Ten is a common and familiar number, the base of our number system. Numbers are rounded to 10 or to multiples of ten or tenths. The resulting distortion, of course, need not have much to do with reality. Is there anything more vapid than explanation by decade?
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Sarcasm and hyperbole aside,victimization and the differential treatment of groups,whether intentional or not,are the basis for many a news story. The percentage of African-American students at elite colleges,the proportion of women in managerial positions,the ratio of Hispanic representatives in legislatures have all been written about extensively. Oddly enough,the shape of normal bell-shaped statistical curves sometimes has unexpected consequences for such situations. For example,even a slight divergence between the averages of different population groups is accentuated at the extreme ends of these curves,and these extremes often receive inordinate attention in the press.
There are other inferences that have been drawn from this fact,some involving social policy issues such as affirmative action and jobs programs. As an illustration,assume that two populations vary along some dimension - height for example. Then even if the average height of one group is only slightly greater than the average height of the other,people from the taller group will constitute a large majority among the very tall the right tail of the curve. Likewise,people from the shorter group will constitute a large majority among the very short the left tail of the curve.
This is true even though the bulk of the people from both groups are of roughly average stature. In general,any differences between the two groups will always be greatly accentuated at the extremes. These simple ideas can be used and misused by people of very different political persuasions.
Many people submit their job applications to a large corporation. Some of these people are Mexican and some are Korean,and the corporation uses a single test to determine which jobs to offer to whom. She may suspect racism,but the result might just as well be an unforeseen consequence of the way the normal distribution works. Paradoxically, if she lowers the threshold for entrance to mid-level jobs,she will actually end up increasing the percentage of Mexicans in the bottom category.
Much of this discussion is valid even if the distribution is not the normal bell-shaped one. Such statistical disparities are not necessarily evidence of racism or ethnic prejudice,although,without a doubt,they sometimes are. Aside from having a questionable rationale,schemes of strict proportional representation are impossible to implement.
Another thought experiment illustrates this point. Unknown to PCI and the community is the fact that only 2 percent of the blacks are homosexual,whereas 6 percent of the whites are. Making a concerted attempt to assemble a work force of 1, that "fairly" reflects the community,the company hires whites and blacks.
However,just 5 of the blacks or 2 percent would be homosexual,whereas 45 of the whites or 6 percent would be totalling 50,5 percent of all workers. Despite these efforts,the company could still be accused by its black employees of being homophobic,since only 2 percent of the black employees would be homosexual,not the community wide 5 percent. White heterosexuals would certainly make similar complaints. To complete the reductio ad absurdum,factor in several other groups: Hispanics, women, Norwegians,even.
Their memberships will likely also intersect to various unknown degrees. The backgrounds and training across these various cross sections and intersections are extremely unlikely to be uniform.
Statistical disparities will necessarily result. Racism and homophobia and all other forms of group hatreds are real enough without making them our unthinking first inference when confronted with such disparities. Assume further that Mr X has a surname 20 percent of whose co-owners have characteristic C. If one knows nothing else about Mr X,then it seems prudent to suppose that there is a 20 percent chance that Mr X possesses C. And what if one subsequently learns that Mr X is also an active member of a nation-wide organization only 3 percent of whose members possess characteristic C?
The enduring obsession with guardian angels and statues that bleed or cry are cases in point. It received wide notice,and both the Washington Post and the New York Times ran profiles on him,which,while not exactly credulous,were not exactly incredulous either.
One might have guessed that such a radical claim would have galvanized scores of reporters. And James Gleick in the New Republic wrote a scathing review of the book and what he termed its weasely dodges and equivocations.
Of course,The Skeptical Enquirer whose reason for existence is the critical examination of such claims,also printed an article on the book. As a fellow of the Committee for the Scientific Investigation of Claims of the Paranormal,which publishes The Skeptical Enquirer ,I may be biased,but in my estimation the publication deserves a Pulitzer Prize for its work on these issues over the years.
Sometimes mathematics is also helpful in uncloaking pseudoscientific claims and explaining their appeal. These remarkable relationships between totally dissimilar items frequently seem to have an air of scientific hypotheses:sunspots and the stock market,hemlines and presidential elections,Super Bowl outcomes and the economy.
Very often there is some personal connection or some element of self-reference involved in these relationships. The sheer number of such possible links and associations should convince one that almost all are merely coincidences. Rather than rehash the skeptical arguments,let me present a mathematical recipe anyone may use to develop his or her very own personal pseudoscience. The method comes from the Dutch physicist Cornelis de Jager,who used it to advance a theory about the metaphysical properties of Dutch bicycles.
The recipe: Take any four numbers associated with you height,weight,birthday,social security number,whatever you like and label them X,Y,Z and W. Since each of the four exponents may be any one of these seventeen numbers,the number of possible choices of a,b,c, and d is,by the multiplication principle, 17x17x17x17x There are thus this many values for the expression XaYbZcWd.
If there are not,the units in which these constants are expressed can be altered. A computer program can easily be written that can determine which of these universal constants is equal to one of the numbers generated from your original four. Or you might discover any of a host of other correspondences between your personal numbers and these universal constants - all without having to undergo the rigors of alien abduction.
Incidentally,the ratio of the height of the Sears Building in Chicago to the height of the Woolworth Building in New York is the same to four significant digits 1. Salacious versions of this game are possible,of course. Compare it also to the coincidental similarities,described in the first section,that can be found to link any two American presidents. That such accidental linkages and their more mundane,non-numerical cousins are extremely common is not fully appreciated.
Consider the foreign con artist whose lure was that he could help students hoping to enter a very competitive national university. The man bragged that he knew the arcane details of the admission process,had contacts with the appropriate officials,and so on.
After gathering detailed information from the students,he collected an exorbitant fee,promising to return it if the student was not admitted.
Every year he threw their information out,yet every year some of the students got in anyway. Their fees he kept. Similar stories can be told about miracle medical interventions whose efficacy is only an illusion.
Many people simply improve on their own. The lesson for all of us is that talking only to people who have a vested interest in some result or linkage can be beguiling especially if that person is a Harvard professor of psychiatry ; gullible journalism is often the result. Of course,coincidences occasionally point up valuable yet overlooked connections or,vastly less often,defective scientific laws.
But,as the philosopher David Hume observed more than years ago,every piece of evidence for a miraculous coincidence - that is ,for a contravention of natural law - is also evidence for the proposition that the regularities that the miracle contravened are not really laws of nature after all.
Any near-instantaneous transmission of voices across great distances might have been considered a miracle centuries ago.
As it turned out,however,the scientific principles prohibiting or seeming to prohibit such transmissions were not,in fact,natural laws. Not only are coincidences not miraculous but -Freudians,tabloids,and popular sentiment to the contrary - an overwhelming majority of them have no significance whatsoever.
Neither,I might add,will the expected numerological fatuities connected to the turn of the millennium in for purists. Since equals 3 times ,the festivities may begin even sooner.
John Allen Paulos
In an interview he described himself as lifelong skeptic. He was also part of the Peace Corps in the seventies. John Allen Paulos, Innumeracy The most amazing coincidence of all would be the complete absence of all coincidences. John Allen Paulos, "Irreligion" His academic work is mainly in mathematical logic and probability theory. His book Innumeracy: Mathematical Illiteracy and its Consequences was a bestseller and A Mathematician Reads the Newspaper extended the critique. In his books Paulos discusses innumeracy with quirky anecdotes, scenarios and facts, encouraging readers in the end to look at their world in a more quantitative way.
A Mathematician Reads the Newspaper