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Innumeracy is the mathematical equivalent of illiteracy, but is far more widespread and socially accepted than illiteracy. Some people will even claim they are glad to be innumerate: they are “big picture” people, artistic, their imaginations are not constrained by mere numbers. Highly numerate people are often dismissed as nerds or “bean-counters.” “Innumerate people often--
Number Numbness Confusion over orders of magnitude creates serious public policy problems. As the late Senator Everett Dirksen quipped, chiding his colleagues in Congress for their loose spending habits, “A billion here, a billion there, pretty soon you’re talking real money!” Most people know that a billion is “more than a million,” but don’t fully appreciate that it’s a thousand times more. Here are some problems that may help you overcome some “number numbness.”The objective is to get the right order of magnitude:
Cultivate your ability to do basic math calculations in your head. It takes a little effort and practice, but it will help you spot erroneous numbers much more easily. When you started doing multi-digit addition, subtraction and multiplication, your elementary school teacher taught you to start with the rightmost digits and work left to right, and you often got bogged down before reaching the most important digits on the left. Don’t! Train yourself to work right to left instead, mentally incrementing digits when you "carry tens" back to the left. A lot of people freeze up at percentages. Quick!--what's forty percent of forty? Seventy percent of thirty? You can use your knowledge of squares to do quick multiplications: 8 x 12 = (10+2)(10-2) = 102 - 22. There are lots of tricks like these for refining your mental math skills. Risk Personalization and Dread Irrational personalization of risk creates odd distortions in social policies. Americans dread relatively rare but bizarre causes of death (terrorism, AIDS, kidnappings, airplane crashes) more than frequent, mundane causes (car accidents, Alzheimer’s disease). The continuing slaughter of citizens on US highways every month is greater than the total number of fatalities from the 9/11 attacks. But the US spends tens of billions more on homeland security than on highway safety, where prevention of deaths would be far more cost-effective. When a middle-class child gets abducted and murdered in Florida, and a bank security camera records the abduction, it gets national news coverage for days—goodness gracious, that could have been my child! When a child in Wilmington is killed by a stray bullet in an ordinary inner-city drug dispute, the story only makes the front of the Local News section of the Wilmington News-Journal—oh well, I don’t live in the inner city. The problem is compounded by news media bias that plays to personalization and dread. News media depend on advertising sales, which depend on how many readers/listeners/viewers they can grab and keep, so they logically focus on the lurid and bizarre. The conventional wisdom in programming local TV news is “If it bleeds, it leads!” A woman finds a live snake in her toilet. Killer bees are spreading across the country. Weird Michael Jackson hosted weird sleepovers for kids. When something shocking happens, such as the 9/11 attacks on the World Trade Center or the December 2004 tsunami, many people compulsively watch news programs that simply repeat the same information and show the same footage over and over. Stranger abductions of children are a major focus of dread, but they are statistically very rare, and have become much rarer in the past 40 years. In the 1960’s abductions of children by strangers generally got only local news coverage; today they often get national coverage precisely because they elicit parent dread. This dread has a high social cost: parents are more reluctant than ever to allow their children to play outside, which contributes to epidemic obesity among kids. Paulos suggests using a logarithmic scale to understand relative risks. The typical American child's annual risk of kidnapping by a stranger is less than 1 in 5 million. The same child's annual risk of being killed riding a bicycle without a helmet is greater than 1 in 5 thousand. The innumerate person is unimpressed by this difference, and isn't really satisfied by anything short of absolute zero risk. Distracted by the lurid nature of kidnappings, the innumerate person dismisses the relative probabilities: "Yeah, but what if it's my kid who's kidnapped?" While bizarre but statistically insignificant risks elicit dread, many mundane but statistically significant risks do not. Americans are remarkably complacent about the high rate of automobile fatalities in the US. Since so many people have inflated opinions of their driving expertise, most states need laws to force drivers purchase car insurance and wear seatbelts. (Dustin Hoffman’s character in “Rain Man” kept insisting “I’m an excellent driver.”) Cars have seat belts, air-bags, safety glass, ABS and crush-resistant frames. They are designed for safety, but more importantly, they are designed to appeal to you. They are supposed to be comfortable and fun to drive, and to make you feel safe at whatever speed you drive. To be extra safe, you can buy a big SUV that will significantly increase your odds of surviving a collision with a car (while decreasing survivability for the car’s occupants). Automobile collisions are very ordinary; most people have been in one at some point in their lives. We all know people who have died in car crashes. It’s kind of expected that high school kids will get killed on prom night, and drunk drivers will get killed. There is no significant stigma attached to an automobile death—it’s simply tragic, it can happen to anybody. It elicits much less dread than it should. (Some single-car accidents are actually suicides, where the victim stages the accident to spare his or her family the stigma of suicide.) Conversely, AIDS elicits far more dread than it should. Most people would much rather die in a car accident than die from AIDS. After all, everyone knows that you get AIDS from sex, and it’s probably not ordinary sex—how embarrassing! Paulos estimates the risk of contracting AIDS from unprotected heterosexual intercourse with an AIDS-infected partner is 1 in 500. The chance of not contracting AIDS from a single sexual contact is 499:500. The chance of not contracting AIDS from two contacts is (499/500) squared, or from N contacts is (499/500) to the Nth power. So you could have daily unprotected sex with that person every day for half a year, with only a 30 percent probability of getting AIDS. If you use a condom, your chance of AIDS from a single encounter with an AIDS-infected person is less than 1 in 5 million. Hypothetical hook-up situation at a college party: that cute guy/girl is very receptive, and you’ve had enough beer to decide to go for it, but you realize you don’t have any condoms! You don't know his/her HIV status, but he/she is not a member of a known high-risk group. The choice is to (a) have unprotected sex, or (b) drive drunk to the drug store for condoms and then have protected sex. What proportion of your friends will choose (a) over (b)? Since your driving skills are above-average (even after a few beers) and it would be unimaginably awful to get AIDS, you may well head out to the drug store. In this context the chance of contracting AIDS from a single sexual contact without protection is about 1 in 5 million. The likelihood of getting maimed or killed in a car accident driving drunk to the drug store for condoms is about a thousand times higher. Especially if you don't wear a seatbelt! Miscalculating probabilities A lot of gambling is motivated by erroneous assumptions about probabilities. Games of chance are constructed to generate independent outcomes: the odds of getting heads on the next toss of a coin are independent of whatever sequence of heads and/or tails has already occurred. After a sequence of heads some people will think tails are “due.” You can see old ladies pumping quarters into slot machines (excuse me—they’re “video lottery” machines!) at Delaware Park. Some of them wear adult diapers so they don’t have to give up their seat at a machine that is “due” to pay off if they need to urinate. Casino games and lotteries are structured to disguise their long odds. Winning a “pick six” lottery looks a lot easier than it is: pay $1 and pick six out of 49 numbers. Your actual odds of hitting the jackpot by picking all six are 1:13,983,816. Your odds of picking five correct numbers are 1:54,201, and the average prize in the NJ Pick-Six for this is only $2,700. Your odds of picking just four correct numbers are about 1:1,000 for a lousy $56 average payout. And you have to pay taxes on your winnings. The lottery is sometimes criticized as a “voluntary tax on stupidity”—the stupider you are, the more you play…and lose. As a revenue source for the state, the lottery functions as a highly regressive tax, extracting more money per-capita from poor (and innumerate) people than wealthy, better-educated people. Its only “virtue” is that it is voluntary, unlike other taxes. Casinos are for suckers too. You often hear people brag about their casino winnings, but they always keep quiet about their losses, so it may seem as if everybody wins at Atlantic City. But you know where those big casino revenues come from. When slot machines pay off, they make lots of noise, but when they don’t, they stay silent. Slots parlors like to have lots of machines running, so that players stay motivated by hearing frequent payoffs. Gambling can be genuinely addictive. In brain scans of gambling addicts, subjects hearing the sound of a slot machine paying off reportedly exhibit the same neural responses that addicts exhibit when given drugs, although the stimulus is purely auditory. Gambling is Delaware’s second largest source of state government revenues after the personal income tax, so the state is addicted to gambling too. And like any addict, it tries to rationalize its addiction, restricting gambling to a few extremely lucrative franchises, and funding a hotline and counseling services for “problem gamblers.” If you want gambling with decent odds, play the stock market. People often misunderstand how probabilities can change. Here's a widely-discussed problem. Suppose you are on the old “Let’s Make a Deal” TV game show and asked to choose one of three curtains, A, B or C. Behind one curtain is a really nice car; behind the other two are live goats. You state your choice. Then the host opens one of the other curtains to reveal…goats! Then he asks you if you really want to stick with your choice, or choose the other unopened curtain instead. It now looks like you have an even chance of winning the car with either curtain. But in reality, if you switch, you will have a 2-in-3 chance of winning the car! In the 3-curtain choice your odds of winning were 1-in-3, and the fact that the host (who knows what’s behind the curtains) showed you another losing choice didn’t change those odds. (It may be more convincing if you think of choosing one of 100 curtains. Your odds of winning are 1-in-100, and if the host then opens 98 losing curtains, the odds that the curtain you didn’t choose will be the winner are 99-in-100.) Other Forms of Innumeracy Randomness vs. Pattern. Pattern recognition is the essence of neural intelligence. Organisms survive by recognizing and responding to different patterns of sensory perception of their environments. They learn to distinguish predators from food, suitable from unsuitable habitat, males of their species from females, etc.; otherwise they die. Pattern recognition is evolved. Most patterns have some meaning, but some perceived patterns do not; they are purely specious, appearing by random chance. At a fundamental level, intelligence is the ability to distinguish meaningful from specious pattern. Correlation vs. Causality. People have a tendency to correlation with causality, and to see spurious correlations in random coincidences. One manifestation of this tendency is in superstitions, which then create expectations that tend to be self-fulfilling. If a black cat crosses in front of me, I will be on the lookout for bad luck. I’m not actually unluckier on such days, but I’ll be more aware of unlucky outcomes. This logical fallacy is known as post hoc, ergo propter hoc (“after it, therefore because of it”). People are often suspicious of coincidences, and some "coincidences" are much more likely than you think: the chance that of 23 randomly-chosen people 2 will share the same birthday is 50 percent. Some coincidences can be engineered not to look like coincidences. At the start of football season you could mail letters to 64,000 people predicting the outcome (by point spread) of a football game: half the letters pick one team, half the other. The next week you mail predictions to the 32,000 people who received correct picks the first week; the third week 16,000 predictions to those who received two correct picks; etc. Near the end of the season you would have dozens of people awed by your ability to make eight or nine correct predictions in a row, and willing to pay you serious money for your prediction on the next big game. If only it were legal…. Precision vs. Accuracy. People also confuse precision with accuracy. A fairly precise (but very inaccurate) estimate of pi is 4.5278403. A more accurate (but less precise) estimate would be 3. False precision makes numerical estimates look more authoritative than they really are. Pay attention to how precision degrades as you carry it through mathematical calculations. Suppose you estimate each dimension of a box with a +/-10% margin of error: 60” (+/-6”) wide by 100” (+/-10”) long by 50” (+/-5”) high. The estimated volume would be 60 x 100 x 50 = 300,000 cubic inches with an error margin of 54 x 90 x 45 = 218,700 to 66 x 110 x 55 = 399,300. Since your errors in estimating the dimensions are probably systematic rather than merely random, your compounded margin of error on the volume estimation is a lot more than 10%; it’s more like +/-33%. Pyramid schemes: An easy (and illegal) way to make money is to start a Ponzi (pyramid) scheme. Suppose I promise you a return of at least $31,000 for an up-front investment of only $100 in my “investment club.” I get 10 of you to pay me $100 each, and then I tell you how the “club” works. Each of you agrees to split all the money you receive with me. Each of you will recruit 10 of your friends to join our “club;” we will then have 10 payments of $100 to split. Then each of these friends will recruit 10 of their friends, and we will get 100 payments of $50 to split. And if they all recruit 10 friends, we will get another 1000 payments of $25 to split. And so on. I would collect $1,000 (10 x $100) from each of you, plus $5,000 (100 x $50) from your friends, plus $25,000 (1,000 x $25) from your friends' friends. Then maybe I’d “retire” and leave you at the top of your own “club” so you’d hopefully finish with $31,000 too. Or maybe I’d try to milk it for another round. (Did I mention I only take cash?) Here are the potential receipts through nine rounds of this game: "dues" rec'd number of total rec'd cumulative round per player players this round total rec'd 1 $ 100.00 10 $ 1,000 $ 1,000 2 $ 50.00 100 $ 5,000 $ 6,000 3 $ 25.00 1,000 $ 25,000 $ 31,000 4 $ 12.50 10,000 $ 125,000 $ 156,000 5 $ 6.25 100,000 $ 625,000 $ 781,000Ponzi schemes like this quickly collapse. Because of the exponential growth in “membership” required, they soon run out of people to recruit. For some entertaining historical accounts of scams such as this, check out the classic book Extraordinary Popular Delusions and the Madness of Crowds by Charles MacKay (1841). The fallacy of averaging averages: Suppose during the first half of baseball season player A bats .300 in 200 at-bats, while player B bats .290 in 100 at-bats; and during the second half of the season, A bats .400 in 100 at-bats while B bats .390 in 200 at-bats. So A had the higher batting average in both halves of the season, and they both had the same number of at-bats. But if you do the math, it turns out that B had the higher batting average for the whole season: A got 60 + 40 = 100 hits in 300 total at-bats; B got 29 + 78 = 107 hits in 300 total at-bats. Framing Problems Context matters a lot. People can be misled by how questions are framed. Which of the two scenarios (taken from Kahneman and Tversky) is more likely? (1) Ann graduated from college in 1988 with a degree in women's studies, has an assertive personality, is currently unmarried, and works in a bookstore. (2) Ann graduated from college in 1988 with a degree in women's studies, has an assertive personality, is currently unmarried, works in a bookstore, and is active in women's rights issues. The first scenario is more likely by definition, simply because it has one less condition. But many survey respondents consider the second scenario to be more "likely" because it conforms better to some stereotype of a feminist. Here’s a framing problem from Barry Schwartz: (The Paradox of Choice: Why More is Less 2004:Harper-Collins) “Imagine you are a physician working in an Asian village, and 600 people have come down with a life-threatening disease. Two possible treatments exist. If you choose Treatment A, you will save exactly 200 people. If you choose Treatment B, there is a one-third chance you will save all 600 people and a two-thirds chance you will save no one. Which treatment would you choose, A or B?” The vast majority of survey respondents said they would choose Treatment A. The same situation was posed to an equivalent group of respondents with differently-framed options: “…If you choose Treatment C, exactly 400 people will die. If you choose Treatment D, there is a one-third chance that no one will die, and a two-thirds chance that all 600 people will die.” In this case the overwhelming majority chose Treatment D. The two versions of the question are the same--A is logically identical to C and B is logically identical to D--but in the first version, we are saving lives, while in the second version we are deciding how many people will die. Suppose you buy an expensive pair of shoes that turn out to be really uncomfortable. Thaler (cited in Schwartz) suggests that the more you paid for them, the more often you’ll try to wear them; eventually you’ll stop wearing them but they’ll sit in your closet longer; and they’ll take longer to “depreciate” psychologically before you throw finally throw them away. Here's another framing problem from Kahneman and Tversky: Choose between a sure $30,000 or an 80% chance of winning $40,000 and a 20% chance of winning nothing (expected win: $32,000). Most people choose the sure $30,000. Now choose between a sure loss of $30,000 or an 80% chance of losing $40,000 and a 20% chance of losing nothing (expected loss: $32,000). Here most people take the gamble. People are risk-averse regarding gains, but risk-taking regarding loss avoidance. This win-loss asymmetry is consistent with what we know about contingent valuation analyses of WTP and WTAC for natural resource amenities (Bishop and Heberlein, et al.) Science and Pseudo-Science Innumerate people are often victims of pseudo-science. True science involves the formulation and testing of refutable hypotheses (Popper). Scientists don't really prove hypotheses; they can only disprove them. Science is falsifiable. Pseudo-science is not falsifiable. Karl Popper criticized Freudianism for its immunity to falsification. (Freudian analyst: "You're obviously in denial about this." Patient: "No, I'm not.") Medicine is particularly prone to quackery, because most medical problems are either: (a) self-correcting; (b) self-stabilizing; or (c) fatal, but exhibit uneven patterns of decline and temporary improvement. Survivors may engage in post hoc, ergo propter hoc thinking, crediting the quack treatment for their cure; non-survivors aren't around to complain. (In one of his funkier free-market riffs, Milton Friedman has criticized the AMA as a “guild” of doctors who profit by restricting entry into their profession. He basically recommends caveat emptor (“buyer beware”). Over time the good doctors will profit as their reputations grow while the quacks will eventually go out of business—perhaps after unsuccessful experiments on you!) “Occam’s razor” is a logical principle favoring the simplest, most “parsimonious” explanation for any phenomenon. The principle is names after William of Occam, a 14-century English philosopher. Economists generally follow this principle, specifying economic hypotheses in the form of simple mathematical “models.” A model should be kept as simple as possible, so that its derived predictions are unambiguous and readily testable. Thus a “good” economic model will often sacrifice descriptive realism for predictive clarity. In some cases, useful models may be based on completely counter-realistic assumptions. For example, economists can derive clear, testable predictions about consumer behavior by modeling consumers as utility-maximizing automatons; there is no practical need to describe the actual thought processes involved in consumer decision-making. One of the weaknesses of economics is the non-exclusivity of its models. The data we have may be consistent with any number of competing hypotheses expressible as alternative models. When we test our chosen model against the data, we either reject it as inconsistent with the data, or we fail to reject it. The process of winnowing out the wrong hypotheses is long and slow. There is an asymmetry inherent in hypothesis testing. In the parlance of statisticians, a researcher poses a “null” hypothesis and either rejects it or fails to reject it. Hypotheses are never actually proved, only disproved. In testing a hypothesis, the researcher can be wrong in either of two ways: (1) she can reject a true hypthesis--a "Type 1" error; or (2) she can fail to reject a false hypothesis--a "Type 2" error. The cost of being wrong may depend on what type of error occurs. Most scientists would much rather be criticized for being overly skeptical (Type 1 error) than too gullible (Type 2 error). Social Statistics Most of the statistics reported in the media are “social statistics.” Best notes that social statistics have two purposes: the overt purpose is to quantify some characteristic of society; the covert purpose is to influence political opinion. Activists use statistics to dramatize social problems. Advocacy groups use statistics to compete for your attention. True or false, social statistics are part of the “social construction of knowledge.” We tend to accept social statistics uncritically because numbers sound authoritative. But we should always consider the source of the statistic (does he have some vested interest in the situation?); its political purpose; and its likely accuracy (which depends on how it was calculated). Unfortunately, many social statistics are quickly divorced from their sources and, right or wrong, take on lives of their own. They become “mutant” statistics, subject to misinterpretation and distortion. Many “statistics” are little more than guesses. For example, crime victimization statistics are notoriously fuzzy. Some proportion of crimes don’t get reported (what criminologists call the “dark figure”). If you want to minimize the crime problem, you assume the dark figure is trivial or zero; if you want more police on the streets, you assume it is large. A lot depends on how the problem is defined. In the 1980’s there was a surge of concern about “missing children,” with kids’ pictures on milk cartons and media claims of up to 2 million children per year going “missing” in the US, and up to 50,000 children per year abducted by strangers. But what is the age range of “children,” and how is “missing” defined? Do we count kids kept too long by non-custodial parents, safe at a known location? Teenage runaways? Schoolchildren who fell asleep on the bus home and wound up at the bus depot? The numbers are easily inflated to elicit predictable parent dread (discussed above). Before you swallow the estimate of 50,000 stranger abductions per year, consider that only about 70 child kidnappings are investigated by the FBI each year. If the 2 million per year “missing” figure is true, then you must certainly know a lot of people who went “missing” at some time in their childhoods. Good statistics are more than guesses. They are based on clear and reasonable definitions, accurate measures and representative samples. A large sample size yields a statistic that is more precise and looks more authoritative, but a representative sample yields a statistic that is more accurate. The old TV ad that claimed "Nine out of ten kids prefer Skippy" was probably based on ten kids of company employees in the marketing department, and maybe the choice was Skippy or nothing, but the statistic sounded vaguely authoritative. They could have stated a confidence interval for this statistic, but that would make them sound unsure of themselves. On a yes-no question like this the confidence interval varies inversely with the square root of the sample size. Here if N=100, you would have a confidence interval of plus-or-minus 3%. Most people don't understand statistics well enough to demand this kind of information, and they don't hold advertisers, politicians, public advocacy groups, etc. fully accountable for the bad statistics they promulgate. Competing interest groups often argue over statistics—how they are collected and what the numbers mean. The 2000 Census included a “multi-racial” category for the first time. African-American activists opposed the inclusion of this category because it would reduce the number of Americans identifying themselves as “black,” while Native American groups supported the multi-racial category because many Americans are fractionally Native American but don’t otherwise identify themselves as such. The Census typically undercounts urban and minority populations more than rural and white populations. So Democrats (with larger urban and minority constituencies) tend to favor statistical corrections to the actual count data, based on post-enumeration surveys. Republicans (with larger rural and white constituencies) oppose such adjustments. The Census determines the decennial reapportionment of the US House of Representatives as well as the annual distribution of Federal monies for various programs. Census numbers are subjected to frequent court challenges by one side or the other because there is a lot riding on them! Americans have something of a fetish for social statistics. Best describes four types of response to social statistics. Some people are simply awestruck or fatalistic, and don’t react to statistics at all. Many people are naïve and innumerate, accept statistics uncritically and disseminate them inaccurately; sometimes they evolve into urban legends. Some people have learned to be cynical about all statistics (“you can prove anything with statistics”). And a few people have the critical capabilities to discern good statistics from bad. You can’t really understand a statistic without understanding the process that created it: what was counted and how, who did the counting and why, etc. |