for the samples (a) Calculate the sample variance as well as standard separately for the two temperatures. Does it appear as deviation in tensile strength for both samples. if an increase in temperature inﬂuences the variability (b) Does there appear to be any evidence that aging in tensile strength? Explain. aﬀects the variability in tensile strength? (See also the plot for Exercise 1.3 on page 13.) 1.5 Discrete and Continuous Data Statistical inference through the analysis of observational studies or designed ex- periments is used in many scientiﬁc areas. The data gathered may be discrete or continuous, depending on the area of application. For example, a chemical engineer may be interested in conducting an experiment that will lead to condi- tions where yield is maximized. Here, of course, the yield may be in percent or grams/pound, measured on a continuum. On the other hand, a toxicologist con- ducting a combination drug experiment may encounter data that are binary in nature (i.e., the patient either responds or does not). Great distinctions are made between discrete and continuous data in the prob- ability theory that allow us to draw statistical inferences. Often applications of statistical inference are found when the data are count data. For example, an en- gineer may be interested in studying the number of radioactive particles passing through a counter in, say, 1 millisecond. Personnel responsible for the eﬃciency of a port facility may be interested in the properties of the number of oil tankers arriving each day at a certain port city. In Chapter 5, several distinct scenarios, leading to diﬀerent ways of handling data, are discussed for situations with count data. Special attention even at this early stage of the textbook should be paid to some details associated with binary data. Applications requiring statistical analysis of binary data are voluminous. Often the measure that is used in the analysis is the sample proportion. Obviously the binary situation involves two categories. If there are n units involved in the data and x is deﬁned as the number that fall into category 1, then n − x fall into category 2. Thus, x/n is the sample proportion in category 1, and 1 − x/n is the sample proportion in category 2. In the biomedical application, 50 patients may represent the sample units, and if 20 out of 50 experienced an improvement in a stomach ailment (common to all 50) after all were given the drug, then 20 50 = 0.4 is the sample proportion for which 18 Chapter 1 Introduction to Statistics and Data Analysis the drug was a success and 1 − 0.4 = 0.6 is the sample proportion for which the drug was not successful. Actually the basic numerical measurement for binary data is generally denoted by either 0 or 1. For example, in our medical example, a successful result is denoted by a 1 and a nonsuccess a 0. As a result, the sample proportion is actually a sample mean of the ones and zeros. For the successful category, x1 + x2 + · · · + x50 1 + 1 + 0 + ··· + 0 + 1 20 = = = 0.4. 50 50 50 What Kinds of Problems Are Solved in Binary Data Situations? The kinds of problems facing scientists and engineers dealing in binary data are not a great deal unlike those seen where continuous measurements are of interest. However, diﬀerent techniques are used since the statistical properties of sample proportions are quite diﬀerent from those of the sample means that result from averages taken from continuous populations. Consider the example data in Ex- ercise 1.6 on page 13. The statistical problem underlying this illustration focuses on whether an intervention, say, an increase in curing temperature, will alter the population mean tensile strength associated with the silicone rubber process. On the other hand, in a quality control area, suppose an automobile tire manufacturer reports that a shipment of 5000 tires selected randomly from the process results 100 in 100 of them showing blemishes. Here the sample proportion is 5000 = 0.02. Following a change in the process designed to reduce blemishes, a second sample of 5000 is taken and 90 tires are blemished. The sample proportion has been reduced 90 to 5000 = 0.018. The question arises, “Is the decrease in the sample proportion from 0.02 to 0.018 substantial enough to suggest a real improvement in the pop- ulation proportion?” Both of these illustrations require the use of the statistical properties of sample averages—one from samples from a continuous population, and the other from samples from a discrete (binary) population. In both cases, the sample mean is an estimate of a population parameter, a population mean in the ﬁrst illustration (i.e., mean tensile strength), and a population proportion in the second case (i.e., proportion of blemished tires in the population). So here we have sample estimates used to draw scientiﬁc conclusions regarding population parameters. As we indicated in Section 1.3, this is the general theme in many practical problems using statistical inference. 1.6 Statistical Modeling, Scientiﬁc Inspection, and Graphical Diagnostics Often the end result of a statistical analysis is the estimation of parameters of a postulated model. This is natural for scientists and engineers since they often deal in modeling. A statistical model is not deterministic but, rather, must entail some probabilistic aspects. A model form is often the foundation of assumptions that are made by the analyst. For example, in Example 1.2 the scientist may wish to draw some level of distinction between the nitrogen and no-nitrogen populations through the sample information. The analysis may require a certain model for 1.6 Statistical Modeling, Scientiﬁc Inspection, and Graphical Diagnostics 19 the data, for example, that the two samples come from normal or Gaussian distributions. See Chapter 6 for a discussion of the normal distribution. Obviously, the user of statistical methods cannot generate suﬃcient informa- tion or experimental data to characterize the population totally. But sets of data are often used to learn about certain properties of the population. Scientists and engineers are accustomed to dealing with data sets. The importance of character- izing or summarizing the nature of collections of data should be obvious. Often a summary of a collection of data via a graphical display can provide insight regard- ing the system from which the data were taken. For instance, in Sections 1.1 and 1.3, we have shown dot plots. In this section, the role of sampling and the display of data for enhancement of statistical inference is explored in detail. We merely introduce some simple but often eﬀective displays that complement the study of statistical populations. Scatter Plot At times the model postulated may take on a somewhat complicated form. Con- sider, for example, a textile manufacturer who designs an experiment where cloth specimen that contain various percentages of cotton are produced. Consider the data in Table 1.3. Table 1.3: Tensile Strength Cotton Percentage Tensile Strength 15 7, 7, 9, 8, 10 20 19, 20, 21, 20, 22 25 21, 21, 17, 19, 20 30 8, 7, 8, 9, 10 Five cloth specimens are manufactured for each of the four cotton percentages. In this case, both the model for the experiment and the type of analysis used should take into account the goal of the experiment and important input from the textile scientist. Some simple graphics can shed important light on the clear distinction between the samples. See Figure 1.5; the sample means and variability are depicted nicely in the scatter plot. One possible goal of this experiment is simply to determine which cotton percentages are truly distinct from the others. In other words, as in the case of the nitrogen/no-nitrogen data, for which cotton percentages are there clear distinctions between the populations or, more speciﬁ- cally, between the population means? In this case, perhaps a reasonable model is that each sample comes from a normal distribution. Here the goal is very much like that of the nitrogen/no-nitrogen data except that more samples are involved. The formalism of the analysis involves notions of hypothesis testing discussed in Chapter 10. Incidentally, this formality is perhaps not necessary in light of the diagnostic plot. But does this describe the real goal of the experiment and hence the proper approach to data analysis? It is likely that the scientist anticipates the existence of a maximum population mean tensile strength in the range of cot- ton concentration in the experiment. Here the analysis of the data should revolve 20 Chapter 1 Introduction to Statistics and Data Analysis around a diﬀerent type of model, one that postulates a type of structure relating the population mean tensile strength to the cotton concentration. In other words, a model may be written μt,c = β0 + β1 C + β2 C 2 , where μt,c is the population mean tensile strength, which varies with the amount of cotton in the product C. The implication of this model is that for a ﬁxed cotton level, there is a population of tensile strength measurements and the population mean is μt,c. This type of model, called a regression model, is discussed in Chapters 11 and 12. The functional form is chosen by the scientist. At times the data analysis may suggest that the model be changed. Then the data analyst “entertains” a model that may be altered after some analysis is done. The use of an empirical model is accompanied by estimation theory, where β0 , β1 , and β2 are estimated by the data. Further, statistical inference can then be used to determine model adequacy. 25 20 Tensile Strength 15 10 5 15 20 25 30 Cotton Percentages Figure 1.5: Scatter plot of tensile strength and cotton percentages. Two points become evident from the two data illustrations here: (1) The type of model used to describe the data often depends on the goal of the experiment; and (2) the structure of the model should take advantage of nonstatistical scientiﬁc input. A selection of a model represents a fundamental assumption upon which the resulting statistical inference is based. It will become apparent throughout the book how important graphics can be. Often, plots can illustrate information that allows the results of the formal statistical inference to be better communicated to the scientist or engineer. At times, plots or exploratory data analysis can teach the analyst something not retrieved from the formal analysis. Almost any formal analysis requires assumptions that evolve from the model of the data. Graphics can nicely highlight violation of assumptions that would otherwise go unnoticed. Throughout the book, graphics are used extensively to supplement formal data analysis. The following sections reveal some graphical tools that are useful in exploratory or descriptive data analysis. 1.6 Statistical Modeling, Scientiﬁc Inspection, and Graphical Diagnostics 21 Stem-and-Leaf Plot Statistical data, generated in large masses, can be very useful for studying the behavior of the distribution if presented in a combined tabular and graphic display called a stem-and-leaf plot. To illustrate the construction of a stem-and-leaf plot, consider the data of Table 1.4, which speciﬁes the “life” of 40 similar car batteries recorded to the nearest tenth of a year. The batteries are guaranteed to last 3 years. First, split each observation into two parts consisting of a stem and a leaf such that the stem represents the digit preceding the decimal and the leaf corresponds to the decimal part of the number. In other words, for the number 3.7, the digit 3 is designated the stem and the digit 7 is the leaf. The four stems 1, 2, 3, and 4 for our data are listed vertically on the left side in Table 1.5; the leaves are recorded on the right side opposite the appropriate stem value. Thus, the leaf 6 of the number 1.6 is recorded opposite the stem 1; the leaf 5 of the number 2.5 is recorded opposite the stem 2; and so forth. The number of leaves recorded opposite each stem is summarized under the frequency column. Table 1.4: Car Battery Life 2.2 4.1 3.5 4.5 3.2 3.7 3.0 2.6 3.4 1.6 3.1 3.3 3.8 3.1 4.7 3.7 2.5 4.3 3.4 3.6 2.9 3.3 3.9 3.1 3.3 3.1 3.7 4.4 3.2 4.1 1.9 3.4 4.7 3.8 3.2 2.6 3.9 3.0 4.2 3.5 Table 1.5: Stem-and-Leaf Plot of Battery Life Stem Leaf Frequency 1 69 2 2 25669 5 3 0011112223334445567778899 25 4 11234577 8 The stem-and-leaf plot of Table 1.5 contains only four stems and consequently does not provide an adequate picture of the distribution. To remedy this problem, we need to increase the number of stems in our plot. One simple way to accomplish this is to write each stem value twice and then record the leaves 0, 1, 2, 3, and 4 opposite the appropriate stem value where it appears for the ﬁrst time, and the leaves 5, 6, 7, 8, and 9 opposite this same stem value where it appears for the second time. This modiﬁed double-stem-and-leaf plot is illustrated in Table 1.6, where the stems corresponding to leaves 0 through 4 have been coded by the symbol and the stems corresponding to leaves 5 through 9 by the symbol ·. In any given problem, we must decide on the appropriate stem values. This decision is made somewhat arbitrarily, although we are guided by the size of our sample. Usually, we choose between 5 and 20 stems. The smaller the number of data available, the smaller is our choice for the number of stems. For example, if 22 Chapter 1 Introduction to Statistics and Data Analysis the data consist of numbers from 1 to 21 representing the number of people in a cafeteria line on 40 randomly selected workdays and we choose a double-stem-and- leaf plot, the stems will be 0, 0·, 1, 1·, and 2 so that the smallest observation 1 has stem 0 and leaf 1, the number 18 has stem 1· and leaf 8, and the largest observation 21 has stem 2 and leaf 1. On the other hand, if the data consist of numbers from $18,800 to $19,600 representing the best possible deals on 100 new automobiles from a certain dealership and we choose a single-stem-and-leaf plot, the stems will be 188, 189, 190,... , 196 and the leaves will now each contain two digits. A car that sold for $19,385 would have a stem value of 193 and the two-digit leaf 85. Multiple-digit leaves belonging to the same stem are usually separated by commas in the stem-and-leaf plot. Decimal points in the data are generally ignored when all the digits to the right of the decimal represent the leaf. Such was the case in Tables 1.5 and 1.6. However, if the data consist of numbers ranging from 21.8 to 74.9, we might choose the digits 2, 3, 4, 5, 6, and 7 as our stems so that a number such as 48.3 would have a stem value of 4 and a leaf of 8.3. Table 1.6: Double-Stem-and-Leaf Plot of Battery Life Stem Leaf Frequency 1· 69 2 2 2 1 2· 5669 4 3 001111222333444 15 3· 5567778899 10 4 11234 5 4· 577 3 The stem-and-leaf plot represents an eﬀective way to summarize data. Another way is through the use of the frequency distribution, where the data, grouped into diﬀerent classes or intervals, can be constructed by counting the leaves be- longing to each stem and noting that each stem deﬁnes a class interval. In Table 1.5, the stem 1 with 2 leaves deﬁnes the interval 1.0–1.9 containing 2 observations; the stem 2 with 5 leaves deﬁnes the interval 2.0–2.9 containing 5 observations; the stem 3 with 25 leaves deﬁnes the interval 3.0–3.9 with 25 observations; and the stem 4 with 8 leaves deﬁnes the interval 4.0–4.9 containing 8 observations. For the double-stem-and-leaf plot of Table 1.6, the stems deﬁne the seven class intervals 1.5–1.9, 2.0–2.4, 2.5–2.9, 3.0–3.4, 3.5–3.9, 4.0–4.4, and 4.5–4.9 with frequencies 2, 1, 4, 15, 10, 5, and 3, respectively. Histogram Dividing each class frequency by the total number of observations, we obtain the proportion of the set of observations in each of the classes. A table listing relative frequencies is called a relative frequency distribution. The relative frequency distribution for the data of Table 1.4, showing the midpoint of each class interval, is given in Table 1.7. The information provided by a relative frequency distribution in tabular form is easier to grasp if presented graphically. Using the midpoint of each interval and the 1.6 Statistical Modeling, Scientiﬁc Inspection, and Graphical Diagnostics 23 Table 1.7: Relative Frequency Distribution of Battery Life Class Class Frequency, Relative Interval Midpoint f Frequency 1.5–1.9 1.7 2 0.050 2.0–2.4 2.2 1 0.025 2.5–2.9 2.7 4 0.100 3.0–3.4 3.2 15 0.375 3.5–3.9 3.7 10 0.250 4.0–4.4 4.2 5 0.125 4.5–4.9 4.7 3 0.075 0.375 Relativ e Frequencty 0.250 0.125 1.7 2.2 2.7 3.2 3.7 4.2 4.7 Battery Life (years) Figure 1.6: Relative frequency histogram. corresponding relative frequency, we construct a relative frequency histogram (Figure 1.6). Many continuous frequency distributions can be represented graphically by the characteristic bell-shaped curve of Figure 1.7. Graphical tools such as what we see in Figures 1.6 and 1.7 aid in the characterization of the nature of the population. In Chapters 5 and 6 we discuss a property of the population called its distribution. While a more rigorous deﬁnition of a distribution or probability distribution will be given later in the text, at this point one can view it as what would be seen in Figure 1.7 in the limit as the size of the sample becomes larger. A distribution is said to be symmetric if it can be folded along a vertical axis so that the two sides coincide. A distribution that lacks symmetry with respect to a vertical axis is said to be skewed. The distribution illustrated in Figure 1.8(a) is said to be skewed to the right since it has a long right tail and a much shorter left tail. In Figure 1.8(b) we see that the distribution is symmetric, while in Figure 1.8(c) it is skewed to the left. If we rotate a stem-and-leaf plot counterclockwise through an angle of 90◦ , we observe that the resulting columns of leaves form a picture that is similar to a histogram. Consequently, if our primary purpose in looking at the data is to determine the general shape or form of the distribution, it will seldom be necessary 24 Chapter 1 Introduction to Statistics and Data Analysis f (x ) 0 1 2 3 4 5 6 Battery Life (years) Figure 1.7: Estimating frequency distribution. (a) (b) (c) Figure 1.8: Skewness of data. to construct a relative frequency histogram. Box-and-Whisker Plot or Box Plot Another display that is helpful for reﬂecting properties of a sample is the box- and-whisker plot. This plot encloses the interquartile range of the data in a box that has the median displayed within. The interquartile range has as its extremes the 75th percentile (upper quartile) and the 25th percentile (lower quartile). In addition to the box, “whiskers” extend, showing extreme observations in the sam- ple. For reasonably large samples, the display shows center of location, variability, and the degree of asymmetry. In addition, a variation called a box plot can provide the viewer with infor- mation regarding which observations may be outliers. Outliers are observations that are considered to be unusually far from the bulk of the data. There are many statistical tests that are designed to detect outliers. Technically, one may view an outlier as being an observation that represents a “rare event” (there is a small probability of obtaining a value that far from the bulk of the data). The concept of outliers resurfaces in Chapter 12 in the context of regression analysis. 1.6 Statistical Modeling, Scientiﬁc Inspection, and Graphical Diagnostics 25 The visual information in the box-and-whisker plot or box plot is not intended to be a formal test for outliers. Rather, it is viewed as a diagnostic tool. While the determination of which observations are outliers varies with the type of software that is used, one common procedure is to use a multiple of the interquartile range. For example, if the distance from the box exceeds 1.5 times the interquartile range (in either direction), the observation may be labeled an outlier. Example 1.5: Nicotine content was measured in a random sample of 40 cigarettes. The data are displayed in Table 1.8. Table 1.8: Nicotine Data for Example 1.5 1.09 1.92 2.31 1.79 2.28 1.74 1.47 1.97 0.85 1.24 1.58 2.03 1.70 2.17 2.55 2.11 1.86 1.90 1.68 1.51 1.64 0.72 1.69 1.85 1.82 1.79 2.46 1.88 2.08 1.67 1.37 1.93 1.40 1.64 2.09 1.75 1.63 2.37 1.75 1.69 1.0 1.5 2.0 2.5 Nicotine Figure 1.9: Box-and-whisker plot for Example 1.5. Figure 1.9 shows the box-and-whisker plot of the data, depicting the observa- tions 0.72 and 0.85 as mild outliers in the lower tail, whereas the observation 2.55 is a mild outlier in the upper tail. In this example, the interquartile range is 0.365, and 1.5 times the interquartile range is 0.5475. Figure 1.10, on the other hand, provides a stem-and-leaf plot. Example 1.6: Consider the data in Table 1.9, consisting of 30 samples measuring the thickness of paint can “ears” (see the work by Hogg and Ledolter, 1992, in the Bibliography). Figure 1.11 depicts a box-and-whisker plot for this asymmetric set of data. Notice that the left block is considerably larger than the block on the right. The median is 35. The lower quartile is 31, while the upper quartile is 36. Notice also that the extreme observation on the right is farther away from the box than the extreme observation on the left. There are no outliers in this data set. 26 Chapter 1 Introduction to Statistics and Data Analysis The decimal point is 1 digit(s) to the left of the | 7 | 2 8 | 5 9 | 10 | 9 11 | 12 | 4 13 | 7 14 | 07 15 | 18 16 | 3447899 17 | 045599 18 | 2568 19 | 0237 20 | 389 21 | 17 22 | 8 23 | 17 24 | 6 25 | 5 Figure 1.10: Stem-and-leaf plot for the nicotine data. Table 1.9: Data for Example 1.6 Sample Measurements Sample Measurements 1 29 36 39 34 34 16 35 30 35 29 37 2 29 29 28 32 31 17 40 31 38 35 31 3 34 34 39 38 37 18 35 36 30 33 32 4 35 37 33 38 41 19 35 34 35 30 36 5 30 29 31 38 29 20 35 35 31 38 36 6 34 31 37 39 36 21 32 36 36 32 36 7 30 35 33 40 36 22 36 37 32 34 34 8 28 28 31 34 30 23 29 34 33 37 35 9 32 36 38 38 35 24 36 36 35 37 37 10 35 30 37 35 31 25 36 30 35 33 31 11 35 30 35 38 35 26 35 30 29 38 35 12 38 34 35 35 31 27 35 36 30 34 36 13 34 35 33 30 34 28 35 30 36 29 35 14 40 35 34 33 35 29 38 36 35 31 31 15 34 35 38 35 30 30 30 34 40 28 30 There are additional ways that box-and-whisker plots and other graphical dis- plays can aid the analyst. Multiple samples can be compared graphically. Plots of data can suggest relationships between variables. Graphs can aid in the detection of anomalies or outlying observations in samples. There are other types of graphical tools and plots that are used. These are discussed in Chapter 8 after we introduce additional theoretical details. 1.7 General Types of Statistical Studies 27 28 30 32 34 36 38 40 Paint Figure 1.11: Box-and-whisker plot for thickness of paint can “ears.” Other Distinguishing Features of a Sample There are features of the distribution or sample other than measures of center of location and variability that further deﬁne its nature. For example, while the median divides the data (or distribution) into two parts, there are other measures that divide parts or pieces of the distribution that can be very useful. Separation is made into four parts by quartiles, with the third quartile separating the upper quarter of the data from the rest, the second quartile being the median, and the ﬁrst quartile separating the lower quarter of the data from the rest. The distribution can be even more ﬁnely divided by computing percentiles of the distribution. These quantities give the analyst a sense of the so-called tails of the distribution (i.e., values that are relatively extreme, either small or large). For example, the 95th percentile separates the highest 5% from the bottom 95%. Similar deﬁnitions prevail for extremes on the lower side or lower tail of the distribution. The 1st percentile separates the bottom 1% from the rest of the distribution. The concept of percentiles will play a major role in much that will be covered in future chapters. 1.7 General Types of Statistical Studies: Designed Experiment, Observational Study, and Retrospective Study In the foregoing sections we have emphasized the notion of sampling from a pop-

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