What Numbers Reveal -- From Sumo Wrestlers to Professors

Perhaps not surprisingly, studies (again of large data sets) in the 1990s showed that companies' earnings were much more likely to come in a penny or two above analysts' average estimate than a penny or two below it. If earnings were figured without regard to analysts' expectations, they'd come in below the average estimate as often as above it. The reason for the asymmetry is probably that some companies "backed in" to their earnings. Instead of determining revenues and expenses and subtracting the latter from the former to obtain earnings (or more complicated variants of this), companies began with the earnings they needed and adjusted revenues and expenses to achieve them.

Math Professors -- Generous to a Fault?

From sumo wrestlers to corporate accountants to (moving closer to my home) university mathematics professors, the tendency to hedge matters in certain crucial situations is almost universal, albeit in this last example rather benign and selfless. Once again, large data sets reveal their secrets when probed with the right questions.

Like many universities, mine requires that a core mathematics course be taken by all students who do not plan on going on in mathematics or the sciences. To pass this particular course, a student is required to do reasonably well and score a C- or higher. Suspecting that the number of C-'s would be much larger than the number of D+'s because of how crucial this small difference is to students, I decided to examine the number of C-'s and D+'s given in this course over the last four years.

In any sort of roughly normal distribution of grades there should not be any sharp divide between the frequencies of the two grades. But for the period and course in question, there were approximately 800 C-'s and 100 D+'s awarded. One might argue that the number of D+'s should be lower than the number of C-'s simply as a result of a normal distribution with an average of C or more, but this drop-off was precipitous, fully eight times as many C-'s as D+'s. (The 400 or so plain D's and roughly 700 F's given out during this period indicate that general grade inflation is not the issue.)

It seems that, at the crucial point between C- and D+, the faculty were likely to give the students in this course a bit of a break. A genuine uncertainty is the likely motivation. Assigning grades is not a cut-and-dried activity, and many professors apparently preferred to give students the benefit of a doubt in these close calls rather than adhere rigidly to standards that are inevitably slightly arbitrary.

Note, finally, that in all three of these examples scrutinizing critical borderline cases leads to the observation in question.

-- Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.

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