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Everyday Misadventures! All rights reserved. All visual and textual content on this site including all names, characters, images, trademarks, and logos is protected by trademarks, copyrights, and other intellectual property rights owned by Seven Seas Entertainment or its subsidiaries, licensors, licensees, suppliers, and accounts. I learned the important distinction between precision and accuracy in a less malicious context.
For Christmas one year my wife bought me a golf range finder to calculate distances on the course from my golf ball to the hole. This is an improvement upon the standard yardage markers, which give distances only to the center of the green and are therefore accurate but less precise. With my Christmas-gift range finder I was able to know that I was I expected the precision of this nifty technology to improve my golf game. Instead, it got appreciably worse. There were two problems.
First, I used the stupid device for three months before I realized that it was set to meters rather than to yards; every seemingly precise calculation The lesson for me, which applies to all statistical analysis, is that even the most precise measurements or calculations should be checked against common sense. To take an example with more serious implications, many of the Wall Street risk management models prior to the financial crisis were quite precise. The problem was that the supersophisticated models were the equivalent of setting my range finder to meters rather than to yards.
The math was complex and arcane. The answers it produced were reassuringly precise. But the assumptions about what might happen to global markets that were embedded in the models were just plain wrong, making the conclusions wholly inaccurate in ways that destabilized not only Wall Street but the entire global economy. Even the most precise and accurate descriptive statistics can suffer from a more fundamental problem: a lack of clarity over what exactly we are trying to define, describe, or explain.
Statistical arguments have much in common with bad marriages; the disputants often talk past one another. Consider an important economic question: How healthy is American manufacturing? One often hears that American manufacturing jobs are being lost in huge numbers to China, India, and other low-wage countries. Which is it?
This would appear to be a case in which sound analysis of good data could reconcile these competing narratives. The British news magazine the Economist reconciled the two seemingly contradictory views of American manufacturing with the following graph.
In terms of output—the total value of goods produced and sold—the U. The United States remains a manufacturing powerhouse. But the graph in the Economist has a second line, which is manufacturing employment. The number of manufacturing jobs in the United States has fallen steadily; roughly six million manufacturing jobs were lost in the last decade.
Together, these two stories—rising manufacturing output and falling employment—tell the complete story. Manufacturing in the United States has grown steadily more productive, meaning that factories are producing more output with fewer workers.
This is good from a global competitiveness standpoint, for it makes American products more competitive with manufactured goods from low-wage countries. But there are a lot fewer manufacturing jobs , which is terrible news for the displaced workers who depended on those wages.
In this case and many others , the most complete story comes from including both figures, as the Economist wisely chose to do in its graph. Even when we agree on a single measure of success, say, student test scores, there is plenty of statistical wiggle room.
Sixty percent of our schools had lower test scores this year than last year. Eighty percent of our students had higher test scores this year than last year.
The unit of analysis is the entity being compared or described by the statistics—school performance by one of them and student performance by the other. Thirty states had falling incomes last year. The thirty states with falling average incomes are likely to be much smaller: Vermont, North Dakota, Rhode Island, and so on. The key lesson is to pay attention to the unit of analysis. Although the examples above are hypothetical, here is a crucial statistical question that is not: Is globalization making income inequality around the planet better or worse?
By one interpretation, globalization has merely exacerbated existing income inequalities; richer countries in as measured by GDP per capita tended to grow faster between and than poorer countries.
Down with globalization! But hold on a moment. The same data can and should be interpreted entirely differently if one changes the unit of analysis. Both countries are huge with a population over a billion ; each was relatively poor in Not only have China and India grown rapidly over the past several decades, but they have done so in large part because of their increased economic integration with the rest of the world.
Given that our goal is to ameliorate human misery, it makes no sense to give China population 1. The unit of analysis should be people, not countries.
What really happened between and is a lot like my fake school example above. Both companies provide cellular phone service. One of the primary concerns of most cell phone users is the quality of the service in places where they are likely to make or receive phone calls. Thus, a logical point of comparison between the two firms is the size and quality of their networks. Since the population is not evenly distributed across the physical geography of the United States, the key to good cell service the campaign argued implicitly is having a network in place where callers actually live and work, not necessarily where they go camping.
As someone who spends a fair bit of time in rural New Hampshire, however, my sympathies are with Verizon on this one. Our old friends the mean and the median can also be used for nefarious ends.
The mean of 3, 4, 5, 6, and is The median is the midpoint of the distribution; half of the observations lie above the median and half lie below. The median of 3, 4, 5, 6, and is 5. Now, the clever reader will see that there is a sizable difference between 24 and 5. If, for some reason, I would like to describe this group of numbers in a way that makes it look big, I will focus on the mean.
If I want to make it look smaller, I will cite the median. Consider the George W. Bush tax cuts, which were touted by the Bush administration as something good for most American families. But was that summary of the tax cut accurate? A relatively small number of extremely wealthy individuals were eligible for very large tax cuts; these big numbers skew the mean, making the average tax cut look bigger than what most Americans would likely receive.
The median is not sensitive to outliers, and, in this case, is probably a more accurate description of how the tax cuts affected the typical household.
Of course, the median can also do its share of dissembling because it is not sensitive to outliers. Suppose that you have a potentially fatal illness. The good news is that a new drug has been developed that might be effective.
The doctor informs you that the new drug increases the median life expectancy among patients with your disease by two weeks. That is hardly encouraging news; the drug may not be worth the cost and unpleasantness. Your insurance company refuses to pay for the treatment; it has a pretty good case on the basis of the median life expectancy figures.
Yet the median may be a horribly misleading statistic in this case. Suppose that many patients do not respond to the new treatment but that some large number of patients, say 30 or 40 percent, are cured entirely. This success would not show up in the median though the mean life expectancy of those taking the drug would look very impressive. In this case, the outliers—those who take the drug and live for a long time—would be highly relevant to your decision. Evolutionary biologist Stephen Jay Gould was diagnosed with a form of cancer that had a median survival time of eight months; he died of a different and unrelated kind of cancer twenty years later.
The definition of the median tells us that half the patients will live at least eight months—and possibly much, much longer than that. In contrast, the mean is affected by dispersion. From the standpoint of accuracy, the median versus mean question revolves around whether the outliers in a distribution distort what is being described or are instead an important part of the message. Once again, judgment trumps math.
Of course, nothing says that you must choose the median or the mean. Any comprehensive statistical analysis would likely present both. Those of a certain age may remember the following exchange as I recollect it between the characters played by Chevy Chase and Ted Knight in the movie Caddyshack.
Suppose you are trying to compare the price of a hotel room in London with the price of a hotel room in Paris. You send your six-year-old to the computer to do some Internet research, since she is much faster and better at it than you are. Your child reports back that hotel rooms in Paris are more expensive, around a night; a comparable room in London is a night. You would likely explain to your child the difference between pounds and euros, and then send her back to the computer to find the exchange rate between the two currencies so that you could make a meaningful comparison.
This example is loosely rooted in truth; after I paid rupees for a pot of tea in India, my daughter wanted to know why everything in India was so expensive. Obviously the numbers on currency from different countries mean nothing until we convert them into comparable units. What is the exchange rate between the pound and the euro, or, in the case of India, between the dollar and the rupee? This seems like a painfully obvious lesson—yet one that is routinely ignored, particularly by politicians and Hollywood studios.
These folks clearly recognize the difference between euros and pounds; instead, they overlook a more subtle example of apples and oranges: inflation. A dollar today is not the same as a dollar sixty years ago; it buys much less. This is such an important phenomenon that economists have terms to denote whether figures have been adjusted for inflation or not.
Nominal figures are not adjusted for inflation. A comparison of the nominal cost of a government program in to the nominal cost of the same program in merely compares the size of the checks that the Treasury wrote in those two years—without any recognition that a dollar in bought more stuff than a dollar in Yes, spending has gone up in nominal terms, but that does not reflect the changing value of the dollars being spent. Real figures, on the other hand, are adjusted for inflation.
Many websites, including that of the U. Bureau of Labor Statistics, have simple inflation calculators that will compare the value of a dollar at different points in time. The federal minimum wage—the number posted on the bulletin board in some remote corner of your office—is set by Congress. Supporters of a minimum wage should care about the real value of that wage, since the whole point of the law is to guarantee low-wage workers some minimum level of consumption for an hour of work, not to give them a check with a big number on it that buys less than it used to.
If that were the case, then we could just pay low-wage workers in rupees. Hollywood studios may be the most egregiously oblivious to the distortions caused by inflation when comparing figures at different points in time—and deliberately so. What were the top five highest-grossing films domestic of all time as of ? Avatar 2. Titanic 3. The Dark Knight 4.
Star Wars Episode IV 5. Shrek 2 Now you may feel that list looks a little suspect. These were successful films—but Shrek 2?
Was that really a greater commercial success than Gone with the Wind? The Godfather? No, no, and no. Hollywood likes to make each blockbuster look bigger and more successful than the last.
Instead, Hollywood studios and the journalists who report on them merely use nominal figures, which makes recent movies look successful largely because ticket prices are higher now than they were ten, twenty, or fifty years ago. The most accurate way to compare commercial success over time would be to adjust ticket receipts for inflation. So what are the top grossing films in the U. Gone with the Wind 2. Star Wars Episode IV 3.
The Sound of Music 4. The Ten Commandments In real terms, Avatar falls to number 14; Shrek 2 falls all the way to 31st. Even comparing apples and apples leaves plenty of room for shenanigans. As discussed in the last chapter, one important role of statistics is to describe changes in quantities over time. Are taxes going up? How many cheeseburgers are we selling compared with last year? By how much have we reduced the arsenic in our drinking water?
We often use percentages to express these changes because they give us a sense of scale and context. We understand what it means to reduce the amount of arsenic in the drinking water by 22 percent, whereas few of us would know whether reducing arsenic by one microgram the absolute reduction would be a significant change or not.
One way to make growth look explosive is to use percentage change to describe some change relative to a very low starting point. I live in Cook County, Illinois. However, I called off my massive antitax rally which was really still in the planning phase when I learned that this change would cost me less than a good turkey sandwich.
The Tuberculosis Sanitarium District deals with roughly a hundred cases a year; it is not a large or expensive organization. Obviously the flip side is true. A small percentage of an enormous sum can be a big number. Suppose the secretary of defense reports that defense spending will grow only 4 percent this year. Great news! In fact, that seemingly paltry 4 percent increase in the defense budget is more than the entire NASA budget and about the same as the budgets of the Labor and Treasury Departments combined.
In a similar vein, your kindhearted boss might point out that as a matter of fairness, every employee will be getting the same raise this year, 10 percent. Any comparison of a quantity changing over time must have a start point and an end point. One can sometimes manipulate those points in ways that affect the message. For his Republican audiences, he would offer the following slide with data on increases in defense spending under Ronald Reagan.
Clearly Reagan helped restore our commitment to defense and security, which in turn helped to win the Cold War. No one can look at these numbers and not appreciate the steely determination of Ronald Reagan to face down the Soviets. Defense Spending in Billions, — For the Democrats, my former professor merely used the same nominal data, but a longer time frame.
For this group, he pointed out that Jimmy Carter deserves credit for beginning the defense buildup. While the main point of statistics is to present a meaningful picture of things we care about, in many cases we also hope to act on these numbers.
NFL teams want a simple measure of quarterback quality so that they can find and draft talented players out of college.
Firms measure the performance of their employees so that they can promote those who are valuable and fire those who are not. But you had better be darn sure that what you are measuring is really what you are trying to manage. Consider school quality. And within each school, we have the similar challenge of measuring teacher quality, for the same basic reason.
The most common measure of quality for both schools and teachers is test scores. If students are achieving impressive scores on a well-conceived standardized test, then presumably the teacher and school are doing a fine job. Conversely, bad test scores are a clear signal that lots of people should be fired, sooner rather than later. These statistics can take us a long way toward fixing our public education system, right?
Any evaluation of teachers or schools that is based solely on test scores will present a dangerously inaccurate picture. Students who walk through the front door of different schools have vastly different backgrounds and abilities.
Students who live in affluent, highly educated communities are going to test well from the moment their parents drop them off at school on the first day of kindergarten.
The flip side is also true. There are schools with extremely disadvantaged populations in which teachers may be doing a remarkable job but the student test scores will still be low—albeit not nearly as low as they would have been if the teachers had not been doing a good job. At first glance, this seems an easy task, as we can simply give students a pretest and a posttest. If we know student test scores when they enter a particular school or classroom, then we can measure their performance at the end and attribute the difference to whatever happened in that school or classroom.
Alas, wrong again. Students with different abilities or backgrounds may also learn at different rates. Some students will grasp the material faster than others for reasons that have nothing to do with the quality of the teaching.
So if students in Affluent School A and Poor School B both start algebra at the same time and level, the explanation for the fact that students at Affluent School A test better in algebra a year later may be that the teachers are better, or it may be that the students were capable of learning faster—or both.
Researchers are working to develop statistical techniques that measure instructional quality in ways that account appropriately for different student backgrounds and abilities. Here is the part that is laugh-out-loud funny from a statistical standpoint: Several of the high schools consistently at the top of the rankings are selective enrollment schools, meaning that students must apply to get in, and only a small proportion of those students are accepted.
One of the most important admissions criteria is standardized test scores. This is the logical equivalent of giving an award to the basketball team for doing such an excellent job of producing tall students.
Even if you have a solid indicator of what you are trying to measure and manage, the challenges are not over. If you can measure the proportion of defective products coming off an assembly line, and if those defects are a function of things happening at the plant, then some kind of bonus for workers that is tied to a reduction in defective products would presumably change behavior in the right kinds of ways.
Each of us responds to incentives even if it is just praise or a better parking spot. Statistics measure the outcomes that matter; incentives give us a reason to improve those outcomes. Or, in some cases, just to make the statistics look better. If school administrators are evaluated—and perhaps even compensated—on the basis of the high school graduation rate for students in a particular school district, they will focus their efforts on boosting the number of students who graduate.
Of course, they may also devote some effort to improving the graduation rate, which is not necessarily the same thing. This is not merely a hypothetical example; it is a charge that was leveled against former secretary of education Rod Paige during his tenure as the Houston school superintendent. Paige was hired by President George W. Bush to be U. Houston reported a citywide dropout rate of 1.
The statistical chicanery with test scores was every bit as impressive. One way to improve test scores in Houston or anywhere else is to improve the quality of education so that students learn more and test better. This is a good thing. Another less virtuous way to improve test scores is to prevent the worst students from taking the test.
If the scores of the lowest-performing students are eliminated, the average test score for the school or district will go up, even if all the rest of the students show no improvement at all. In Texas, the statewide achievement test is given in tenth grade.
There was evidence that Houston schools were trying to keep the weakest students from reaching tenth grade.
In one particularly egregious example, a student spent three years in ninth grade and then was promoted straight to eleventh grade—a deviously clever way of keeping a weak student from taking a tenth-grade benchmark exam without forcing him to drop out which would have showed up on a different statistic.
The state of New York learned this the hard way. So is this a good policy? Yes, other than the fact that it probably ended up killing people. 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. A statistical index has all the potential pitfalls of any descriptive statistic—plus the distortions introduced by combining multiple indicators into a single number. By definition, any index is going to be sensitive to how it is constructed; it will be affected both by what measures go into the index and by how each of those measures is weighted.
For example, why does the NFL passer rating not include any measure of third down completions? In the end, the important question is whether the simplicity and ease of use introduced by collapsing many indicators into a single number outweighs the inherent inaccuracy of the process.
Sometimes that answer may be no, which brings us back as promised to the U. The benefit of the USNWR rankings is that they provide lots of information about thousands of schools in a simple and accessible way.
Of course, providing meaningful information is an enterprise entirely different from that of collapsing all of that information into a single ranking that purports to be authoritative. To critics, the rankings are sloppily constructed, misleading, and detrimental to the long-term interests of students.
According to U. In his general critique of rankings, Malcolm Gladwell offers a scathing though humorous indictment of the peer assessment methodology. He cites a questionnaire sent out by a former chief justice of the Michigan Supreme Court to roughly one hundred lawyers asking them to rank ten law schools in order of quality.
At the time, Penn State did not have a law school. News about whether the education they got during those four years actually improved their talents or enriched their knowledge. For example, one statistic used to calculate the rankings is financial resources per student; the problem is that there is no corresponding measure of how well that money is being spent.
An institution that spends less money to better effect and therefore can charge lower tuition is punished in the ranking process.
Colleges and universities also have an incentive to encourage large numbers of students to apply, including those with no realistic hope of getting in, because it makes the school appear more selective. This is a waste of resources for the schools soliciting bogus applications and for students who end up applying with no meaningful chance of being accepted. Since we are about to move on to a chapter on probability, I will bet that the U.
What is the best place? Number 1. If anything, impressive calculations can obscure nefarious motives. Judgment and integrity turn out to be surprisingly important. A detailed knowledge of statistics does not deter wrongdoing any more than a detailed knowledge of the law averts criminal behavior.
How does Netflix do that? Is there some massive team of interns at corporate headquarters who have used a combination of Google and interviews with my family and friends to determine that I might like a documentary about a former Pakistani prime minister? Of course not. Netflix has merely mastered some very sophisticated statistics.
Using that information, along with ratings from other customers and a powerful computer, Netflix can make shockingly accurate predictions about my tastes. Bhutto was recommended because of my five-star ratings for two other documentaries, Enron: The Smartest Guys in the Room and Fog of War. Correlation measures the degree to which two phenomena are related to one another.
For example, there is a correlation between summer temperatures and ice cream sales. When one goes up, so does the other. Two variables are positively correlated if a change in one is associated with a change in the other in the same direction, such as the relationship between height and weight.
Taller people weigh more on average ; shorter people weigh less. A correlation is negative if a positive change in one variable is associated with a negative change in the other, such as the relationship between exercise and weight. The tricky thing about these kinds of associations is that not every observation fits the pattern.
Sometimes short people weigh more than tall people. Still, there is a meaningful relationship between height and weight, and between exercise and weight. But a pattern consisting of dots scattered across the page is a somewhat unwieldy tool. If Netflix tried to make film recommendations for me by plotting the ratings for thousands of films by millions of customers, the results would bury the headquarters in scatter plots. Instead, the power of correlation as a statistical tool is that we can encapsulate an association between two variables in a single descriptive statistic: the correlation coefficient.
The correlation coefficient has two fabulously attractive characteristics. First, for math reasons that have been relegated to the appendix, it is a single number ranging from —1 to 1. A correlation of 1, often described as perfect correlation, means that every change in one variable is associated with an equivalent change in the other variable in the same direction.
A correlation of —1, or perfect negative correlation, means that every change in one variable is associated with an equivalent change in the other variable in the opposite direction. The closer the correlation is to 1 or —1, the stronger the association. A correlation of 0 or close to it means that the variables have no meaningful association with one another, such as the relationship between shoe size and SAT scores.
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