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<li><a href="#variability-and-small-samples" id="toc-variability-and-small-samples" class="nav-link active" data-scroll-target="#variability-and-small-samples"><span class="header-section-number">14.1</span> Variability and small samples</a></li>
<li><a href="#regression-to-the-mean" id="toc-regression-to-the-mean" class="nav-link" data-scroll-target="#regression-to-the-mean"><span class="header-section-number">14.2</span> Regression to the mean</a></li>
<li><a href="#summary-and-conclusion" id="toc-summary-and-conclusion" class="nav-link" data-scroll-target="#summary-and-conclusion"><span class="header-section-number">14.3</span> Summary and conclusion</a></li>
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<h1 class="title"><span id="sec-sampling-variability" class="quarto-section-identifier"><span class="chapter-number">14</span> <span class="chapter-title">On Variability in Sampling</span></span></h1>
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<blockquote class="blockquote">
<p>[Debra said]: “I’ve had such good luck with Japanese cars and poor luck with American...”</p>
</blockquote>
<blockquote class="blockquote">
<p>The ’65 Ford Mustang: “It was fun, but I had to put two new transmissions in it.”</p>
</blockquote>
<blockquote class="blockquote">
<p>The Ford Torino: “That got two transmissions too. That finished me with Ford.”</p>
</blockquote>
<blockquote class="blockquote">
<p>The Plymouth Horizon: “The disaster of all disasters. That should’ve been painted bright yellow. What a lemon.”</p>
</blockquote>
<p>(From <em>Washington Post Magazine</em>, May 17, 1992, p. 19)</p>
<p>Do the quotes above convince you that Japanese cars are better than American? Has Debra got enough evidence to reach the conclusion she now holds? That sort of question, and the reasoning we use to address it, is the subject of this chapter.</p>
<p>More generally, how should one go about using the available data to test the hypothesis that Japanese cars are better? That is an example of the questions that are the subject of statistics.</p>
<section id="variability-and-small-samples" class="level2" data-number="14.1">
<h2 data-number="14.1" class="anchored" data-anchor-id="variability-and-small-samples"><span class="header-section-number">14.1</span> Variability and small samples</h2>
<p>Perhaps the most important idea for sound statistical inference — the section of the book we are now beginning, in contrast to problems in probability, which we have studied in the previous chapters — is recognition of the <em>presence of variability in the results of small samples</em>. The fatal error of relying on too-small samples is all too common among economic forecasters, journalists, and others who deal with trends and public opinion. Athletes, sports coaches, sportswriters, and fans too frequently disregard this principle both in their decisions and in their discussion.</p>
<p>Our intuitions often carry us far astray when the results vary from situation to situation — that is, when there is variability in outcomes — and when we have only a small sample of outcomes to look at.</p>
<p>To motivate the discussion, I’ll tell you something that almost no American sports fan will believe: There is no such thing as a slump in baseball batting. That is, a batter often goes an alarming number of at-bats without getting a hit, and everyone — the manager, the sportswriters, and the batter himself — assumes that something has changed, and the probability of the batter getting a hit is now lower than it was before the slump. It is common for the manager to replace the player for a while, and for the player and coaches to change the player’s hitting style so as to remedy the defect. But the chance of a given batter getting a hit is just the same after he has gone many at-bats without a hit as when he has been hitting well. A belief in slumps causes managers to play line-ups which may not be their best.</p>
<p>By “slump” I mean that a player’s probability of getting a hit in a given at-bat is lower during a period than during average periods. And when I say there is no such thing as a slump, I mean that the chances of getting a hit after any sequence of at-bats without a hit is <em>not</em> different than the long-run average.</p>
<p>The “hot hand” in basketball is another illusion. In practical terms, the hot hand does not exist — or rather — if it does, the effect is weak.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> The chance of a shooter scoring is more or less the same after they have just missed a flock of shots as when they have just sunk a long string. That is, the chance of scoring a basket is not appreciably higher after a run of successes than after a run of failures. But even professional teams choose plays on the basis of who supposedly has a hot hand.</p>
<p>Managers who substitute for the “slumping” or “cold-handed” players with other players who, in the long run, have lower batting averages, or set up plays for the shooter who supposedly has a hot hand, make a mistake. The supposed hot hand in basketball, and the slump in baseball, are illusions because the observed long runs of outs, or of baskets, are statistical artifacts, due to ordinary random variability. The identification of slumps and hot hands is superstitious behavior, classic cases of the assignment of pattern to a series of events when there really is no pattern.</p>
<p>How do statisticians ascertain that slumps and hot hands are very weak effects, or do not exist? In brief, in baseball we simulate a hitter with a given average — say .250 — and compare the results with actual hitters of that average, to see whether they have “slumps” longer than the computer. The method of investigation is roughly as follows. You program a computer or other machine to behave the way a player would, given the player’s long-run average, on the assumption that each trial is a random drawing. For example, if a player has a .250 season-long batting average, the machine is programmed like a bucket containing three black balls and one white ball. Then for each simulated at bat, the machine shuffles the “balls” and draws one; it then records whether the result is black or white, after which the ball is replaced in the bucket. To study a season with four hundred at-bats, a simulated ball is drawn four hundred times.</p>
<p>The records of the player’s real season and the simulated season are then compared. If there really is such a thing as a non-random slump or streak, there will be fewer but longer “runs” of hits or outs in the real record than in the simulated record. On the other hand, if performance is independent from at-bat trial to at-bat trial, the actual record will change from hit to out and from out to hit as often as does the random simulated record. I suggested this sort of test for the existence of slumps in my 1969 book that first set forth the resampling method, a predecessor of this book.</p>
<p>For example, <a href="#tbl-at-bats" class="quarto-xref">Table <span>14.1</span></a> shows the results of one 400 at-bat season for a simulated .250 hitter. (H = hit, O = out, sequential at-bats ordered vertically) Note the “slump” — 1 for 24 — in columns 7 & 8 (in bold).</p>
<div id="tbl-at-bats" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-at-bats-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 14.1: 400 simulated at-bats (ordered vertically)
</figcaption>
<div aria-describedby="tbl-at-bats-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<tbody>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
</tr>
<tr class="odd">
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
</tr>
<tr class="even">
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;"><strong>H</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
</tr>
<tr class="odd">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">O</td>
</tr>
<tr class="even">
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;">H</td>
<td style="text-align: left;"><strong>O</strong></td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">O</td>
<td style="text-align: left;">H</td>
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<td style="text-align: left;">H</td>
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<td style="text-align: left;"><strong>O</strong></td>
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<td style="text-align: left;">O</td>
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</figure>
</div>
<p>Harry Roberts investigated the batting records of a sample of major leaguers.<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> He compared players’ season-long records against the behavior of random-number drawings. If slumps existed rather than being a fiction of the imagination, the real players’ records would shift from a string of hits to a string of outs less frequently than would the random-number sequences. But in fact the number of shifts, and the average lengths of strings of hits and outs, are on average the same for players as for player-simulating random-number devices.</p>
<p>Over long periods, averages may vary systematically, as Ty Cobb’s annual batting averages varied non-randomly from season to season, Roberts found. But in the short run, most individual and team performances have shown results similar to the outcomes that a lottery-type random number machine would produce.</p>
<p>Thomas Gilovich, Robert Vallone and Amos Twersky <span class="citation" data-cites="gilovich1985hot">(<a href="references.html#ref-gilovich1985hot" role="doc-biblioref">1985</a>)</span> performed a similar study of basketball shooting. They examined the records of shots from the floor by the Philadelphia 76’ers, foul shots by the Boston Celtics, and a shooting experiment of Cornell University teams. They found that “basketball players and fans alike tend to believe that a player’s chance of hitting a shot are greater following a hit than following a miss on the previous shot. However, detailed analyses…provided no evidence for a positive correlation between the outcomes of successive shots.”</p>
<p>To put their conclusion differently, knowing whether a shooter has scored or not scored on the previous shot — or in any previous sequence of shots — is of absolutely no use in predicting whether the shooter will or will not score on the next shot. Similarly, knowledge of the past series of at-bats in baseball does not improve a prediction of whether a batter will get a hit this time.</p>
<p>Of course a batter <em>feels</em> — and intensely — as if she or he has a better chance of getting a hit at some times than at other times. After a series of successful at-bats, both sandlot players and professionals feel confident that this time will be a hit, too. And after you have hit a bunch of baskets from all over the court, you feel as if you can’t miss.</p>
<p>But notice that card players get the same poignant feeling of being “hot” or “cold,” too. After a poker player “fills” several straights and flushes in a row, s/he feels s/he will hit the next one too. (Of course there are some players who feel just the opposite, that the “law of averages” is about to catch up with them.)</p>
<p>You will agree, I’m sure, that the cards don’t have any memory, and a player’s chance of filling a straight or flush remains the same no matter how he or she has done in the last series of hands. Clearly, then, a person can have a strong feeling that something is about to happen even when that feeling has no foundation. This supports the idea that even though a player in sports “feels” that s/he is in a slump or has a hot hand, this does not imply that the feeling has any basis in reality.</p>
<p>Why, when a batter is low in his/her mind because s/he has been making a lot of outs or for personal reasons, does her/ his batting not suffer? And why the opposite? Apparently at any given moment there are many influences operating upon a player’s performance in a variety of directions, with none of them clearly dominant. Hence there is no simple convincing explanation why a player gets a hit or an out, a basket or a miss, on any given attempt.</p>
<p>But though science cannot provide an explanation, the sports commentators always are ready to offer their analyses. Listen, for example, to how they tell you that Joe Zilch must have been trying extra hard <em>just because of</em> his slump. There is a sportswriter’s explanation for <em>anything</em> that happens.</p>
<p>Why do we believe the nonsense we hear about “momentum,” “comeback,” “she’s due this time,” and so on? The adult of the human species has a powerful propensity to believe that he or she can find a pattern even when there is no pattern to be found. Two decades ago I cooked up series of numbers with a random-number machine that looked as if they were prices on the stock market. Subjects in the experiment were told to buy and sell whichever stocks they chose. Then I gave them “another day’s prices,” and asked them to buy and sell again. The subjects did all kinds of fancy figuring, using an incredible variety of assumptions — even though there was no way for the figuring to help them. That is, people sought patterns even though there was no reason to believe that there were any patterns to be found.</p>
<p>When I stopped the game before the ten buy-and-sell sessions the participants expected, people asked that the game continue. Then I would tell them that there was no basis for any patterns in the data. “Winning” or “losing” had no meaning. But the subjects demanded to continue anyway. They continued believing that they could find patterns even after I told them that the numbers were randomly looked up and not real stock prices.</p>
<p>The illusions in our thinking about sports have important counterparts in our thinking about such real-world phenomena as the climate, the stock market, and trends in the prices of raw materials such as mercury, copper and wheat. And private and public decisions made on the basis of faulty understanding of these real situations, caused by illusory thinking on the order of belief in slumps and hot hands, are often costly and sometimes disastrous.</p>
<p>An example of the belief that there are patterns when there are none: Systems for finding patterns in the stock market are peddled that have about the same reliability as advice from a racetrack tout — and millions buy them.</p>
<p>One of the scientific strands leading into research on variability was the body of studies that considers the behavior of stock prices as a “random walk.” That body of work asserts that a stock broker or chartist who claims to be able to find patterns in past price movements of stocks that will predict future movements should be listened to with about the same credulity as a racetrack tout or an astrologer. A second strand was the work in psychology in the last decade or two which has recognized that people’s estimates of uncertain events are systematically biased in a variety of interesting and knowable ways.</p>
<p>The U.S. government has made — and continues to make — blunders costing the public scores of billions of dollars, using slump-type fallacious reasoning about resources and energy. Forecasts are issued and policies are adopted based on the belief that a short-term increase in price constitutes a long-term trend. But the “experts” employed by the government to make such forecasts do no better on average than do private forecasters, and often the system of forecasting that they use is much more misleading than would be a random-number generating machine of the sort used in the baseball slump experiments.</p>
<p>Please look at the data in <a href="#fig-nile_height" class="quarto-xref">Figure <span>14.1</span></a> for the height of the Nile River over about half a century. Is it not natural to think that those data show a decline in the height of the river? One can imagine that if our modern communication technology existed then, the Cairo newspapers would have been calling for research to be done on the fall of the Nile, and the television anchors would have been warning the people to change their ways and use less water.</p>
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<img src="diagrams/nile_height.svg" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-nile_height-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 14.1: Height of the Nile River Over Half of a Century
</figcaption>
</figure>
</div>
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<p>Let’s look at <a href="#fig-nile_long" class="quarto-xref">Figure <span>14.2</span></a> which represents the data over an even longer period. What now would you say about the height of the Nile? Clearly the “threat” was non-existent, and only appeared threatening because the time span represented by the data was too short. The point of this display is that looking at too-short a segment of experience frequently leads us into error. And “too short” may be as long as a century.</p>
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<img src="images/nile_levels.png" class="img-fluid figure-img">
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<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-nile_long-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 14.2: Variations in the height of Nile Flood in centimeters. The sloping line indicates the secular raising of the bed of the Nile by deposition of silt. From <span class="citation" data-cites="brooks1928periodicities">Brooks (<a href="references.html#ref-brooks1928periodicities" role="doc-biblioref">1928</a>)</span>
</figcaption>
</figure>
</div>
<p>Another example is the price of mercury, which is representative of all metals. <a href="#fig-mercury_prediction" class="quarto-xref">Figure <span>14.3</span></a> shows a forecast made in 1976 by natural-scientist Earl Cook <span class="citation" data-cites="cook1976limits">(<a href="references.html#ref-cook1976limits" role="doc-biblioref">1976</a>)</span>. He combined a then-recent upturn in prices with the notion that there is a finite amount of mercury on the earth’s surface, plus the mathematical charm of plotting a second-degree polynomial with the computer. <a href="#fig-mercury_reserves" class="quarto-xref">Figure <span>14.4</span></a> and <a href="#fig-mercury_price_indexes" class="quarto-xref">Figure <span>14.5</span></a> show how the forecast was almost immediately falsified, and the price continued its long-run decline.</p>
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<img src="images/mercury_prices.png" class="img-fluid figure-img">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-mercury_prediction-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 14.3: The Price of Mercury from <span class="citation" data-cites="cook1976limits">Cook (<a href="references.html#ref-cook1976limits" role="doc-biblioref">1976</a>)</span>
</figcaption>
</figure>
</div>
<div class="cell" data-layout-align="center">