NATURAL HISTORY, STATISTICS, AND APPLIED MATHEMATICS 253 



let US invent a ridiculously simple example which could yield two such lines. 

 Therefore (as a basis for discussion) let us imagine a big punch press which 

 cuts squares out of sheets of plastic. For the purposes of our problems we 

 will suppose that the cutting edge has a right-angled bend in it, that one 

 square is cut from each sheet of plastic, and that the machine is oriented at 

 the corner of the plastic sheet as in iig. 2. Furthermore, let us suppose that 

 at the beginning of the day's run the press is set up by a supervisor using 

 precision equipment so that A is exactly equal to B. We will also suppose 

 that so long as A equals B and the punched corners are square, there is no 

 need in this particular process for any great precision and that the operator 

 merely uses his eye and turns out a set of squares of somewhere near the 

 same size. If we examine the squares from a day's run and actually measure 

 A and B on each, we will find that a scatter diagram of them looks something 

 like fig. 3. Having been set up with precision equipment, on the scale of our 

 measurement they vary in size but not appreciably in shape. 



Now, consider what will happen when something goes wrong with the pre- 

 cise setting up of the machine. It may be that the supervisor is having 

 trouble with his wife and stops for a couple of drinks on his way to work. 

 Under these conditions he will set up the press some mornings so that A is 

 much shorter than B and on others so that it is longer. The press of course 

 is oriented on the corner of the plastic sheets so that it will turn out a series 

 of rectangles of varying size, all of them somewhat wider than high in the 

 first case and somewhat higher than wide in the other. If we take a few 

 samples from a series of runs on two different days and plot A against B, 

 we will get a scatter diagram something like that of fig. 4. 



This diagram is the nub of the argument. Given the facts as shown on 

 this diagram, what are the chances that the two sets of squares were punched 

 on different days? What would be the odds if you knew about the machine 

 but not about the supervisor's home life; what would they be if you under- 

 stood the supervisor and the operator as well as the machine? What would 

 your opinion be if you knew nothing about the machine but had merely had 

 considerable experience with pattern data? Or to ask the question a more 

 significant way: under each of these conditions how many of the punched- 

 out rectangles would you want to measure before you could be reasonably 

 certain that they were produced on different days? The actual answers to 

 these questions by scientists with different backgrounds are illuminating. 

 One research physicist replied immediately, "Well, if you know that the 

 punch is oriented on the corner, then the line through the measurements for 

 each day must go through the origin. You'll need only one sample from each 

 day's run." Other physicists thought about the problem in the same way but 

 wanted two samples from each run to be reasonably certain. One of my former 

 students, familiar with various kinds of pattern data, said that he would want 



