The Seven Pillars of Statistical Wisdom by Stephen M. Stigler

Author:Stephen M. Stigler [Stigler, Stephen M.]
Language: eng
Format: epub
ISBN: 9780674970212
Publisher: Harvard University Press
Published: 2016-03-07T05:00:00+00:00

5.6 The original quincunx, constructed in 1873 for use in an 1874 public lecture. The lead shot at the bottom give the impression of a bell-shaped curve. (Stigler 1986a)

5.7 Galton’s 1877 version of the quincunx, showing how the inclined chutes near the top compensate for the increase in dispersion below to maintain constant population dispersion, and how the offspring of two of the upper-level stature groups can be traced through the process to the lowest level. (Galton 1877)

To understand the solution Galton finally arrived at, and the wonderful device that carried him there, consider Figure 5.8 (focusing first on the left panel), an embellishment of one he published in 1889.10 It shows a quincunx that has been interrupted in the middle (A). The lead shot are now stopped halfway down; the outline they would have produced had they not been interrupted is shown at the bottom (B). The two outlines of distribution at levels A and B are similar; they differ only in that the midlevel (A) is more crudely drawn (it is my addition) and is more compact than the lower level (B), as would be expected, since the shot at level A are only subjected to about half the variation.

Galton observed the following paradox. If you were to release the shot in a single midlevel compartment, say, as indicated by the arrow on the left panel, they would fall randomly left or right, but on average they would fall directly below. Some would vary to the left, some to the right, but there would be no marked propensity one way or the other. But if we look at, say, a lower compartment on the left, after all midlevel shot have been released and permitted to complete their journey to level B, and ask where the residents of that lower compartment were likely to have descended from, the answer is not “directly above.” Rather, on average, they came from closer to the middle! (see Figure 5.8, right panel). The reason is simple: there are more level A shot in the middle that could venture left toward that compartment than there are level A shot to the left of it that could venture right toward it. Thus the two questions, asked from two different standpoints, have radically different answers. The simple reciprocity we might naively expect is not to be found.

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