Why (n-1)?

To my fellow criminologists (primarily)

Why (n-1)?

Hal Pepinsky, pepinsky@indiana.edu, “peacemaking” at pepinsky.blogspot.com

April 14, 2014

                My first day in my office in Albany, I made a point of introducing myself to Leslie T. Wilkins.  Shortly thereafter, Les invited me to team teach a seminar in one of the school of crimjus’s core areas: Philosophical Issues of Law and Social Control.  He proposed over lunch that we invite another philosopher from the university.  He first picked Stefan Temesvary, one of only several members of the Astronomy Department (later abolished entirely by a colleague who led the university “task force” in making the decision, as he explained to me at yet another lunch meeting).  Stefan had been forced as a draftee German physicist to develop the V-2 rocket at Peenemunde that was to rain bombs on London; he described watching a rocket “…go up…., and then it came down” with an ironic grin.  Next came Jim McClelland, a philosopher in the Historical and Philosophical Foundations of Education in which my wife Jill received her Sociology Ph.D.  I vividly Jim using his inaugural lecture in our seminar to fill 2 blackboards with symbolic logic as “proof” that only anarchism works.  That seminar remains the highlight of my teaching experiences.  Les, Barb, Jill and I became close friends, and visited them twice in England after Les’s retirement.  I dedicated my 1980 book, Crime Control Strategies, a systems analysis of what it meant when each major measure of crime and criminality, from convictions of crime, through arrests and self-reports, on through measures of criminality down to “recidivism.”

                Les mentored me personally and enthusiastically (often as I sat with him after hours in his living room) throughout our times together.  One thing he pointed out to me was that regression models simply made analyses of variance ordinal rather than descriptive.  And regression was indeed what he brought to his collaboration with Don Gottfredson to create the statistical system that became the first U.S. federal parole guidelines, which as I predicted to Les, soon became perverted into the notorious federal sentencing guidelines, to justify further incarceration rather than to promote safe and early release from prison.

                What struck me as most peculiar about analyses of variance was that the numerator was not simply divided by the total number of cases, but by (n-1).  I accepted implicitly that as Les said, that any 1 case in itself had no variance, and so “n” by itself had no variance, and Les’s insistence that you had to have at least 3 cases to have enough variation to infer variance, but it is only as I write today that I think I can see why, in Euclidean geometric terms.

                In purely mathematical material terms, 1 case equals a dot, a location that has purely unique significance in itself, or in plain English, doesn’t matter at all, or in scientific terms, has no significance.  You have to place one dot in relation to a second dot in order to show what direction they are headed.  You have to have three dots for them together in the “third” dimension in order for them as a group to have any substance at all in common.  And you have to move the three dots toward a fourth in order to show any movement of any substance, the “fourth” dimension commonly known as “time.”

                In social science, dots are called “variables.”  I learned in chapter 2 of my grad statistics text that it is a “fallacy” to “affirm a null hypothesis.”  Suppose we compare a group of people who have never been convicted of a felony, say murder, and find that the people who are black in one population have on average murdered more people than the people who are white.  Individually, the fact that someone happens to be black or white is insignificant—the variable black or white is fixed for each individual who happens to be black or white; it only signifies anything when it is some average of 3 blacks and 3 whites; in itself, the fact that blacks “commit” more “murders” than whites doesn’t even tell you anything but which group of any 3 blacks versus group of 3 whites is more likely include someone convicted of murder.  As for comparing 2 blacks to 2 whites, the odds of any single white or black being among all those convicted of murder is simply fifty-fifty.  The odds of any single black or white person being convicted of murder in my “violent” country are less than one in ten thousand, period.

                My fellow criminologists, please refrain from thinking you know anything about how dangerous or crime-free any individual is by statistical profile.  If this be your “best evidence” of what to expect from any human being you have no firsthand knowledge of, you know nothing.  Love and peace--hal