Part 3:
Let us assume the existence of a quantity that measures the collection of attributes that enables an athlete to perform in his sport. We call it the athletic quotient or AQ. It will not be necessary to define AQ precisely, only to specify a few of its properties. Each sport has its own set of physical demands, so that AQ is not a universal measure of athletic ability, but rather a specific measure for a specific sport. Good sprinters are generally not good swimmers. Blacks with their high body density have to work just to stay afloat. Thus, we can expect some racial/ethnic groups to have high AQs in one sport and low AQs in others. Black domination of basketball and sprinting suggests that the AQ for these sports may be closely related. We shall see.
We assume that the distribution of AQ in the black and white populations are of the same form but displaced from one another along the AQ axis by the quantity Δ. (Standard units are used throughout.) That is, if PB(x) is the probability distribution of AQ for blacks, and PW(x) that for whites, then PW(x) = PB(x+Δ). The quantity, Δ, is the black-white AQ gap.
Basketball AQ. Consider the pool of NBA aspirants to be all American men between the ages of 20 and 35. From this pool no more than 348 will make the NBA active list at any one time.
Let fB and fW be the fractions of blacks and whites, respectively, with AQ greater than or equal to some value, λ. (For example, λ might be the AQ needed to get into the circle of 150 players who put the most points per game on the board.) These fractions are given by
and
We note the following transformation:
and write for fW,
What value of the gap, Δ, will account for the racial composition of the 150 NBA high scorers? We first find the threshold AQ, λ, that selects them out. The number of blacks who made the cutoff (135) divided by the number of blacks in the pool of aspirants, is the quantity, fB. The Census Bureau estimates 4.12 million black men between the ages of 20 and 35 in the U.S. in 1996. Using this value to compute fB, and assuming PB is Gaussian, (1) may be solved numerically for λ, with the result: λ = 4.0 SD. That is, the threshold for this inner circle of 150 NBA players is about 4 SD from the mean of the black population at large. This is very rarefied territory indeed. The fraction of whites, fW, who make the cutoff is the number of whites (15) divided by the 25.7 million white men between 20 and 35 estimated by the Census Bureau. Using this value of fW, and the value of 4.0 for λ, (4) may be solved numerically for Δ, yielding D=0.87 SD. That is, the black-white basketball AQ gap is 0.87 SD. A white must have a minimum AQ almost 5 SD above the white mean to make the inner circle of 150 NBA top scorers.
If we were to look at the circle of 150 again a week or a month later, we would likely find its composition to be different from the 135/15 black-white breakdown we found one day in late December. Each look could produce a different value for Δ. We want to establish a range where we would expect to find Δ most of the time. The number of blacks in the circle of 150 is a binomial variable. Its variance is given by np(1-p), where n is the number of players (150) and p is the probability of a player being black. Estimating p as 135/150, the variance of the sampling error is 13.5 and the corresponding standard error is 3.67. The central limit theorem tells us that the sampling distribution of blacks (or whites) that we find in our various surveys will be approximately normal. Consequently, the 95% confidence interval for the number of blacks in the circle of 150 is 135 ± (1.96)(3.67). From the extremes of this interval we can map a 95% confidence interval for the gap, Δ. Doing so, we find that repeated sampling of the circle of 150 should produce values of Δ that 95% of the time will fall in the interval [0.78, 1.01] with a most probable value of 0.87.
Sprinting AQ. Running is a solitary undertaking, which lends it a certain nobility. The best athletes compete against the clock as much as against other runners, and the available data reflect this uncommon aspect of the sport. We focus on the apotheosis of speed: the 100-meter dash. The record holder in this event is known as the world's fastest man. On Peter Larsson's website,
http://www.algonet.se/~pela2, we found a remarkable compilation of the 819 all-time best 100m performances from October 1964 to September 1999. There were 111 runners on the list, many having contributed more than one of the best performances. Fifteen of the 111 were native West Africans. Most of them were relative newcomers, 14 of the 15 making the list in the last five years.
We can speculate why native West Africans are latecomers to the ranks of the very best sprinters. West Africa has a combined population of 200 million. Seven of its 19 nations have illiteracy rates over 70% and six others over 60%. Côte d'Ivoire, the richest country in the West Africa, has a per capita income of about $850. In such an environment, overlooking concealed talent is easy. In the U.S., high-school coaches are quick to spot athletic promise. In West Africa many boys never even make it to high school. West Africans and African Americans have entirely different access to athletic opportunity. In fact, it is remarkable that of the 111 all-time best 100m runners, 15 were West African. It is even more remarkable when we note that native West Africans comprise only about 1/30 of the world's population. Their recent emergence among the ranks of the world's best sprinters reflects improved opportunity. Still, if we seek a black-white (more properly a West African-European) AQ gap for sprinting, comparing groups with historically similar access to the big time is best, especially since the data stretch back to 1964. Accordingly, we eliminated native Africans from the list of 111. We also removed two Japanese, leaving us with 94 runners, 80 of West African descent and 14 of European descent.
The fractions of blacks and whites, fB and fW, needed to compute Δ, were estimated by assuming a world population containing 1 billion whites of European descent and 100 million in the West African Diaspora. Twelve percent of these were assumed to be between the ages of 20 and 35. The calculation yielded a most probable AQ gap for sprinting of 0.82 with a 95% confidence interval of [0.72, 0.96]. Table 1 summarizes the estimated black-white gaps for both basketball and sprinting.
Population sample Most probable Δ (SD) 95% confidence interval (SD)
NBA high scorers 0.87 [0.78, 1.01]
Top sprinters (no Africans) 0.82 [0.72, 0.96]
Table 1. Black-white AQ gaps for basketball and sprinting.
The AQ gap values for basketball and sprinting are conspicuously close. Their 95% confidence intervals span almost the same part of gap space. Coincidence? We doubt it, but cannot prove it either way. Both sports rely heavily on fundamental physical attributes. Jumping ability, prized in the NBA, is not in the sprinter's bag of tricks. But the same powerful legs and fast-twitch muscles that drive a man forward can also propel him upward. In the NFL blacks dominate the ranks of wide receivers and corner backs, where both speed and leaping talent are valued. The case that AQ for basketball and sprinting stems from a common set of physical attributes bestowed more abundantly upon blacks is circumstantial but strong.
PLAYING WITH THE GAP
We turn now to the questions posed previously. We asked for the probability that a young adult white American will be faster than his African American counterpart, both being selected randomly. The question reduces to finding the probability that the white has a higher sprinting AQ. The probability that the black has an AQ between y and y + dy is PB(y)dy. Let g(y) be the probability that the white has an AQ > y. That is,
The conditional probability that the white has the higher AQ given that the black has an AQ between y and y + dy is g(y)PB(y)dy. Integration over all values of y gives the probability, p, that the white has the higher AQ. That is,