From the archives:
David Brooks Gets Burned by Trusting Charles Murray: Archive Entry From Brad DeLong's Webjournal: David Brooks gets burned by trusting the American Enterprise Institute's Charles Murray:
The Atlantic | September 2003 | People Like Us | Brooks: My favorite illustration of this latter pattern comes from the first, noncontroversial chapter of The Bell Curve. Think of your twelve closest friends, Richard J. Herrnstein and Charles Murray write. If you had chosen them randomly from the American population, the odds that half of your twelve closest friends would be college graduates would be six in a thousand. The odds that half of the twelve would have advanced degrees would be less than one in a million...
Ummm... No. Definitely not. Back when The Bell Curve was published, according to the Statistical Abstract, 22.2% of Americans over 25 had bachelor's degrees (an additional 7% had associate's degrees) and 7.5% of Americans over 25 had advanced degrees. Draw 12 people at random from this set, and if my hasty back-of-the-envelope calculation is correct* the odds that half of them will have college degrees is 2.5% (7.2% if we are counting associate's degrees)--not "six in a thousand." The odds that half of 12 people drawn at random from this set will have advanced degrees is 0.1%--not "less than one in a million." I can't for the life of me figure out what calculations Murray was trying to make that would produce his numbers. But whatever calculations he made, he is off by a factor of 4 (or 12, if we are counting associate's degrees) for the college-educated and off by a factor of 100 for those with advanced degrees.
"Does being off by a factor of a hundred (or four) really matter?" you ask. "2.5% or 0.6%, 0.1% or 0.001%, the odds are still low--and the point that American society is not well-mixed is still true. " But Murray's (and Brooks's) point is not that American society is not well mixed. Their point is that American society is totally stratified--and that is surely false.
And there is another point. Brooks's reference to the "first, noncontroversial chapter of The Bell Curve" is hard to read as anything other than a partial attempt to try to rehabilitate the reputation that Charles Murray shattered by writing The Bell Curve. It is worth noting that nothing Charles Murray writes can be trusted without being independently verified, and that even the first chapter of The Bell Curve is "controversial"--that is, flat-out wrong.
*Suppose we draw twelve people at random. The chance that all of the first six we draw will have college degrees is 0.222^6. The chance that all of the last six we draw will not have college degrees is 0.778^6. The chance that both of these things will happen together is the product of those two numbers--0.0000265. But we don't care about the order: we would be perfectly happy if numbers 2, 4, 7,8,9, and 12 had college degrees. So we need to multiply 0.0000265 by the number of possible ways in which six college and six non-college graduates can be ordered. There are (12!)/((6!)(6!)) such ways--924 such ways. Multiplying 0.0000265 by 924 gives us 0.025--our 2.5% number.