(TAN.1) Thinking About Numbers: The Epicurean Frame-Of-Mind

(Questions)

(A.1) Enter the world of Epicurus

You do not need to hold Epicurian beliefs. You must, however, think like Epicurus when you analyze data and report its findings. That is, you must be objective in your interpretation of data and acknowledge the challenge that stochasticity (randomness) poses.

An Athenean philosopher of the ancient world, Epicurus held revolutionary views. His school did the unheard of in 250 BC: admit women and slaves. In an age of superstitution that would not end until 2,000 years later, Epicurus resembled modern atheists, though today he is officially classified as a classical humanist.

If there are gods, they play no role in creating the world and do not interact with human affairs, he asserted. No god would intervene in response to a prayer, nor would the gods flood the earth as punishment for man's sins. That is what he believed.

Figure 1—Greek Philosopher, Epicurus (342-270 BC)

His theory of the physical universe was oddly accurate. Believing in an objective, physical world, Epicurus asserted that all matter was constituted from small, identical particles he called atoms. The physical laws he asserted described how these atoms interacted with one another, and these laws were universal. He was even able to argue, using simple logic and casual observation alone, that world contained not only matter but large amounts of empty space, an argument that in the development of quantum physics would be proven correct.

It was not a deterministic world in which he lived, though, because the atoms had both a deterministic and a stochastic (meaning random) component. This means that when observing the world we will always be challenged with determining whether something is caused by a stochastic or deterministic factor. It would be two millenia before humans discovered quantum physics, where they saw the true stochasticity behind the physical world and could marvel at Epicurus' prescience.^{(A1),(E1),(E2),(J1)}

What does this have to do with data analysis? Consider the following example, but do not jump to conclusions about what I am trying to say, for you may be offended at first. Give me a fair hearing, and hopefully when I am done you will take no offense.

(A.2) Does prayer work?

The Templeton Foundation recently spent over two million dollars to determine whether praying for people helped them recover from coronary bypass surgery. A total of 1,802 patients received the surgery, and were assigned to one of three categories.

1. Group 1 received prayers from three churches but did not know it.
2. Group 2 received no prayers.
3. Group 3 received prayers and were made aware of the fact.

This is an example of an empirical study, which means the results are derived solely from observation (not theory). Empirical research is only possible if we all live in the same physical universe, guided by the same laws, just as Epicurus proposed. If Sam and Bill's bodies are identical and have identical surgeries, their recovery should be the same, for they both live in the same universe.

Pretending to be an Epicurian, you must start from the position of being skeptical of prayer, even if you ardently believe in its power. If you are given the data from this experiment to analyze, and are told you would have to communicate the results to a diverse audience, you cannot admit to the efficacy of prayer unless the data show people in Group 1 recover better than Groups 2 and 3.

It turns out that prayer did not help. Groups 1 and 2 exhibited no noticeable difference. Group 3 actually did worse, which was unexpected, as knowing they were being prayed for should have had a placebo effect.^(D1)

Would you be able to accept these findings and report on the inefficacy of prayer? You must, if you are to be respected as an analyst and a person. Regardless of your beliefs, you would have to report that the data show no indication that prayer helps, even if prayer is performed three times a day by monks. You must pretend to hold Epicurus' beliefs, because the data say Epicurus is right, and in empirical work data are all that matter.

This doesn't mean you can't criticize how the experiment was performed or how the data were collected. If you believe the data are flawed due to a defect in how the research was performed, you should say so, but you can't search for flaws simply because you want to find evidence that prayer works. One can search for flaws out of a dedication to truth, though. Unless you believe the research has a deficiency and the data are bias, your integrity requires you to say prayer did not help, at least in this study.

(A.3) When prayer does work

Integrity also requires a person to communicate the benefits of religion, if the data warrant, even if you believe religion to be silly superstition. Studies have shown that religious people are happier, healthier, and recover better from trauma. Even if you are an atheist, so long as you have confidence in how these studies are conducted, you must acknowledge the benefits of religion.^(L1)

Video 1—Religious people are happier than atheists, according to Twitter
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(B.1) My prediction is not perfect: stochastic errors

If we are able to deny our most fervent beliefs when reporting an analysis of data, we must consider Epicurus' claim that the world is partly deterministic and partly stochastic. Think back to the prayer study. The data analysis concludes prayer does not work because the predicted recovery rate was the same for people with and without prayer. Yet, when observing the exact, precise recovery rates it may have been the case that those receiving prayer had slightly better recovery rates. Did I just contradict myself?

No, I didn't. Those receiving prayer may have recovered better simply due to the natural, random fluctuation among recovery rates. The researcher might see a slight improvement in recovery for those receiving prayer, but determined (using statistical tests) that the improvement is so small it is likely due to random chance.

The most important task in a statistical analysis is determining which patterns in data are due to stochasticity and which patterns cannot be attributable to chance, but must be caused by something real (like prayer).

(B.2) Satan, and the 9/11 Terrorist Attacks

While the World Trade Center burned, the black smoke took on a number of different forms, all of them simple manifestations of randomness. Because the smoke took on random appearances, it looked random...except for one part of the smoke that looked like a face—Satan's face? Could it really be Satan? After all, if Satan were to become active in the world, 9/11 is something he might do.

Figure 2—Satan's Face During 9/11?

Most of us dismissed the possibility that it was really Satan . Yet, if a Medieval Englishman saw Figure 2 he would have had little doubt it was Satan. What makes us so different from our ancestors? The main difference is that we believe in randomness. Most of the world may still be religious, but even the most devote Christian believes some things in the world are not directly caused by God, even if God lets it happen. Our ancestors believed everything that happened was a deliberate choice by the gods, such that there was no such thing as stochasticity. To understand what makes the modern world different from the times before, we must understand how views on stochasticity have changed.

(B.3) Talking to the gods: the case of ancient Jews

   Then the Lord hurled a violent wind on the sea, and such a violent storm arose on the sea that the ship threatened to break apart. The sailors were afraid, and each cried out to his god. They threw the ship's cargo into the sea to lighten the load. Meanwhile, Jonah had gone down to the lowest part of the vessel and had stretched out and fallen into a deep sleep.
The captain approached him and said, "What are you doing sound asleep? Get up! Call to your god. Maybe this god will consider us, and we won't perish."
    Come on!" the sailors said to each other. "Let's cast lots. Then we'll know who is to blame for the trouble we're in." So they cast lots, and the lot singled out Jonah. Then they said to him, "Tell us who is to blame for this trouble we're in. What is your business and where are you from? What is your country and what people are you from?"
   He [Jonah] answered them, "Pick me up and throw me into the sea so it may quiet down for you...".
   ...Then they picked up Johan and threw him into the sea, and the sea stopped raging...".
    Now the Lord had appointed a huge fish to swallow Jonah, and Jonah was in the fish three days and three nights.
—Holman Christian Standard Bible. Book of Jonah.

Most readers have heard the story of Jonah and the whale, and for those who actually read the story the part about casting lots must have been confusing. Unfortunately for us, little historical evidence exists to tell us exactly how lot-casting was performed, but we do know it was basically like rolling dice. For example, suppose sailors were caught in a frightening storm, and they believed it was God punishing one of them for his sins. They would need to identify that person and throw him overboard—maybe then God would stop the storm. However, if no sailor admitting to being the reprobate, how could they identify him? They would cast lots. If there were only twelve sailors on the ship, they could assign each man a number, roll the dice, and the resulting number indicated who was to blame. They would them sacrifice him so that the others might be saved.

If this sounds odd then you are understanding correctly, and the reason why the casting of lots seems to arbitrary is because—regardless of how religious you are—you view the world like a devote Epicurian. What I mean is that you do not assume every single thing that happens in life is determined by gods. There is some randomness in life, and the reason gambling is fun is because there is randomness. If we lose money in Vegas, it's not because god spites us, we believe (we hope?).

Many people in ancient history did believe that the will of the gods is manifested in what happens in the physical world. For Jews, who incorporated lot-casting extensively into their culture (to see for yourself, acquire an e-book of the Bible and search for the word "lots" or "cast lots"), God could be consulted directly by the casting of lots. In battle, if the Jews were conflicted about whether to attack or retreat, they may very well cast lots. Meaning, they flipped a coin: "heads", they attack; "tails", they retreat.

If you were an ancient Jew, almost everything statisticians do would be considered heresy because it assumes there is randomness in the world independent of God's will.

(B.4) Talking to the gods: Beware the Ides of March

The Romans were always keen to observe the world around them, interpreting odd events like natural disasters as the manifestation of gods' emotions. They would especially observe the behavior of birds. The English word "auspicious" is derived from the Latin word "auspice", which is an interpretation of bird behavior as a sign of favor from the gods. The Roman army would take sacred chickens with them, and if they were unsure whether they should perform an action—should we attack now or tomorrow?—they would place crums of cake on the bottom of the chickens' cages, and the gods would answer "yes" or "no" depending on whether the chickens ate the cake. This is the same as the ancient Jew casting lots, or a modern American rolling the dice.

In plays and movies about Julius Caesar, his murder is usually foretold by a soothsayer who says, "Beware the Ides of March." This is another way of saying, "Beware of March 15." Of course, Caesar is murdered on this day. The soothsayer would not receive direct prophecy from the gods, but would infer the gods' communication through omens and auspices. The video below contains clips from three movies where the ancient Greeks and Romans divine the future based on the guts of their livestock.

Video 2—Beware the Ides of March, Caesar!
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The Romans' superstitious beliefs were exploited by the ruling classes. Great men would always invent stories of auspices that occurred on the day of their birth to signify their favor from the gods. Such men might also lie about an omen they saw to make others think that the gods spoke to them. This is not all that unusual. Rome's most formidable enemy, Hannibal of Carthage, acquired much of his power by convincing others that he was ordained by gods to emerge victorious in all battles.

Dio attributes Hannibal's ability to predict future events to the fact that he understood divination by the inspection of entrails. At those critical moments when confidence in their mission had begun to ebb away from his troops, Hannibal seems to have ensured that some evidence of divine favor was presented by which the stock of Carthaginian self-belief was replenished and the troops were reminded that they were literally following in the footsteps of Heracles [or, Hercules] and his army.
—Miles, Richard. 2011. Carthage Must Be Destroyed. Chapter 10.

To my amazement, some cultures still portend the future by observing entrails. In Aka, India, marriages are not recognized until a particular type of cattle called Mithan has been slaughtered and its liver "read." For example, a couple is predicted to experience an accident but to live a largely happy life if the liver contains a small spot in a certain area.

Figure 3—Fortelling the future by observing a cow liver (Aka, India)

Consider one more contemporary example. Americans believe that we die from natural causes or accidents, both of which appear randomly. The people of the Trobriand Islands (near Papua New Guinea) do not share this view. They believe that death cannot derive from accident or natural causes. Every death is caused by sorcery. If an islander dislikes someone or is envious of them, those feelings are believed to release some sort of magic which can kill the person, so after a death they engage in various sorts of elaborate rituals, where each person attempts to prove that they are not the source of the person's death.^(F1)

Figure 4—Trobriand Islanders
 

(C) We believe in stochasticity (randomness)

If two identical twins receive the same surgery and are treated the same during their recovery, we do not expect their recoveries to be identical. Nor would we expect those differences to be a sign from God that one is favored over the other. There are so many things that influence a person's health that we don't expect to know the exact outcome from any surgery. We treat these unpredictable fluctuations in recovery rates as randomness, even though those fluctuations were caused by real factors. Because we allow for randomness in the world, we must acknowledge this randomness when we study data. Determining whether a pattern exhibited by data is due to randomness or a cause we can observe is the domain of statistics.

Our predictions almost always have some inaccuracy, what we call an "error" component. One can never predict how all 1,000 people will recover from a surgery with perfect accuracy, nor could you correctly predict the exact salaries of 500 men and women. The world is simply too complex to account for all the things that affect human health and salaries. Because we remain in ignorance of many things in our world, this ignorance is manifested in imperfect predictions. Our ignorance results in "stochastic errors."

The prayer study mentioned earlier requires researchers to determine how variations in recovery rates are affected by prayer, as opposed to variations influenced by the stochastic errors. If the patients prayed for recover better, but that improved rate could easily be caused by stochastic errors, we say prayer has no discernible effect. If the recovery rates of those prayed for are much higher than those not prayed for, and that difference cannot be caused by stochastic errors alone, we would say it is prayer, not randomness, that cause better recoveries.

Virtually all of statistics is dedicated to determining whether outcomes are the result of stochastic errors or deterministic (non-random) factors.

An analogy can be found in a series of coin flips. Suppose I tell you I have a technique of flipping coins that almost always results in heads. Interested, you ask me to prove it. I then flip a coin three times, resulting in a heads each time. You are not convinced, you say, because with only three flips, that could happen just by random chance. If I then flip a coin 1,000 times and it lands on heads 98% of the time, then you are convinced that there is something special about my coin flip, because the probability that 998 out of 1,000 coin flips could randomly land on heads is very low.

(D.1) The Middle Ages had no perspective on real life

Even if you are not a fan of art, you have probably seen art in really old churches and know they don't look realistic. Contrast that art to those from the Renaissance, where the scene is shockingly realistic. Most Medieval Art was for churches by anonymous artists, and they tend to lack something called perspective. In particular, the depth of the picture seems awkward, unrealistic. Objects in the distance should look smaller than they are depicted in Medieval art (see Figure 5).

Artists in the Middle Ages did not know how to draw pictures using perspective. The Greeks knew perspective, and so did the Romans. As Europe crumbled to many small duchies after Barbarians conquered Rome, much artistic knowledge was lost. What is more important, though, is the fact that artists did not want pictures to look realistic.

The form of Christianity that emerged in Medieval times saw the physical world as a shadow of the "true" world, and so pictures of God and angels were not supposed to look like the physical world we inhabit. The physical world was a fallen world: who would want to draw that? Most Medieval painters worked for the church, and since the church did not want art to look realistic, the artists themselves placed little importance on realism.

An example is given below, where the artist tries to represent depth but does so poorly. Notice that all the objects do not vanish towards the distance in the same proportion, which they would if they had a properly constructed central vanishing point. For example, the men in the background should be smaller than they are.

Figure 5—Fourteenth Century Illustration of William of Tyre's Histoire d'Outremer
(notice the lack of perspective—no central vanishing point)

Then came the Renaissance (French for "rebirth"). People were still religious, but they began to show an interest in Greek and Roman art and placed more emphasis on understanding the physical world they inhabited. They thought this "fallen world" to be more lovely and worthy of admiration than their Medieval ancestors. In their efforts to reclaim an Epicurian past—a desire partially derived from the discovery of an ancient Epicurean poem—they began to study the use of mathematics to help make their pictures look more realistic.

To reflect depth, they incorporated a central vanishing point where all objects receive to the distance in such a proportion to make the pictures look real. About this time people started framing pictures, with the idea that the picture should be like a window one uses to see real world.^(G1) In Medieval times, a picture was not supposed to be considered a window to the real, physical world, so they would not want to frame it.

Like our Renaissance forefathers, we yearn to see the world as it really exists, and we will need mathematics to do so.

Figure 6—St. Peter Healing a Cripple and the Raising of Tabitha by Masolino in 1425
(with central vanishing point)

(D.2) It is our ethical duty to understand the real world as it really is

This discussion may seem pointless. Who doesn't want to understand the world as it truly is? Many people.

(D.2.i) Karl Marx spent his life trying to convince the world that the Industrial Revolution forced people into poverty, yet whenever he gathered wage data, hoping to document a downward trend in wages, he found the opposite. Average incomes rose throughout the Industrial Revolution—in direct contradiction with the philosophy he attempted to spread. Did Marx acknowledge the reality of rising incomes? No, because he didn't care about reality, and nations like China are still attempting to escape the misery that Marx's philosophy wrought.

Marx wasn't alone in his shunning of reality.

(D.2.ii) A number of American and British journalists were communist sympathizers, and when they visited the Soviet Union they expected to find Utopia—but instead they found poverty, suffering, political oppression, religious oppression, alienation of minorities, tyranny, injustice, corruption, and even famine. Their findings were not reported to their American and British readers as they should have been, because these journalists assumed the misery they saw was temporary. They deliberately decided to lie about what they saw, for if they told the truth they were fearful America would not consider Communism as a serious political option. Their lies were full of good intentions, for they did not want to discourage anyone from what would eventually [they thought would] be a superior political philosophy. Though well-intentioned, their lies had bad outcomes, because America remained clueless about the true state of Russia for many years.

During the Great Depression, many families considered emigrating to Russia, where they thought everyone was given a job at a decent wage. Those that did emigrate left an American depression to enter a Russian graveyard, and suffered for the lies the journalists told.

Thousands of immigrants forsook the fabled American land of promise and returned to their old countries. Some one hundred thousand American workers in 1931 applied for jobs in what appeared to be a newly promising land, Soviet Russia.
—David M. Kenneddy. 1999. Freedom From Fear. Oxford University Press: NY, NY. Page 164.

Americans sympathetic to the Soviet cause continued to misrepresent the true state of Russia's economy—even economics professors. Paul Samuelson is one of the most famous economists of all time. A Nobel Prize winner, he is often credited with the incorporation of mathematics into economics, helping to create what is now one of the most mathematical scientific fields. He also wrote the most popular economics textbook of all time.

Samuelson believed in a strong central government, and so favored government involvement in the economy that he defended the centrally planned economy of the U.S.S.R. In his textbook he would include a chart showing the growth rate of the U.S. and Soviet economy. These were not real growth rates for the Soviet Union, but projected growth rates, and for some reason Samuelson never allowed the dismal rate of actual growth in Russia to change his view on it's projected growth rate. Every overly-optimistic growth rate he would justify as due to something like bad weather. Even in 1989, just as the walls of the U.S.S.R began to shake and fall, he wrote, "The Soviet economy is proof that, contrary to what many skeptics had earlier believed, a socialist command economy can function and even thrive."^(W1)

Samuelson was like a Medieval painter, in that he placed too little emphasis on what the world really looked like.

(D.2.iii) American scientists in the nineteenth century told their naive audiences that black people were intellectually inferior due to their smaller skulls, yet it would later be shown that these scientists did a horrible job of collecting data, for they confused black children (with smaller skulls) with black adults (who had larger skulls). Instead of comparing the skulls of white and black adults, they compared white adults to black children. The reason these esteemed researchers were so careless about the data is that they didn't care about the data. They had already decided white people were more intelligent, and they only cared that the data support their a priori beliefs. It is so sad to think about the misery their fraudulence has inflicted on the black race.

What is even sadder is the fact that bad science continues to encourage racism. Neo-Nazi Skinheads, for instance, still trade stories about how science proves the white race superior. Every now and then a Skinhead encounters real science—sees the world as it really is!—and changes his ways.

The story of how a Skinhead reformed his ways
    One of the first things that did change me was OJ Simpson. I watched these [stories] about DNA—I’d never heard of DNA before…the Philly newspaper would do these long articles on what DNA is…I started reading on how, like, science is breaking us all down and how we all have the same DNA except for one strand, except for like this one strand that makes us get caught for doing crimes, and then like we’ll have these old strands that prove we used to be Neanderthals and gorillas together…I was like, “Okay, I can see this.”
    Cause one thing I was always taught was science proved the [Skinhead] movement right, cause we are different and “science” had proved that…certain people get certain diseases, because of your ethnic background, our skulls are shaped a little bit different…our makeup is a little bit different, so… “science” had always backed-up my [racist] beliefs before: well, their science. And now [real science is proving the Skinhead’s science wrong to me].
    So I started to open my mind to other things…like how there’s only two different types of humans: women and men [not white humans, black humans, etc.].
—Frank Meenik (interviewee, quoted) and Phoebe Judge (host). August 15, 2012. “Anatomy of a Skinhead.” The Story. American Public Radio.

(D.3) Statistics is a window to the real world

(D.3.i) A business which does not integrate data into its strategies is threatened with extinction, because most large businesses collect data on a regular basis, especially now, where online accounts and credit cards provide new sources of data. The retailer Target collects volumes of data about consumers, using it to better understand their consumers, determine product variety, and set prices. They even use data on your purchases to predict whether you are pregnant, and can then make you special offers on baby supplies, as shown in the video below.

Video 3—Colbert Discusses Target's Statistical Strategies
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(D.3.ii) Sports team now hire statisticians in addition to players and managers. For years, baseball teams assumed the best measure of a hitter's value is their slugging percentage, which does not account for walks. Then came the Oakland A's in the 2000's. On a tiny budget they assembled a superior team by basing their hiring decisions on statistical analyses. The on-base percentage is a better measure of a hitter's worth, the A's discovered, because it gives players credit for having a "good eye" and reaching first base by being walked. Who cares if you reach first base by a hit or being walked? The A's then went out and sought players who had a high on-base percentage but could be purchased cheaply, allowing them to assemble a superior team on an inferior budget. They went from a losing team to becoming a contender for the World Series. Eventually, Boston copied their statistical strategy and did win the World Series. Now, every team uses the on-base percentage to determine a player's worth, and most major sports teams hire statisticians. The story of the A's dominance through statistical strategy became so famous a movie was made about it: Moneyball (and a The Simpsons parady of Moneyball).

In this course, we will use the same data the Oakland A's use, and perform the same statistical analysis, allowing us an intimate view of the true story behind Moneyball.

Video 4—Scenes from Moneyball
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Video 5—Parody of Moneyball by The Simpsons
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Using data other people do not have has long been used to better predict prices, and make profits from those predictions. What makes today different is the enormous amount of data we have and advanced technologies for crunching these numbers.

They say that Thales, perceiving by his skill in astrology (astronomy) that there would be great plenty of olives that year, while it was yet winter, hired at a low price all the oil presses in Miletus and Chios, there being no one to bid against him. But when the season came for making oil, many persons wanting them, he all at once let them upon what terms he pleased; and raising a large sum of money by that means, convinced them that it was easy for philosophers to be rich if they chose it.
—Will Durant. 1939. The Story of Civilization Part II: The Life of Greece. Chapter 3: The Heroic Age. Page 136. This story was told by Aristotle.

(D.3.iii) The current global warming debate is fought by one side convinced of imminent global collapse and the other side who believes carbon dioxide is not a greenhouse gas. One side neglects to acknowledge that water levels have been rising for decades with minimum harm (and possible benefits) and that there are more pressing human concerns, while the other side ignores the fact that the greenhouse gas effect is undeniably real. After all, Venus is much hotter than Mercury despite being further from the sun—simply because Venus' atmosphere contains greenhouse gases and Mercury's doesn't. Both sides ignore reality because it gives their arguments less nuance. For reasons I do not understand or like, moderation in opinion has become a sign of weakness. Our unwillingness to use data to understand the world may come back to bite us if we respond too little or too much to climate change.

Video 6—Why Is Venus Hotter Than Mercury? (from Carbon Nation, 2012)

We need a serious debate about whether the Bell-Shaped Curve describing current temperatures truly differs from the temperatures of past. How much has it shifted to the right? How much of that shift is caused by carbon emissions? That debate has been somewhat serious, and seems to have reached a conclusion that global warming is a real threat. What has not been debated is whether money spent to curb emissions is perhaps better spent on other causes, like micronutrient provision and Malaria prevention in developing countries (see the Copenhagen Consensus).

Figure 7—Global Temperatures and the Bell-Shaped Curve^(E3)
(Note: y-axis is probability; x-axis is temperature, with the right denoting hotter temperatures)

(D.3.iv) Statistics are used to help us see the real, physical world, as it really exists. Like the Renaissance painters, we value accurate numbers because we want to understand how the world works. Like the Renaissance painting held within a frame, it is the window we use to see what the world looks like, except that "looks like" does not refer to a static picture, but how the physical world changes over time, or how people interact with each other in war or in markets. That makes it sound like statistics are akin to a video, but that is not the case, because statistics help us see aspects of the world our eyes cannot—perspectives of the world that can be represented in the form of numbers and logic.

(E) See what is not there

To illustrate the window into the world statistics can provide, consider the following problem. This was a real statistical problem encountered in World War II, one that helped us win the war. You don't need to crunch numbers to answer it, just a clear and creative mind.

Abraham Wald was part of a research team in the last world war who was given the task of protecting Allied bomber planes from German shots fired from large guns located on the ground. Some parts of the plane could be reinforced with stronger steel, but because steel is heavy, the whole plane couldn't be reinforced. His team was asked to study the planes that returned from bombing campaigns, collect statistics on which parts of the planes were pierced from enemy bullets, and make recommendations on which parts of the plane should be reinforced with stronger steel.

The data they collected showed that most of the returning planes were hit on the wings and the tails. This wasn't always the case, but when the statistical averages were compiled, the average plane had more wing and tail punctures than anywhere else. Understandably, Abraham's team suggested reinforcing the wings and tails, but Abraham disagreed. Instead, he recommended the plane be reinforced in the parts of the plane which were typically not hit by bullets.

"What?" you say, surprised. I said the same thing when I first heard the story.

Wald was right, and he convinced the others on his team (and me). Remember the team was only able to study the planes which flew into battle and returned safely. This meant that planes hit in the tail and wings often returned safely. What about the planes they couldn't study—the ones shot down? If you imagine a German gunner firing large guns high up into the sky, there is no way they can aim for the tail and wings. They can barely see the plane, so they just start firing bullets in the plane's vicinity. This means that as the bullets approach the plane they are scattered, uniform. Each square inch of the plane exposed to the guns is just as likely to get hit as another part. This means that a plane is equally likely to get hit in the tail and wings than any other part of the plane. Those hit in the tail and wings made it home but planes hit in other locations did not, which meant that when planes were hit in places other than the tail and wings, they usually crashed.

Abraham's logic was praised and his advice heeded—and this illustration has been an excellent teaching tool ever since.^(J2) It illustrates that statistics are not about just numbers. It involves a marrying of numbers with an open, creative, organized, and logical mind. Most of us are not born with this mind. We develop it in classes and our own study. We will be developing that mind in this class.

(F) We are all Epicurians now

Throughout this course we will remain in the Epicurian frame-of-mind, where we report only what the data say (regardless of our personal beliefs) and we recognize that our predictions always have a stochastic error. We insist on seeing the world as it really exists, through a statistical window, because doing so provides practical benefits and because it is our moral obligation.

Being an Epicurian puts you in good company, for the same man who wrote, "We hold these truths to be self-evident, that all men are created equal,", also declared himself an Epircurean!

I TOO AM AN EPICUREAN.
—Jefferson, Thomas. Letter to William Short on October 31, 1819.

References

(A1) Amicus, Cassius. A Life Worthy of the Gods: The Life and Work of Epicurus.

(D1) Dawkins, Rickard. The God Delusion. 2006. Houghton Mifflin Company: NY, NY.

(E1) "Epicureanism." Encyclopedia Britannica. Encyclopædia Britannica Online Academic Edition. Encyclopedia Britannica Inc., 2012. Web. 27 April, 2012.

(E2) "Epicurus." Encyclopedia Britannica Online Academic Edition. Encyclopedia Britannica Inc., 2012. Web. 27 April, 2012.

(E3) The Economist. August 11, 2012. "Bell Weather." Page 69.

(F1) Fischer, Edward. 2004. "Lecture 12: Multiple Spouses and Matrilineality." Peoples and Cultures of the World. The Teaching Company.

(G1) Goldman, Steven L. "Renaissance Painting and the Techno-Science." Lecture Eleven. Great Scientific Ideas That Changed the World. The Teaching Company.

(J1) Johnston, Ian. Lucretius: On the Nature of Things. Introduction. Richer Resources Publications: Arlington, Virginia. 2010.

(J2) Jack Uldrich. January 29, 2010. "See What Isn't There." Unlearning 101.

(L1) Lyubomirsky, Sonja. The How of Happiness 2008. The Penguin Press: NY, NY.

(W1) White, Larry. 2012. The Clash of Economic Ideas.