Descriptive Statistics Quotes

What we call ‘normal’ is a product of repression, denial, splitting, projection, introjection and other forms of destructive action on experience. It is radically estranged from the structure of being. The more one sees this, the more senseless it is to continue with generalized descriptions of supposedly specifically schizoid, schizophrenic, hysterical ‘mechanisms.’ There are forms of alienation that are relatively strange to statistically ‘normal’ forms of alienation. The ‘normally’ alienated person, by reason of the fact that he acts more or less like everyone else, is taken to be sane. Other forms of alienation that are out of step with the prevailing state of alienation are those that are labeled by the ‘formal’ majority as bad or mad.

Descriptive statistics exist to simplify, which always implies some loss of nuance or detail.

The good news is that these descriptive statistics give us a manageable and meaningful summary of the underlying phenomenon. That’s what this chapter is about. The bad news is that any simplification invites abuse. Descriptive statistics can be like online dating profiles: technically accurate and yet pretty darn misleading.

But psychology is passing into a less simple phase. Within a few years what one may call a microscopic psychology has arisen in Germany, carried on by experimental methods, asking of course every moment for introspective data, but eliminating their uncertainty by operating on a large scale and taking statistical means. This method taxes patience to the utmost, and could hardly have arisen in a country whose natives could be bored. Such Germans as Weber, Fechner, Vierordt, and Wundt obviously cannot ; and their success has brought into the field an array of younger experimental psychologists, bent on studying the elements of the mental life, dissecting them out from the gross results in which they are embedded, and as far as possible reducing them to quantitative scales. The simple and open method of attack having done what it can, the method of patience, starving out, and harassing to death is tried ; the Mind must submit to a regular siege, in which minute advantages gained night and day by the forces that hem her in must sum themselves up at last into her overthrow. There is little of the grand style about these new prism, pendulum, and chronograph-philosophers. They mean business, not chivalry. What generous divination, and that superiority in virtue which was thought by Cicero to give a man the best insight into nature, have failed to do, their spying and scraping, their deadly tenacity and almost diabolic cunning, will doubtless some day bring about. No general description of the methods of experimental psychology would be instructive to one unfamiliar with the instances of their application, so we will waste no words upon the attempt.

To me the most troubling statistics focus on our friendships. In a survey given in 1985, people were asked to list their friends in response to the question “Over the last six months, who are the people with whom you discussed matters important to you?” The most common number of friends listed was three; 59 percent of respondents listed three or more friends fitting this description. The same survey was given again in 2004. This time the most common number of friends listed was zero. And only 37 percent of respondents listed three or more friends. Back in 1985, only 10 percent indicated that they had zero confidants. In 2004, this number skyrocketed to 25 percent. One out of every four of us is walking around with no one to share our lives with. Being social makes our lives better. Yet every indication is that we are getting less social, not more.

The implication that the change in nomenclature from “Multiple Personality Disorder” to “Dissociative Identity Disorder” means the condition has been repudiated and “dropped” from the Diagnostic and Statistical Manual (DSM) of the American Psychiatric Association is false and misleading. Many if not most diagnostic entities have been renamed or have had their names modified as psychiatry changes in its conceptualizations and classifications of mental illnesses. When the DSM decided to go with “Dissociative Identity Disorder” it put “(formerly multiple personality disorder)” right after the new name to signify that it was the same condition. It’s right there on page 526 of DSM-IV-R. There have been four different names for this condition in the DSMs over the course of my career. I was part of the group that developed and wrote successive descriptions and diagnostic criteria for this condition for DSM-III-R, DSM–IV, and DSM-IV-TR. While some patients have been hurt by the impact of material that proves to be inaccurate, there is no evidence that scientifically demonstrates the prevalence of such events. Most material alleged to be false has been disputed by someone, but has not been proven false. Finally, however intriguing the idea of encouraging forgetting troubling material may seem, there is no evidence that it is either effective or safe as a general approach to treatment. There is considerable belief that when such material is put out of mind, it creates symptoms indirectly, from “behind the scenes.” Ironically, such efforts purport to cure some dissociative phenomena by encouraging others, such as Dissociative Amnesia.

Rather than saying, "Median survival is eleven months" or "You have a ninety-five percent chance of being dead in two years," I'd say, "Most patients live many months to a couple of years." This was, to me, a more honest description. The problem is that you can't tell an individual patient where she sits on the curve: Will she die in six months or sixty? I came to believe that it is irresponsible to be more precise than you can be accurate. Those apocryphal doctors who gave specific numbers ("The doctor told me I had six months to live"): Who were they, I wondered, and who taught them statistics?

The concern is so acute that the political scientist Jacqueline Stevens has even suggested that research and even emails discussing biological differences across populations should be banned, and that the United States “should issue a regulation prohibiting its staff or grantees…from publishing in any form—including internal documents and citations to other studies—claims about genetics associated with variables of race, ethnicity, nationality, or any other category of population that is observed or imagined as heritable unless statistically significant disparities between groups exist and description of these will yield clear benefits for public health, as deemed by a standing committee to which these claims must be submitted and authorized.

After surgery, we talked again, this time discussing chemo, radiation, and prognosis. By this point, I had learned a couple of basic rules. First, detailed statistics are for research halls, not hospital rooms. The standard statistic, the Kaplan-Meier curve, measures the number of patients surviving over time. It is the metric by which we gauge progress, by which we understand the ferocity of a disease. For glioblastoma, the curve drops sharply until only about 5 percent of patients are alive at two years. Second, it is important to be accurate, but you must always leave some room for hope. Rather than saying, “Median survival is eleven months” or “You have a ninety-five percent chance of being dead in two years,” I’d say, “Most patients live many months to a couple of years.” This was, to me, a more honest description. The problem is that you can’t tell an individual patient where she sits on the curve: Will she die in six months or sixty? I came to believe that it is irresponsible to be more precise than you can be accurate. Those apocryphal doctors who gave specific numbers (“ The doctor told me I had six months to live”): Who were they, I wondered, and who taught them statistics?

The two hit it off well, because de Broglie was trying, like Einstein, to see if there were ways that the causality and certainty of classical physics could be saved. He had been working on what he called “the theory of the double solution,” which he hoped would provide a classical basis for wave mechanics. “The indeterminist school, whose adherents were mainly young and intransigent, met my theory with cold disapproval,” de Broglie recalled. Einstein, on the other hand, appreciated de Broglie’s efforts, and he rode the train with him to Paris on his way back to Berlin. At the Gare du Nord they had a farewell talk on the platform. Einstein told de Broglie that all scientific theories, leaving aside their mathematical expressions, ought to lend themselves to so simple a description “that even a child could understand them.” And what could be less simple, Einstein continued, than the purely statistical interpretation of wave mechanics! “Carry on,” he told de Broglie as they parted at the station. “You are on the right track!” But he wasn’t. By 1928, a consensus had formed that quantum mechanics was correct, and de Broglie relented and adopted that view. “Einstein, however, stuck to his guns and continued to insist that the purely statistical interpretation of wave mechanics could not possibly be complete,” de Broglie recalled, with some reverence, years later.

Depression” is a problematic word. We all believe we know what it means because we toss it off so easily: “Oh, I’m depressed; I got a run in my stocking.” At the same time, when we are describing severe psychopathology, we presume that because the word is descriptive, it offers a definition as well. We move to the next step and presume that because we can take a picture of the brain and “see” depression, it therefore is real. It has been occurring to me more and more, not just from these conversations, but also from my work, that when the brain is in clearly different states—and the Diagnostic and Statistical Manual of Mental Disorders80 says they are the same pathology—maybe our definition of the psychopathology is too broad. We need to redefine the nature of suffering to understand how it may be a condition more like dukkha, instead of a disease with a physiological cause as specific as something like a brain lesion. That is not to deny that true psychopathology exists, or that the patients I take care of do not suffer from a brain disease. I believe very strongly that they do. But I also see patients who, with focused attention and by acquiring new skill sets, can bring themselves out of it in the same way that William James did when he decided to focus his attention from inside to outside. The ability to focus attention means your brain is in a different state. Maybe we ought to understand those as different definitions of illness. What I’ve learned from all of you is that maybe we have to start making those distinctions more strongly. That will allow us to focus attention on how to handle ourselves in a world with natural levels of suffering, and help us not stigmatize people who don’t have the brain capacity to even start. Those are two separate items.

Remedies exist for correcting substantial departures from normality, but these remedies may make matters worse when departures from normality are minimal. The first course of action is to identify and remove any outliers that may affect the mean and standard deviation. The second course of action is variable transformation, which involves transforming the variable, often by taking log(x), of each observation, and then testing the transformed variable for normality. Variable transformation may address excessive skewness by adjusting the measurement scale, thereby helping variables to better approximate normality.8 Substantively, we strongly prefer to make conclusions that satisfy test assumptions, regardless of which measurement scale is chosen.9 Keep in mind that when variables are transformed, the units in which results are expressed are transformed, as well. An example of variable transformation is provided in the second working example. Typically, analysts have different ways to address test violations. Examination of the causes of assumption violations often helps analysts to better understand their data. Different approaches may be successful for addressing test assumptions. Analysts should not merely go by the result of one approach that supports their case, ignoring others that perhaps do not. Rather, analysts should rely on the weight of robust, converging results to support their final test conclusions. Working Example 1 Earlier we discussed efforts to reduce high school violence by enrolling violence-prone students into classes that address anger management. Now, after some time, administrators and managers want to know whether the program is effective. As part of this assessment, students are asked to report their perception of safety at school. An index variable is constructed from different items measuring safety (see Chapter 3). Each item is measured on a seven-point Likert scale (1 = strongly disagree to 7 = strongly agree), and the index is constructed such that a high value indicates that students feel safe.10 The survey was initially administered at the beginning of the program. Now, almost a year later, the survey is implemented again.11 Administrators want to know whether students who did not participate in the anger management program feel that the climate is now safer. The analysis included here focuses on 10th graders. For practical purposes, the samples of 10th graders at the beginning of the program and one year later are regarded as independent samples; the subjects are not matched. Descriptive analysis shows that the mean perception of

safety at the beginning of the program was 4.40 (standard deviation, SD = 1.00), and one year later, 4.80 (SD = 0.94). The mean safety score increased among 10th graders, but is the increase statistically significant? Among other concerns is that the standard deviations are considerable for both samples. As part of the analysis, we conduct a t-test to answer the question of whether the means of these two distributions are significantly different. First, we examine whether test assumptions are met. The samples are independent, and the variables meet the requirement that one is continuous (the index variable) and the other dichotomous. The assumption of equality of variances is answered as part of conducting the t-test, and so the remaining question is whether the variables are normally distributed. The distributions are shown in the histograms in Figure 12.3.12 Are these normal distributions? Visually, they are not the textbook ideal—real-life data seldom are. The Kolmogorov-Smirnov tests for both distributions are insignificant (both p > .05). Hence, we conclude that the two distributions can be considered normal. Having satisfied these t-test assumptions, we next conduct the t-test for two independent samples. Table 12.1 shows the t-test results. The top part of Table 12.1 shows the descriptive statistics, and the bottom part reports the test statistics. Recall that the t-test is a two-step test. We first test whether variances are equal. This is shown as the “Levene’s test for equality of variances.” The null hypothesis of the Levene’s test is that variances are equal; this is rejected when the p-value of this Levene’s test statistic is less than .05. The Levene’s test uses an F-test statistic (discussed in Chapters 13 and 15), which, other than its p-value, need not concern us here. In Table 12.1, the level of significance is .675, which exceeds .05. Hence, we accept the null hypothesis—the variances of the two distributions shown in Figure 12.3 are equal. Figure 12.3 Perception of High School Safety among 10th Graders Table 12.1 Independent-Samples T-Test: Output Note: SD = standard deviation. Now we go to the second step, the main purpose. Are the two means (4.40 and 4.80)

different from 3.5. However, it is different from larger values, such as 4.0 (t = 2.89, df = 9, p = .019). Another example of this is provided in the Box 12.2. Finally, note that the one-sample t-test is identical to the paired-samples t-test for testing whether the mean D = 0. Indeed, the one-sample t-test for D = 0 produces the same results (t = 2.43, df = 9, p = .038). In Greater Depth … Box 12.2 Use of the T-Test in Performance Management: An Example Performance benchmarking is an increasingly popular tool in performance management. Public and nonprofit officials compare the performance of their agencies with performance benchmarks and draw lessons from the comparison. Let us say that a city government requires its fire and medical response unit to maintain an average response time of 360 seconds (6 minutes) to emergency requests. The city manager has suspected that the growth in population and demands for the services have slowed down the responses recently. He draws a sample of 10 response times in the most recent month: 230, 450, 378, 430, 270, 470, 390, 300, 470, and 530 seconds, for a sample mean of 392 seconds. He performs a one-sample t-test to compare the mean of this sample with the performance benchmark of 360 seconds. The null hypothesis of this test is that the sample mean is equal to 360 seconds, and the alternate hypothesis is that they are different. The result (t = 1.030, df = 9, p = .330) shows a failure to reject the null hypothesis at the 5 percent level, which means that we don’t have sufficient evidence to say that the average response time is different from the benchmark 360 seconds. We cannot say that current performance of 392 seconds is significantly different from the 360-second benchmark. Perhaps more data (samples) are needed to reach such a conclusion, or perhaps too much variability exists for such a conclusion to be reached. NONPARAMETRIC ALTERNATIVES TO T-TESTS The tests described in the preceding sections have nonparametric alternatives. The chief advantage of these tests is that they do not require continuous variables to be normally distributed. The chief disadvantage is that they are less likely to reject the null hypothesis. A further, minor disadvantage is that these tests do not provide descriptive information about variable means; separate analysis is required for that. Nonparametric alternatives to the independent-samples test are the Mann-Whitney and Wilcoxon tests. The Mann-Whitney and Wilcoxon tests are equivalent and are thus discussed jointly. Both are simplifications of the more general Kruskal-Wallis’ H test, discussed in Chapter 11.19 The Mann-Whitney and Wilcoxon tests assign ranks to the testing variable in the exact manner shown in Table 12.4. The sum of the ranks of each group is computed, shown in the table. Then a test is performed to determine the statistical significance of the difference between the sums, 22.5 and 32.5. Although the Mann-Whitney U and Wilcoxon W test statistics are calculated differently, they both have the same level of statistical significance: p = .295. Technically, this is not a test of different means but of different distributions; the lack of significance implies that groups 1 and 2 can be regarded as coming from the same population.20 Table 12.4 Rankings of

For comparison, we use the Mann-Whitney test to compare the two samples of 10th graders discussed earlier in this chapter. The sum of ranks for the “before” group is 69.55, and for the “one year later group,” 86.57. The test statistic is significant at p = .019, yielding the same conclusion as the independent-samples t-test, p = .011. This comparison also shows that nonparametric tests do have higher levels of significance. As mentioned earlier, the Mann-Whitney test (as a nonparametric test) does not calculate the group means; separate, descriptive analysis needs to be undertaken for that information. A nonparametric alternative to the paired-samples t-test is the Wilcoxon signed rank test. This test assigns ranks based on the absolute values of these differences (Table 12.5). The signs of the differences are retained (thus, some values are positive and others are negative). For the data in Table 12.5, there are seven positive ranks (with mean rank = 6.57) and three negative ranks (with mean rank = 3.00). The Wilcoxon signed rank test statistic is normally distributed. The Wilcoxon signed rank test statistic, Z, for a difference between these values is 1.89 (p = .059 > .05). Hence, according to this test, the differences between the before and after scores are not significant. Getting Started Calculate a t-test and a Mann-Whitney test on data of your choice. Again, nonparametric tests result in larger p-values. The paired-samples t-test finds that p = .038 < .05, providing sufficient statistical evidence to conclude that the differences are significant. It might also be noted that a doubling of the data in Table 12.5 results in finding a significant difference between the before and after scores with the Wilcoxon signed rank test, Z = 2.694, p = .007. Table 12.5 Wilcoxon Signed Rank Test The Wilcoxon signed rank test can also be adapted as a nonparametric alternative to the one-sample t-test. In that case, analysts create a second variable that, for each observation, is the test value. For example, if in Table 12.5 we wish to test whether the mean of variable “before” is different from, say, 4.0, we create a second variable with 10 observations for which each value is, say, 4.0. Then using the Wilcoxon signed rank test for the “before” variable and this new,

when does an electron know when to jump, and how does it decide where to jump? Reasonably enough, Rutherford wanted to know what underlying process controlled the quantum jumping: "Bohr's answer was remarkable. Bohr suggested that the whole process was fundamentally random, and could only be considered by statistical methods: every change in the state of an atom should be regarded as an individual process, incapable of more detailed description. We are here so far removed from a causal description that an atom may in general even be said to possess a free choice between various possible transitions.

The standard deviation is the descriptive statistic that allows us to assign a single number to this dispersion around the mean.

Rather than taking the time to understand the interesting parts of scientific research, armchair statisticians snipe at news articles, using the vague description of the study regurgitated from some overenthusiastic university press release to criticize the statistical design of the research.[

The Mantle of Science For a decade or so, A.A. grew modestly. But, lacking scientific confirmation, it remained a relatively small sectarian movement, occasionally receiving a boost in popular magazines. The great surge in the popularity of the A.A. disease concept came when it received what seemed to be impeccable scientific support. Two landmark articles by E. M. Jellinek, published in 1946 and 1952, proposed a scientific understanding of alcoholism that seemed to confirm major elements of the A.A. view.12 Jellinek, then a research professor in applied physiology at Yale University, was a distinguished biostatistician and one of the early leaders in the field of alcohol studies. In his first paper he presented some eighty pages of elaborately detailed description, statistics, and charts that depicted what he considered to be a typical or average alcoholic career. Jellinek cautioned his readers about the limited nature of his data, and he explicitly acknowledged differences among individual drinkers. But from the data's "suggestive" value, he proceeded to develop a vividly detailed hypothesis.

With a digital display having few pixels, symmetries are common but there is very little meaning because the image is very course grained. As we reiterate and begin breaking the symmetry of the individual pixels an image will begin to appear. Time is related to the process of reiteration and truth is related to the symmetry, with meaning being related to the image created. we do not know where the symmetry or the reiterations come from but the image is emergent. The idea of a quantum random walk in state space says that every complex event is statistically impossible and even though the probability space is very large, it is navigated and expressed, as I understand it, in a tree like structure or a fractal structure. This is a computational expression of the material world that looks very much like a display on a monitor. The decision engine is generating value. There is a bifurcation of the fitness into different dimensions and like the human brain which is said to have at least eleven dimensions, the dimensions are not constrained by a physical geometry, they are computational. Another way we can look at this would be to say that every behavior we can measure is constrained by a network of associations just like the nodes on the internet and the conservation laws become approximately true because of levels of description. All material expressions are constructed from a network of associations and are only reducible to some degree of resolution. If we are talking about information, then it is only reducible to some approximate explanation.

Statistics became what it still is today: a form of political rhetoric. The categories that statisticians had to develop were reified in the hands of government bureaucracies. Categories that statistics made technically necessary—classes, strata, castes, ethnic groups—acquired the power to mold reality for administrative departments and, indeed, in society’s perception of itself. Statistics had two faces: a tool for sociological description and explanation, and a powerful mechanism for stereotyping and labeling people

I described research showing that people who grow up in Western, educated, industrial, rich, and democratic (WEIRD) societies are statistical outliers on many psychological measures, including measures of moral psychology. I also showed that: • The WEIRDer you are, the more you perceive a world full of separate objects, rather than relationships. • Moral pluralism is true descriptively. As a simple matter of anthropological fact, the moral domain varies across cultures. • The moral domain is unusually narrow in WEIRD cultures, where it is largely limited to the ethic of autonomy (i.e., moral concerns about individuals harming, oppressing, or cheating other individuals). It is broader—including the ethics of community and divinity—in most other societies, and within religious and conservative moral matrices within WEIRD societies. • Moral matrices bind people together and blind them to the coherence, or even existence, of other matrices. This makes it very difficult for people to consider the possibility that there might really be more than one form of moral truth, or more than one valid framework for judging people or running a society. In the next three chapters I’ll catalogue the moral intuitions, showing exactly what else there is beyond harm and fairness. I’ll show how a small set of innate and universal moral foundations can be used to construct a great variety of moral matrices. I’ll offer tools you can use to understand moral arguments emanating from matrices that are not your own. SIX

Heat, entropy, and the lower entropy of the past are notions that belong to an approximate, statistical description of nature.

Or I can just tell you that at the end of the 2011 season Derek Jeter had a career batting average of .313. That is a descriptive statistic, or a “summary statistic.

Statistically, the incidence of ‘true statements’—definitional, demonstrative, tautological—in any given mass of discourse is probably small. The current of language is intentional, it is instinct with purpose in regard to audience and situation. It aims at attitude and assent. It will, except on specialized occasions of logically formal, prescriptive, or solemnized utterance, not convey ‘truth’ or ‘information of facts’ at all. We communicate motivated images, local frameworks of feeling. All descriptions are partial. We speak less than the truth, we fragment in order to reconstruct desired alternatives, we select and elide.

Cardiologists obviously care about their “scorecard.” However, the easiest way for a surgeon to improve his mortality rate is not by killing fewer people; presumably most doctors are already trying very hard to keep their patients alive. The easiest way for a doctor to improve his mortality rate is by refusing to operate on the sickest patients. According to a survey conducted by the School of Medicine and Dentistry at the University of Rochester, the scorecard, which ostensibly serves patients, can also work to their detriment: 83 percent of the cardiologists surveyed said that, because of the public mortality statistics, some patients who might benefit from angioplasty might not receive the procedure; 79 percent of the doctors said that some of their personal medical decisions had been influenced by the knowledge that mortality data are collected and made public. The sad paradox of this seemingly helpful descriptive statistic is that cardiologists responded rationally by withholding care from the patients who needed it most.

All of these amplifiers—our tendency to subtract certain emotions from our self-description, to see missteps as situational rather than personality-driven, and to focus on our good intentions rather than our impact on others—add up. And so we get statistics like this: 37 percent of Americans report being victims of workplace bullies, but fewer than 1 percent report being bullies. It’s true that one bully can have many victims, but it’s unlikely that each averages thirty-seven.11 What’s more likely is that at least some percentage of those feeling bullied are receiving ill treatment from people who are unaware of their impact. They judge themselves by their intentions (“I was just trying to get the job done right!”) and attribute others’ reactions to their hypersensitivity (character) or the context (“Look, it was a tense situation. Anyone would have reacted that way”). Telling this latter group not to bully others is no solution, because they don’t realize that they’re doing so.

suffered greater wetland loss than watersheds with smaller surrounding populations. Most watersheds have suffered no or only very modest losses (less than 3 percent during the decade in question), and few watersheds have suffered more than a 4 percent loss. The distribution is thus heavily skewed toward watersheds with little wetland losses (that is, to the left) and is clearly not normally distributed.6 To increase normality, the variable is transformed by twice taking the square root, x.25. The transformed variable is then normally distributed: the Kolmogorov-Smirnov statistic is 0.82 (p = .51 > .05). The variable also appears visually normal for each of the population subgroups. There are four population groups, designed to ensure an adequate number of observations in each. Boxplot analysis of the transformed variable indicates four large and three small outliers (not shown). Examination suggests that these are plausible and representative values, which are therefore retained. Later, however, we will examine the effect of these seven observations on the robustness of statistical results. Descriptive analysis of the variables is shown in Table 13.1. Generally, large populations tend to have larger average wetland losses, but the standard deviations are large relative to (the difference between) these means, raising considerable question as to whether these differences are indeed statistically significant. Also, the untransformed variable shows that the mean wetland loss is less among watersheds with “Medium I” populations than in those with “Small” populations (1.77 versus 2.52). The transformed variable shows the opposite order (1.06 versus 0.97). Further investigation shows this to be the effect of the three small outliers and two large outliers on the calculation of the mean of the untransformed variable in the “Small” group. Variable transformation minimizes this effect. These outliers also increase the standard deviation of the “Small” group. Using ANOVA, we find that the transformed variable has unequal variances across the four groups (Levene’s statistic = 2.83, p = .41 < .05). Visual inspection, shown in Figure 13.2, indicates that differences are not substantial for observations within the group interquartile ranges, the areas indicated by the boxes. The differences seem mostly caused by observations located in the whiskers of the “Small” group, which include the five outliers mentioned earlier. (The other two outliers remain outliers and are shown.) For now, we conclude that no substantial differences in variances exist, but we later test the robustness of this conclusion with consideration of these observations (see Figure 13.2). Table 13.1 Variable Transformation We now proceed with the ANOVA analysis. First, Table 13.2 shows that the global F-test statistic is 2.91, p = .038 < .05. Thus, at least one pair of means is significantly different. (The term sum of squares is explained in note 1.) Getting Started Try ANOVA on some data of your choice. Second, which pairs are significantly different? We use the Bonferroni post-hoc test because relatively few comparisons are made (there are only four groups). The computer-generated results (not shown in Table 13.2) indicate that the only significant difference concerns the means of the “Small” and “Large” groups. This difference (1.26 - 0.97 = 0.29 [of transformed values]) is significant at the 5 percent level (p = .028). The Tukey and Scheffe tests lead to the same conclusion (respectively, p = .024 and .044). (It should be noted that post-hoc tests also exist for when equal variances are not assumed. In our example, these tests lead to the same result.7) This result is consistent with a visual reexamination of Figure 13.2, which shows that differences between group means are indeed small. The Tukey and

others seek to create and predict classifications through independent variables. Table 18.4 Factor Analysis Note: Factor analysis with Varimax rotation. Source: E. Berman and J. West. (2003). “What Is Managerial Mediocrity? Definition, Prevalence and Negative Impact (Part 1).” Public Performance & Management Review, 27 (December): 7–27. Multidimensional scaling and cluster analysis aim to identify key dimensions along which observations (rather than variables) differ. These techniques differ from factor analysis in that they allow for a hierarchy of classification dimensions. Some also use graphics to aid in visualizing the extent of differences and to help in identifying the similarity or dissimilarity of observations. Network analysis is a descriptive technique used to portray relationships among actors. A graphic representation can be made of the frequency with which actors interact with each other, distinguishing frequent interactions from those that are infrequent. Discriminant analysis is used when the dependent variable is nominal with two or more categories. For example, we might want to know how parents choose among three types of school vouchers. Discriminant analysis calculates regression lines that distinguish (discriminate) among the nominal groups (the categories of the dependent variable), as well as other

Something, which the police called a bomb, had exploded in his shed. Investigations were begun, and the efforts of the authorities were soon to be categorized by the appropriate officals as "feverish", for bombs began to go off all over the place. The police collected fragments of the exploded bombs, and the press, anxious to help the police in their work, published impressive pictures of the fragments as well as a drawing of a reconstructed bomb together with a very detailed description of how it had been made.The police had done a really first-rate job. Even my brother and myself, both of us extremely untalented men in technical matters, could easily grasp how the bomb makers had gone to work. A large quantity of ordinary black gunpowder, such as is the be found in the cartridges sold for shoutguns, was encased in plasticine; in it was embedded an explosive cap, of the type used in hand grenades during the war, at the end of a thin wire; the other end of the wire was joined to the battery of a pocket flashlight -- obtainable at any village store -- and thence to the alarm mechanism of an ordinary alarm clock. The whole contratation was packed into a soapbox. Of course my brother did his duty as a journalist.He published the police report, together with the illustrations, on page one. It was not my brother's doing that this issue of the paper had a most spectacular success and that for weeks men were still buying it; no. the credit for that must go to the police; they had done their bit to ensure that the peasantry of Schleswig-Holstein would have a healthy occupation during the long winter evenings. Instead of just sitting and indulging in stupid thoughts, or doing crossword puzzles, or assembling to hear inflamatory speeches, the peasantery was henceforth quetly and busily engaged in procuring soapboxes and alarm clock and flashlight batteries. And then the bombs really began to go of.... Nobody ever asked me what I was actually doing in Schleswig=Holstein, save perhaps Dr. Hirschfeldt, a high official in the Prussian Ministry of the Interior, who had recently taken to frequenting Salinger's salon. Occasionally, and casually, he would glance at ne with his green eyes an honour me with a question, such as: "And what are the peasants up to in the north?" To which I would usually only reply: "Thank you for your interest. According to the statistics, the standard of living is going up -- in particular, there has been in increased demand for alarm clocks.

In nearly every episode of fear mongering I discussed in the previous chapters as well, people with fancy titles appeared. Hardly ever were they among the leading figures in their field. Often they were more akin to the authorities in “War of the Worlds”: gifted orators with elevated titles. Arnold Nerenberg and Marty Rimm come immediately to mind. Nerenberg (a.k.a. “America’s road-rage therapist”) is a psychologist quoted uncritically in scores of stories even though his alarming statistics and clinical descriptions have little scientific evidence behind them. Rimm, the college student whom Time glorified in its notorious “cyberporn” issue as the “Principal Investigator” of “a research team,” is almost totally devoid of legitimate credentials.

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The first descriptive task is often to find some measure of the “middle” of a set of data, or what statisticians might describe as its “central tendency.” What is the typical quality experience for your printers compared with those of the competition?

Descriptive statistics can be like online dating profiles: technically accurate and yet pretty darn misleading.

processed, and transformed into a format that is suitable for analysis. This often involves removing duplicate data, correcting errors, and dealing with missing values. After data is prepared, exploratory data analysis is performed to better understand the data and identify patterns, trends, and outliers. Descriptive statistics, data visualization, and data clustering techniques are often used to explore data. Once the data is understood, statistical methods such as hypothesis testing and regression analysis can be applied to identify relationships and make predictions.