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We call these means conditional as they are conditioning on a Öxed value of the variable sex.
While you might not think of a personís sex as a random variable, it is random from the viewpoint
of econometric analysis. If you randomly select an individual, the sex of the individual is unknown
and thus random. (In the population of U.S. workers, the probability that a worker is a woman
happens to be 43%.) In observational data, it is most appropriate to view all measurements as
random variables, and the means of subpopulations are then conditional means.
As the two densities in Figure 2.3 appear similar, a hasty inference might be that there is not
a meaningful di§erence between the wage distributions of men and women. Before jumping to
this conclusion let us examine the di§erences in the distributions more carefully. As we mentioned
above, the primary di§erence between the two densities appears to be their means. This di§erence
equals
E (log(wage) j sex = man) E (log(wage) j sex = woman) = 3:05 2:81
= 0:24:
A di§erence in expected log wages of 0.24 implies an average 24% di§erence between the wages
of men and women, which is quite substantial. (For an explanation of logarithmic and percentage
di§erences see Section 2.4.)
Consider further splitting the men and women subpopulations by race, dividing the population
into whites, blacks, and other races. We display the log wage density functions of four of these
groups on the right in Figure 2.3. Again we see that the primary di§erence between the four density
functions is their central tendenc
CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 16
Focusing on the means of these distributions, Table 2.1 reports the mean log wage for each of
the six sub-populations.
Table 2.1: Mean Log Wages by Sex and Race
men women
white 3.07 2.82
black 2.86 2.73
other 3.03 2.86
The entries in Table 2.1 are the conditional means of log(wage) given sex and race. For example
E (log(wage) j sex = man; race = white) = 3:07
and
E (log(wage) j sex = woman; race = black) = 2:73