Galton’s fallacy (based on Quah 1993). During the late 1800s, Sir Francis Galton, a famous statistician in England, studied the distribution of heights in the British population and how the distribution was evolving over time. In particular, Galton noticed that the sons of tall fathers tended to be shorter than their fathers, and vice versa. Galton worried that this implied some kind of regression toward “mediocrity.” Suppose that we have a population of ten mothers who have ten daughters. Suppose that their heights are determined as follows.

Place ten sheets of paper in a hat labeled with heights of 5$1&, 5$2&, 5$3&, . . . 5$10&. Draw a number from the hat and let that be the height for a mother. Without replacing the sheet just drawn, continue. Now suppose that the heights of the daughters are determined in the same way, starting with the hat full again and drawing new heights. Make a graph of the change in height between daughter and mother against the height of the mother. Will tall mothers tend to have shorter daughters, and vice versa? Let the heights correspond to income levels, and consider observing income levels at two points in time, say 1960 and 1990. What does Galton’s fallacy imply about a plot of growth rates against initial income? Does this mean the fi gures in this chapter are useless?13