• the proportion of recruits with the characteristic before and after changing the standard, p₁ and p₂ (before changing standards, p₁ is assumed to be the same for the recruit population and the larger youth population).

• the size of the eligible youth population before and after changing standards, POP₁ and POP₂.

• the size of the force needed at 12 months (posttraining), F.

• the number of recruits needed before and after changing standards, N₁ and N₂, in order to attain a given force size F (N₂ will be less than N₁).

• recruiting costs as a function of the number recruited and the recruit-eligible population, which we denote by the function R(N,POP) with dR/dN > 0 and dR/d POP < 0.

The enlistment standard, in this analysis, is used to screen out populations that have higher expected attrition rates in the first year of service than the general population. The benefit associated with screening out a population is that attrition will be lower. The cost is that, because some in the population will no longer be able to enlist, recruiting costs may be higher.

Because there are lower attrition rates, the number of recruits leaving prior to completing a year will be lower. Let a₁ be the average attrition rate before the standard is in place and let a₂ be the average attrition rate after the standard is in place. Then, if N₁ is the number recruited and C is the unit cost of attrition, then is, to a first-order approximation, the savings in attrition costs. However, because attrition rates are lower, fewer individuals must be recruited in the first place to fill a given number of spaces at the end of the year. Let the number of spaces to be filled equal F. Then, N₁, the number originally recruited, is equal to F/(1 − a₁). If expected average attrition were to fall to a₂ with the enlistment standard in effect, the number of recruits needed would be N₂, equal to F/ (1 − a₂). Hence, a more precise estimate of the savings in attrition costs is calculated as the difference in costs before and after the enlistment standard is introduced:

Now, we ask, how does a₁, the average attrition rate prior to the introduction of the enlistment standard, differ from a₂, the average attrition rate after the introduction of the enlistment standard? Let the expected attrition rate (probability of attrition) for those without the characteristic to be subject to the enlistment standard be denoted a_wo and the expected