analytical models for minority representation in academic departments

Research in Higher Education, Vol. 41, No. 4, 2000

ANALYTICAL MODELS FORMINORITY REPRESENTATIONIN ACADEMIC DEPARTMENTS

Kurt N. Johnson and J. D. Wiley

::: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :We present mathematical models for the evolution over time of the proportion ofminorities in an academic department or similarly selected group of constant size.Using the models, we analyze both the steady-state and time-dependent behavior ofthe proportion of minorities, and also obtain a means of evaluating the effectivenessof a department’s hiring history. We derive a number of surprising results with impor-tance to institutional hiring policy and affirmative action; and we also present meth-ods, suggested by the model and its behavior, to improve departmental hiring prac-tices.

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INTRODUCTION

Institutional hiring processes aimed at achieving racial and gender equity havebeen under intense scrutiny recently. This is particularly true in academic insti-tutions, but it also holds for private companies and other organizations. Thescrutiny is typically directed at the results of the hiring process. If the proportionof women or minorities in a given department is below some estimate of theirrespective proportion in the potential applicant pool, this may be viewed asevidence of adverse discrimination in hiring. Contrariwise, if the proportion ofwomen or minorities is above their estimated proportion in the applicant pool,accusations of overly aggressive affirmative action—or of reverse bias—maybe heard.

Consider, for example, a hypothetical academic department of 20 facultymembers that hires one member a year to replace an exiting member, and sup-pose that the applicant pool from which the department hires consistently con-

Kurt N. Johnson and J. D. Wiley, University of Wisconsin–Madison.Address correspondence to: J. D. Wiley, Office of the Provost, 150 Bascom Hall, University of

Wisconsin–Madison, Madison, WI 53706–1380.

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0361-0365/00/0800-0481$18.00/0 2000 Human Sciences Press, Inc.

482 JOHNSON AND WILEY

tains 10% minorities. If the hiring process is completely free from bias, thenone might expect that most of the time two department members, or 10% of the20 faculty total, will be minorities. Conversely, if the department has no minori-ties or only one minority for 10 or 15 years in a row, one might expect that thiswould constitute persuasive evidence of bias in the department’s hiring process.

In fact, neither of these is the case. In this article we analyze the mathematicsof hiring from a mixed minority/majority pool and show that a completely unbi-ased selection process results in frequent and extended periods during which theminority population deviates considerably from its expected value. This fact hassignificant implications for public policy regarding affirmative action.

Related Literature

A number of authors have presented mathematical models for hiring and dis-crimination. Becker and Williams (1986) presented a model for the time-evolu-tion of the expected number of minorities in a department, assuming the ratesat which employees leave and are hired are constant (but not necessarily equal),and the rate at which the composition of the selection pool changes is alsoconstant. As illustration they applied their model to an actual case of allegedgender discrimination in hiring.

Carroll and Rolph (1973) analyzed hiring discrimination in a macroscopicfashion by assuming a multitude of employers as well as employees, with bothgroups acting according to Markov processes. These authors also included aperceived quality or productivity difference between minority and majority em-ployees. Their model resulted in a critical value of majority unemploymentabove which discrimination occurs and below which it does not.

Wise (1975) presented a stochastic behavioral model of promotion in a sys-tem of graded employment. Henderson (1980), Forbes (1971), and Uebe (1971)also analyzed graded employment using Markov chains. Albright (1976) gave aMarkov model of a hiring problem that attempts to optimize employee quality.Other analytic studies of discrimination from non-Markov standpoints includethose of Dempster (1988), who used a Bayesian statistical model, and of Kolpinand Singell (1993), who presented a game-theoretic model.

The model we use is that of Hopkins (1980),1 which is Markov with the statescorresponding to the number of minorities in the department. Hopkins used hismodel to analyze the steady-state behavior of the portion of minorities in adepartment, including determining the mean and standard deviation of the num-ber of minorities and the probability that the department at steady state willinclude no minority employees. Hopkins also examined the transient responseof the number of minorities in the department. He assumed that the numberstarted out at zero but that then hiring followed the Markov chain law, and

ANALYTIC MODELS FOR MINORITY REPRESENTATION 483

calculated the probability that the number of minorities remains zero after agiven number of years.

In this article we present the model that Hopkins used, which is simple yetdisplays surprisingly rich behavior. We then expand on Hopkins’s work, andthat elsewhere in the literature, in several ways. We quantify the variability ofminority representation; compute the ex post facto probability of hiring biasbased on a department’s hiring history; and present two more-detailed modelsof hiring that are variations on the basic Markov model. We conclude with anexamination of the implications of our results for hiring and present possibleprocedural improvements suggested by these results.

Definitions and Assumptions

We will use the term “minority” throughout as shorthand to denote a memberof any given subset or small fraction of the general population, and the term“majority” to denote a nonmember of the subset. From a mathematical stand-point the subset could equally well be left-handers or people whose middlename contains a “q” as it could be racial minorities or women, but of coursethe associated legal and political issues are much larger in the latter two cases.We will use the term “department” for any group of individuals selected inaccordance with our model, but our primary focus is on academic faculty units:departments (in the traditional sense) and to a lesser degree larger units such asschools and colleges.

Our models are based on the idealized assumption of applicant pools com-prised of only two groups: a majority group and a minority group. We assume,further, that all significant characteristics and qualifications are distributed iden-tically among the members of these two groups except for the distinguishingcharacteristic that defines minority status. That characteristic, whatever it maybe, is present in the minority group, absent in the majority group, and (exceptin the section discussing hiring history) postulated to be completely irrelevantfor purposes of selection. Thus, if the fraction of minority candidates in theapplicant pool is f, then the probability of selecting a minority hire is exactly f.

A MARKOV-CHAIN MODEL FOR DEPARTMENT CHANGE

The model we use is that of Hopkins (1980). Consider a department of Nmembers. Suppose that at regular time intervals a randomly selected departmentmember is replaced with a new member whose probability of being a minorityis f, 0 < f < 1; thus the size of the entire department never changes, only itscomposition changes. This process can be modeled as a Markov chain (see forexample Ethier and Kurtz, 1986), which is a stochastic process with a finitenumber of states and with transition probabilities between the states that are


independent of the history of the process. Here the process has N + 1 states,where the state i (i ∈ {0, 1, . . . , N}) corresponds to i members of the depart-ment belonging to the minority group and N − i members belonging to the ma-jority group. The state-to-state transition probabilities in a Markov chain can beencoded in a transition matrix T(N, f ), such that the (i, j) element of T is theprobability that the process moves to state j at the next transition if it is currentlyin state i. (We index the matrix entries, like the states, starting at 0 rather thanat the usual 1.) Thus in our model, Tij is the probability that after the next hiringthe department will have j minorities, given that it currently has i minorities. Thetransition matrix for the process being considered here is particularly simple: Ateach selection cycle, since only one department member is changing, the numberof minorities can change by at most ±1; therefore, the transition matrix is allzeros except for its diagonal and immediate off-diagonal elements. A Markovprocess with a transition matrix of this form is known in the literature as a birth-and-death process. It is also a form of random walk, with “rests” due to thenonzero elements on the diagonal.

It is simple to calculate the transition probabilities for the matrix. Supposethe process is in state i, with 1 ≤ i ≤ N. The next state will be i − 1 if a minorityis removed from the department and replaced with a majority member. Theprobability of the former event is i/N and the probability of the latter is 1 − f,and since they are independent, the probability of transition from state i to statei − 1 is

Ti,i−1 =i

N(1 − f).

Similarly, for 0 ≤ i ≤ N − 1 the probability of transition from state i to statei + 1 is

Ti,i+1 = �1 −i

N� f.

Since each row of the transition matrix must sum to 1, the diagonal ele-ments are

Tii = 1 −i

N+ f � 2i

N− 1�.

We will call the resulting matrix for this idealized model of the selection pro-cess the canonical department transition matrix. As an example, if N = 5 andf = 0.2, the canonical department transition matrix is


The state of a Markov chain at a given time is expressed by the state row vectorx, where the ith element of x is the probability that the process is in state i atthe given time. Thus, the sum of the elements in any state vector must be 1. Ifthe state vector at time n (that is, n selection cycles after the process started) isx(n), then the state vector at time step n + 1 is x(n + 1) = x(n)T. Suppose, forexample, that the five-member department whose transition matrix is given in(1) is known initially to have exactly two minority members. Then the statevector at time n = 0 is x(0) = [0 0 1 0 0 0], and x(1) is

x(0)T = [0 .32 .56 .12 0 0].

That is, the probability that the number of minorities at time n = 1 is one is0.32, the probability that the number is two is 0.56, the probability that thenumber is three is 0.12, and any other number of minorities is impossible.

It is immediately apparent from the form of the canonical department transi-tion matrix that it is irreducible: every state can be reached from every otherstate. Since it is also finite, one can show (Ethier and Kurtz, 1986) there existsa corresponding unique stationary distribution x, which gives the limiting—orlong-term—probabilities of the process being in each of its respective states.That is, x gives the limit as n → ∞ of the probabilities that at time step n thenumber of minority members in the department is 0, 1, . . . , N.

The stationary distribution can be computed in several ways. Since the pro-cess is irreducible, the long-term probability distribution is independent of theinitial state vector, so one can choose any valid state vector x and obtain thelong-term probability distribution from x = lim

n→∞xT n. Alternatively, since x is

indeed a steady-state vector, it is a fixed point of the transition matrix T; thatis, xT = x. Therefore, x can be obtained as the left eigenvector of T correspond-ing to the eigenvalue 1. Or most directly, for the canonical department transitionmatrix one can compute x by means of a simple combinatorial argument: Theith element of x is the probability of finding i minorities (whose a priori densityin any population is f ) in a department of size N, so x is given by the binomialdistribution


xi = � Ni � f i(1 − f )n−i. (2)

One can show using (2) that the maximal element of x is xNf if Nf is an integer.That is, the most probable number of minorities in the department (the mode)is Nf. This is also the expected number (the mean).

For the N = 5, f = 0.2 case shown above, equation 2 yields approximately

x = [0.328 0.410 0.205 0.0512 0.0064 0.0003].

So in steady state the department has no minorities 32.8% of the time, exactlyone minority 41.0% of the time, and so on. Observe that the most probablenumber of minorities is 1, which is also the expected number. However, it issignificantly more likely that the number of minority department members iszero than it is that the number is two. In fact, it is significantly more likely thatthe number of minorities is zero than it is that the number is two or above(0.328 vs. 0.262). Put another way, in a long period of time one would expecta department of 5 people to have no minorities about 33% of the time, and morethan one minority only about 26% of the time. More generally, if Nf is aninteger, then ΣNf − 1

i = 0 xi > ΣNi = Nf + 1 xi.

If i � N and Nf � 1, equation 2 is approximated by the Poisson density func-tion

xi �(Nf )i

e−Nf

i!= : QN, f(i), (3)

which can be easier to work with than (2). In fact, for practical ranges of N andf, this is a useful approximation even if Nf � 10. Again the asymmetry aboutthe mean Nf is apparent: ΣNf − 1

i = 0 QN, f(i) > ΣNi = Nf + 1QN, f(i), which implies that one

can expect the number of minorities in the department to spend more time belowthe mean than above the mean.

DYNAMICS OF DEPARTMENT CHANGE

So far, this discussion has addressed primarily the steady-state behavior ofthe department selection process. However, since the process is a random walkit exhibits significant variation about its mean state. Figure 1 illustrates this. Theplots show typical graphs of the fraction of minority members (which we willcall the minority fraction or minority ratio) as a function of time, in a departmentof size N = 50 and with various proportions of minorities f in the hiring pool.


FIG. 1. Typical plots of minority fraction as a function of selection cycle (or number offaculty hires) for a large department (50 members). It is demonstrated later that for adepartment of this size, there are about 2.5 selection cycles per year. Thus each plot

covers about 400 years’ duration.

The plots in the figure were generated by first randomly choosing a startingvalue for the minority ratio from the distribution (2). The starting value resultsin an initial state vector x 0. The value of the minority ratio at the next hiringcycle is obtained by randomly choosing a value from the distribution x 0T. Thisvalue induces a state vector x 1. x 2 is obtained by choosing a value from x 1T,and so on for 1000 cycles. Observe that in each example, the minority fractiondisplays prolonged and significant excursions from its expected value Nf. (Com-pare with Feller, 1957, p. 84.)

One can compute the standard deviation of the minority fraction from thesteady-state distribution:

σ = √ ∑N

i = 0

xi(i − Nf )2.

For the example case N = 5, f = 0.2, the standard deviation is 0.89. For thePoisson approximation (3) the standard deviation is √Nf, which in the examplecase would yield 1.


The standard deviation quantifies the amplitude of the variation of the minor-ity ratio. There are several ways to obtain in addition a measure of the timeperiod of this variation. One involves computing the autocorrelation (Papoulis,1991) of the time-dependent minority fraction; if the minority fraction after selec-tion cycle n is x(n), then the autocorrelation is defined as y(n) = Σix(i)x(i − n). Forthe example N = 50, f = 0.1 shown in Figure 1, the autocorrelation appears asshown on the left in Figure 2. (The plot on the right shows the same process, butsimulated for 100 000 cycles, and demonstrates that the noise surrounding thecentral peak disappears as the simulation time gets longer.) The width of thecentral peak apparent in the graph provides a measure of the persistence offluctuations in the minority ratio: a wider peak means the ratio tends to persistlonger around a given level. For the example shown in the figure, the centralpeak has a full width at half-maximum (FWHM) of about 100 selection cycles.This can be considered the characteristic time scale for change in this example.

Alternatively, assuming the mean minority ratio Nf is an integer, one cancompute the time, in selection cycles, between consecutive returns of the minor-ity ratio to the value Nf. That is, starting when the minority ratio changes fromNf − 1 or Nf + 1 to Nf, measure the number of cycles required for the ratio tochange to a different value and then return to Nf. The expected value of thisfirst-return time will be a measure of the oscillation time of the minority ratiofor excursions from the mean value, and can be computed as follows. In thesteady state, the process spends on average a portion xi of its time in state i. Ifit is in state i at a given step, the probability of it staying in state i is Tii (whereT is the transition matrix as before), and it is easy to show that on average itpersists in state i for a dwell time

tD(i) = 1/(1 − Tii)

FIG. 2. (Left) Autocorrelation of the minority fraction shown in Figure 1 for N = 50,f = 0.1. (Right) The central 1000 points of the autocorrelation for the same process,simulated for 100000 cycles instead of 1000 cycles. In both plots, the units on the

horizontal axis are selection cycles.


cycles. In particular, tD(0) = 1/f, tD(N) = 1/(1 − f ), tD(N/2) = 2; and tD(Nf ) = 1/[2f(1 − f )]. The average time tR(i) between returns to state i must satisfy tR(i)xi = 1/(1 − Tii), or tR(i) = 1/[xi(1 − Tii)]; and the average time spent away fromstate i without returning is tW(i) = tR(i) − tD(i). Also of interest are the averagedurations tA and tB of excursions above and below the mean respectively. If themean Nf is an integer, then it is easy to see that TNf,Nf−1 = TNf,Nf+1. Hence thefrequencies of excursions above and below the mean must be identical, and sothese excursions’ mean durations must be proportional to the expected fractionsof time spent above and below the mean. That is,

tB = 2ΣNf − 1

i = 0 xi

1 − xNf

tW(Nf )

tA = 2ΣN

i = Nf + 1xi

1 − xNf

tW(Nf )

where we use the fact that (again since TNf,Nf − 1 = TNf,Nf + 1) tW(Nf ) = (tB + tA)/2.The units of these characteristic times can be converted from selection cycles

to years by postulating an annual departmental turnover rate s. Human life spansprovide a practical lower bound for departmental turnover. If a department hiresonly recent Ph.D. graduates at age 30, and if all such hires remain in the depart-ment until retirement at age 65, then each faculty member would remain in thedepartment for 35 years. Thus, annual turnover rates smaller than 1/35, or about3%/year, are unrealistic. Although there is no similarly plausible upper bound,we estimate that turnover rates higher than 8% or so would be sufficiently dis-ruptive and expensive to cause a department to take serious steps to reduce it.Actual turnover data for the University of Wisconsin for the last 20 years showaverage annual turnover of about 90 to 110 faculty/year for a total faculty sizethat has been in the 2200 to 2400 range over that period. So taking s to be about5% seems reasonable.

This means there are about N/20 selection cycles per year. Using this conver-sion factor, Table 1 shows (20/N)tD(Nf ) (the average persistence time at themean minority ratio, in years), (20/N)tB(Nf ) (the average duration of excursionsbelow the mean, in years), and (20/N)tA(Nf ) (the average duration of excursionsabove the mean, in years) for a selection of values of N and f such that Nf is aninteger.

So looking for example at the N = 20, f = 0.1 entries of Table 1, we see theexpected times for a department of size 20 and a minority group whose preva-lence in the candidate pool is 10%. If the number of minorities in the departmentis initially either one or three, and then a department member is replaced result-ing in exactly two minorities in the department, one can expect there to continueto be exactly two minorities for (on average) 5.56 years. If the number of minor-


TABLE 1. Average Time in Years at Mean Ratio (top) and Average Time in Yearsof Excursions Below the Mean (bottom left) and Above the Mean (bottom right)

N

tD(Nf ) � 20N 5 10 20 40

0.05 10.5 5.26

f0.1 11.1 5.56 2.780.2 12.5 6.25 3.13 1.560.4 8.33 4.17 2.08 1.04

N N

tB(Nf ) � 20N 5 10 20 40 tA(Nf ) � 20

N 5 10 20 40

0.05 20.0 15.1 0.05 14.7 12.3

f0.1 20.0 15.3 11.4

f0.1 15.1 12.6 10.0

0.2 20.0 15.6 11.8 8.76 0.2 16.0 13.3 10.6 8.150.4 16.3 12.7 9.64 7.17 0.4 15.3 12.2 9.38 7.03

ities changes from two to one, it will take on average 15.3 years before thenumber of minorities returns to two; if the number of minorities changes fromtwo to three, it will take on average 12.6 years before the number returns totwo.

Finally, suppose a department starts at time zero with no minority members.It is of interest to know the timescale of the approach to the steady-state valueof Nf. It turns out that the timescale depends on the second-largest eigenvalueλ1 of T. In particular,

�x(n) − x� � C(1 − λn1) (4)

(Ethier and Kurtz, 1986), where � � � is a vector norm. For 0 < ε < 1 define therise time τε to be the minimum number of selection cycles necessary for theexpected number of minorities in the department to rise from zero to (1 − ε)Nf.For example, if ε = 0.2, then the rise time would be the expected number ofcycles necessary for the minority ratio to rise from 0 to 80% of its long-termmean value. Then from (4) one obtains τε � logε/logλ1. Letting ε = 1/e andusing2 λ1= 1 − 1/N for the canonical department transition matrix, the rise time(in years) becomes

τ1/e �−1

sN log(1 − 1/N), (5)


where s is the turnover rate. For N large τ1/e is asymptotically 1/s. Since s isabout 5%, the rise time is about 20 years. The precise values, measured in bothselection cycles and years, as given by equation 5 for typical values of N areshown in Table 2. For the mathematical details skipped in this paragraph, seethe website Johnson and Wiley (1998).

From the above analysis and the table, we conclude that a department withinitially no minorities should take, on average, about 20 years for its fraction ofminorities to reach 1 − 1/e � 63% of the fraction of minorities in the applicantpool, assuming a completely unbiased selection process. Similarly, it wouldtake, on average, about 2.3 time constants, or nearly 50 years, for the fractionof minorities to reach 90% of the expected steady-state value Nf. Keep in mind,of course, that the very notion of affirmative action has existed in the politicalconsciousness for less than 50 years.

WHAT CAN BE DEDUCED FROM A DEPARTMENT’SHIRING HISTORY?

The model we have presented assumes a completely “blind” selection pro-cess—a process that involves neither discrimination against minorities nor anyform of favoritism in hiring. Nevertheless, discrimination could occur inadver-tently if the department advertised its vacant positions or recruited applicants inways that resulted in an unrepresentative applicant pool. By unrepresentativepool, we mean a pool exhibiting a minority fraction that is different from thehypothetical “true” value. In practice it is difficult, if not impossible, to deter-mine what the true value should be, no matter how precisely the position andthe required qualifications are defined. Nevertheless, some true value for theminority fraction must exist—in principle. Some finite number of individualswill meet any specified set of job qualifications and be available for recruitment.Of those, a finite fraction will belong to the target minority group. Thus, a truef exists, even if the value is not known accurately. We have demonstrated thatit is not unusual for the minority ratio to differ from its expected value by asignificant amount and for a significant length of time. Suppose, however, that

TABLE 2. Rise Times for Department Minority Ratios

N

5 10 20 50 100

cycles 4.48 9.49 19.5 49.5 99.5years 17.9 19.0 19.5 19.8 19.9


it were known that a particular department’s minority ratio was lower than ex-pected for a long duration, and suppose also that it were plausible that thedepartment is not, in fact, tapping the full talent pool. (For the sake of brevitywe will refer to such unrepresentative hiring as “discrimination” even though itmay well be the sort of inadvertent discrimination described above.) Then it isof interest to know the probability that discrimination had indeed occurred,given the departmental minority ratio as observed over a period of time.

Since the process is stochastic, in a given time span any sequence of minorityratios that could result from the Markov chain model, however apparently un-likely, is plausible. So for example, 10 consecutive hiring cycles with no minori-ties in the department or even 100 consecutive cycles with all minorities areboth possible, though improbable, even if the hiring process is completely biasfree. Thus, one cannot obtain the probability of discrimination having occurredwithout making some additional assumption. One possible solution would be tochoose some suitably low level of probability for a given time span and assumethat, if the sequence of minority ratios over that time span has a probability ofoccurrence that is below the chosen cutoff probability, then discrimination“must have occurred.”

We find another approach inherently more satisfying. We use Bayes’ Rule(Feller, 1957)

P(A*B) =P(A)P(B*A)

P(B), (6)

and assume a value c for the a priori probability of discrimination in the depart-ment. Admittedly, the value of c could be difficult to obtain with any kind ofaccuracy, but we will demonstrate that meaningful results can be obtained evenif one is extremely conservative in estimating c.

Suppose that the given department has N members and the fraction of minori-ties in its hiring pool is f̂. We assume also that hiring always proceeds accordingto the Markov model presented previously, but that if discrimination occurs, thevalue of f used in the model is not f̂ but rather a value selected from a uniformdistribution over [0, f̂ ]. Suppose that for the last n hiring cycles the minorityratio has been zero—that is, there have been no minorities in the department.Then one can show (Johnson and Wiley, 1998) that

P(“discrimination” occurred*no minorities for n cycles) =

1

1 + f̂(N + n) � 1 − cc � (1 − f̂ )N + n − 1

1 − (1 − f̂ )N + n

. (7)


TABLE 3. Conditional Probability of Discrimination as Given inEquation 7, for f̂ = 0.1 and Various Values of N � n and c

c

0.000001 0.0001 0.01 0.1 0.25 0.5

5 0.00000125 0.000125 0.0125 0.122 0.294 0.55510 0.00000168 0.000168 0.0167 0.157 0.359 0.62720 0.00000325 0.000325 0.0318 0.265 0.520 0.765

N + n 50 0.0000347 0.00346 0.260 0.794 0.921 0.972100 0.00338 0.253 0.972 0.997 0.999 1.0200 0.985 1.0 1.0 1.0 1.0 1.0500 1.0 1.0 1.0 1.0 1.0 1.0

Note that the department size N and the length n of the zero sequence appearabove only in the combination N + n.

Table 3 shows the conditional probability of discrimination, as given by (7),for f̂ = 0.1 and assorted values of c and N + n. Table 4 shows the value of N +n that makes the conditional probability that discrimination has occurred, asgiven by (7), equal to 90%. Consider as an example a department of 10 membersand a minority population whose prevalence in the applicant pool is 5%, andsuppose the a priori probability of discrimination is estimated at 1%. Then read-ing off the table, it would be necessary that there have been 166 = 176 − 10consecutive selection cycles of no members of this minority group in the depart-ment in order to conclude, in the absence of any other evidence, that with 90%probability discrimination had occurred. Using the approximate conversion fac-tor N/20 cycles per year, this corresponds to 332 years of no minorities neces-

TABLE 4. Value of N � n Yielding 90% Probability that Discrimination Occurred,According to Equation 7

c

0.000001 0.0001 0.01 0.1 0.25 0.5

0.01 1890.0 1400.0 895.0 620.0 487.0 346.00.02 939.0 696.0 445.0 309.0 243.0 173.0

f̂ 0.05 370.0 274.0 176.0 122.0 96.0 68.40.10 180.0 134.0 85.9 59.7 47.0 33.60.20 85.5 63.5 40.9 28.5 22.5 16.2


sary for a 90% certainty that the lack of minorities was due to discriminationrather than happenstance.

Viewing equation 7 from one more numerical perspective, Table 5 shows thevalue of the a priori probability of discrimination c required to be able to con-clude with 90% certainty that discrimination has occurred, for various values ofN + n and f̂. Consider again a department of 10 members, a minority groupwhose prevalence is 5%, and suppose that the department has had no minoritiesfor 10 selection cycles (about 20 years). Then N + n = 20, and reading off thetable we see that to be 90% confident that this long period of no minoritiesresults from discrimination, one would need to consider the probability of dis-crimination occurring in this department (before having any knowledge of thedepartment’s hiring history) to be at least 84%. This might, for example, entailknowing that at least 84% of similar departments engage in unrepresentativehiring of the sort considered here.

The results in Tables 3–5 can be conveniently summarized in the followingapproximate rule of thumb:

If (N + n) f < 3, then the observation of n successive majority hires is not, by itself,sufficient to demonstrate (at the 90% confidence level) that the department is recruit-ing and hiring inappropriately.

In particular, when (N + n) f < 3, it requires c > 0.5 (that is, it requires anindependent, a priori reason to believe that discrimination is more likely thannot) in order to conclude that the observed absence of minority hires is convinc-ing further evidence of discrimination. If one desires confidence at the 95%,98%, or 99% level, then the 3 in the equation should be replaced by 4, 5, or 6,respectively.

Finally, we wish to make two remarks regarding the results presented in this

TABLE 5. A Priori Probability of Discrimination Necessary to Conclude with 90%Certainty that Discrimination has Occurred, According to Equation 7

f̂

0.01 0.02 0.05 0.10 0.20

5 0.8982 0.8963 0.8901 0.8782 0.845810 0.8958 0.8914 0.8761 0.8426 0.730220 0.8909 0.8806 0.8411 0.7346 0.3442

N + n 50 0.8744 0.8403 0.6638 0.2057 0.0016100 0.8400 0.7374 0.2200 0.0026 0.0000200 0.7377 0.3967 0.0033 0.0000 0.0000500 0.2311 0.0038 0.0000 0.0000 0.0000


section. First, while readers might quibble with the details of our model ofdiscrimination, we believe that other reasonable models would give results qual-itatively similar to those presented here. This is because the principal conclusionfrom this analysis—that it is extremely difficult to prove bias (or, for that mat-ter, lack thereof) based solely on an ex post facto analysis of a hiring history ofany reasonable length—is due to the large fluctuations in the minority ratio seenin the previous section, and that central fact will appear regardless of the specif-ics of the model. Second, we emphasize that the type of “evidence of apparentdiscrimination” considered in detail here (that is, no minorities whatsoever inthe department for an extended period of time) is the worst-case scenario in thesense that it is the strongest possible evidence of discrimination. Weaker evi-dence of discrimination, such as an extended period of time during which thedepartment is below the expected minority ratio but still has a few minorities,would make it even more difficult to prove that discrimination had indeed oc-curred, and all else being equal, would hence require even longer observationtimes—by a large factor in many instances—than those given in Table 4. (De-termining the probability that such weaker evidence of discrimination is due tounrepresentative hiring rather than happenstance is probably best done by MonteCarlo simulation.)

VARIATIONS ON THE BASIC MODEL

In the hiring model presented earlier, exactly one department member is re-placed at each hiring cycle. With N members in the department, the probabilitythat a given member is replaced is therefore 1/N at each cycle, regardless ofhow long that individual has been a department member. Also, the replacementprobabilities for different members in a single department are not independent.Two possible improvements to the model are thus as follows:

A. Since in an actual department more than one member may be replaced atonce (or in a very short period) and sometimes no member is replaced at allfor a long time, make the replacement probabilities for the department mem-bers independent. This allows any number of members (including none) tobe replaced in a single hiring cycle.

B. Allow the probability of a member’s replacement to vary as a function ofthe time since hiring. For example, in academic departments the probabilityof a faculty member leaving is fairly high (perhaps around 10% per year)for the first five years or so of employment in the department, until tenureis obtained. After that it drops considerably, to perhaps 2% per year, risingagain more slowly and hitting perhaps 8% per year 30 years after hiring asretirement age is approached.


Variation A is possible still within the realm of Markov chains; it merelyaffects the transition matrix (which, in particular, is then no longer tridiagonal).Under variation B, however, the model is no longer a Markov chain since thetransition probabilities become dependent on the history of the process. How-ever, it can be implemented using a Monte Carlo simulation.

To address these modifications, let the probability that a department memberis replaced after j hiring cycles in the department be p( j). Suppose first that p( j)is constant, p( j) ≡ p. Then a member’s expected duration in the department is1/p cycles. In the basic hiring model, p = 1/N so the expected duration in adepartment is N cycles, or about 20 years. This is a reasonable figure for anacademic department.

If in addition the department member replacement probabilities are indepen-dent (variation A), then it is not difficult to obtain an analytic expression for thetransition probabilities. One finds that the probability of transition from i minori-ties to j minorities is

Tij = ∑i

k = max(0,i − j)�� ik �pk(1 − p)i − k ×

∑N − i

l = max(0,j − i)� N − i

l �p l(1 − p)N − i − l� l + kj − i + k �f j − i + k(1 − f )l − j + i�. (8)

Here k counts the possible numbers of minority members leaving and l countsthe possible numbers of majority members leaving at a single hiring cycle.

For example, for N = 5, f = 0.2, and p = 0.2,

Compare this with equation 1.Allowing the probability of replacement to depend on the time since hiring,

one can obtain a still more-detailed hiring model. As mentioned previously insuch a case the model is no longer a Markov process, but can still be imple-mented numerically. Table 6 compares numerical output from three models,each with N = 20 and f = 0.2: the basic model; the Markov model with transition


TABLE 6. Comparison of Characteristics Among Three Model Variations.(In each case, N = 20 and f = 0.05.)

Basic Model A Model B

Mean minority ratio 0.201 0.200 0.200Standard deviation of minority ratio 0.089 0.090 0.089Mean employment duration (years) 20.00 19.95 20.00Minority ratio autocorrelation FWHM (years) 27.6 27.4 32.3

matrix as given in (8) and p = 1/N = 0.05 (item A above); and the non-Markovmodel in which the replacement probabilities are 10% per year for the first 5years, 2% per year for the next 25 years, and 8% per year thereafter (item Babove). Each model was run for 800,000 cycles.

The most important lines in the table are the second and fourth, with theothers included primarily to verify consistency among the three models. Thetable data indicate that there is little difference in performance among the mod-els. All three resulted in nearly identical values for the standard deviation of theminority ratio, implying that they are similar in the extent to which the minorityratio varies from its mean value. The basic model and model A also had nearlyidentical minority ratio autocorrelation widths; model B’s autocorrelation widthwas slightly larger, but this was at least in large part because the autocorrelationpeak in model B has a different shape than the peaks in the other models.Specifically, model B’s peak is more triangular, those of the basic model andmodel A more gently curved. This in turn is due to the inclusion in model B ofretirement after 30 years.

Other possible variations to the model include:

• Allowing f to vary over time, to reflect changing demographics in the appli-cant pool. For example, to accurately model the supply of certain minoritiesin certain fields, it may be appropriate to let f rise to twice its current valueover the next 50 years. This variation would still admit a Markov representa-tion, with the transition matrix now a function of time. Hopkins (1980) al-lowed for this possibility in his original model, and Becker and Williams(1986) considered it in some detail, so we will not take it further here.

• Modeling a finite universe of job candidates and institutions, so that the moreminorities are hired by one institution, the fewer remain in the pool for possi-ble hire by another institution and so, in effect, the lower f becomes. To beuseful, this model would probably have to be highly complex and have a vastnumber of adjustable parameters.


CONCLUSION

We have presented three different analytical models of selection processesthat contain neither discrimination against, nor selection quotas that favor thehiring of minorities. Although the models are highly idealized, they demonstratetheoretical bounds on the performance of practical hiring processes. That is tosay, any actual hiring process may approach the goal of being “completely bias-free,” but, by definition, cannot exceed that goal. All three models exhibit thefollowing characteristics:

1. A long-term average of Nf minorities in the department if a department offixed size N is hiring from a pool that has a constant fraction f of minorities.This is, of course, the intuitive expectation, and the standard against whichdepartments are judged.

2. Large and long-lasting fluctuations away from the average number Nf ofminorities in the department. The fluctuations are more pronounced forsmaller Nf values, but the fluctuations are very significant for practical valuesof N and f. For example, the lower left graph in Figure 1 illustrates thedynamics for a department of size 50, hiring from a pool that is 10% minor-ity, and exhibits a period of at least 60 selection cycles (about 25 years)during which the departmental minority fraction is less than half the expectedvalue of 10%.

3. An inherent asymmetry in unbiased hiring. Namely, when the number ofminorities in a department fluctuates away from the expected number, Nf, ittakes longer to return to Nf from below than from above. That is to say,although negative and positive deviations are equally likely to occur, thenegative deviations last longer. Thus, paradoxically, nominally unbiased(“minority-blind”) hiring processes are inherently biased against minoritiesin this sense.

4. Long rise and return times necessary for the department to initially achievea goal or return to that goal after a fluctuation. In particular, we showed thatthe 1/e risetime from zero is quite accurately approximated by the inverse ofthe annual turnover rate for departments of any size. For typical turnoverrates of 5%/year, this implies a 20-year risetime (and 50 years to reach 90%of the asymptotic value).

In the last section we showed that these very large and persistent fluctuationsmake it exceedingly difficult to detect discrimination simply by examining theresults of the departmental hiring. In particular:

5. To obtain convincing evidence that a department is either hiring from aninappropriate applicant pool or actively discriminating against minority ap-plicants solely by observing the results of the hiring process would require


impractically long observation times (100 or more years in many realisticcases). The rarer the minority group is in the selection pool (the smaller f is),the longer the observation time necessary.

It is interesting to compare the time scales associated with changing the mi-nority ratio in a department with other relevant time scales. For example, in adepartment having a 5% annual turnover rate, the average faculty career lengthis 20 years, which, not entirely coincidentally, is the same as the 1/e rise timediscussed above. Thus, individual faculty members can easily spend their entirecareers in one of the peaks or valleys shown in Figure 1, fruitlessly seekingspecial causes or explanations for the seemingly anomalous minority fraction intheir department. In addition, affirmative action policies have in most instancesbeen in place for only about 30 years, which again is well within the timescaleof natural fluctuations in the minority ratio.

Another factor to consider is that many academic departments in existencetoday were formed or experienced most of their growth after 1960, and havebeen at their present sizes for considerably less than 50 years. To cite an exam-ple, the UW–Madison Department of Electrical and Computer Engineering wasestablished (under a different name!) in 1889, and grew slowly but steadily untilit reached a size of about 15 faculty in the late 1950s. Then, between 1960 and1980, it grew rapidly to about 45 faculty, and has been in the 45–50 range sincethat time. Because we seldom see or think about the 100-year growth curve,however, most people on campus think of this department simply as as “a de-partment of about 50 faculty.” In fact, in its entire history, the department hashired only about 150 faculty members, more than half of whom are or wereknown personally by the present faculty. This dramatically emphasizes the con-sequences of low turnover rates, even in a growing department.

It is also of interest to note that turnover rates among the student body are inthe 20% per year range. Thus, during periods in which the demographics arechanging, the gender, ethnic, and racial mix of the faculty will always lag farbehind that of the student body. This is strikingly illustrated in schools of phar-macy, where men are now frequently a distinct minority among the studentbody, but still constitute an overwhelming majority among the faculty.

These results and observations have important implications for institutionaland governmental policymakers. Although the importance, or even the rele-vance, of gender, ethnic, and racial diversity in faculty hiring may be a contro-versial subject within and outside the academy, assume provisionally that “fair-ness” is a universally accepted and uncontroversial goal. It is, after all, stillillegal to engage in overt discrimination, even in jurisdictions that have curtailedor eliminated traditional practices of affirmative action. That being the case,how are fairness and equal opportunity to be demonstrated and judged? Thefirst step, obviously, is to design and implement search, screening, and hiring


processes that, to the extent humanly possible, “everyone” agrees are fair (interms of the notation used in the section on hiring history, perhaps c � 0.01).Suppose that has been done, and consider now a department of 20 faculty thatis, and to this point has always been, all-male. Suppose, further, that this depart-ment is currently hiring from a pool that has been determined to be 10% female.If over the subsequent 20 years the department remains all male, is it likely that“everyone” will continue to believe in the fairness of their hiring processes?According to Table 4, it would require at least 65 years of all-male hiring beforewe might be justified in concluding (at the 90% confidence level) that the pro-cess is gender-biased. We submit that this is a sufficiently counterintuitive find-ing that it undermines our ability to rely on process design for demonstrationsof fairness. Indeed, we conclude that if demonstrating fairness is an importantgoal, then visible differences such as race and gender cannot be irrelevant crite-ria in selection processes, independently of any other reasons that may be ad-vanced for the importance of diversity.

Although we speak casually about the “random” events or the “chaotic” as-pects of our lives, our day-to-day experiences—the basis for whatever intuitionwe may develop—are overwhelmingly characterized by clear cause-and-effectand by substantial predictability. Even coin-tossing, which is the simplest pro-cess commonly used for making random choices, can lead to results that appearto be astoundingly “biased.” In the introduction to his famous chapter on coin-tossing and random walks, Feller (1957, p. 65) promises to demonstrate “conclu-sions which are not only unexpected but actually come as a shock to intuitionand common sense.” Even experienced statisticians would likely agree that Fel-ler delivers on this promise.

Hiring from a pool of similarly qualified candidates of whom 10% are froman identifiable minority population is analogous to tossing a ten-faced coin thathas nine tails and one head. The probability of flipping a “head” with this coinremains exactly 10%, regardless of how long the preceding string of “tails” maybe. Such a coin, flipped at most a few times a year, will necessarily producemany strings of tails that are so long as to demand explanation for why the coin“always” comes up tails.

Faculty hiring is largely dominated by processes and decisions at the individ-ual departmental level, and departments typically contain only a few tens offaculty positions at most. As we have demonstrated, a rigorously blind hiringprocess that selects from pools of identically qualified candidates will resultin departments that fail to reflect the diversity of the availability pool forlong periods of time. The long-term distribution for one department in ourmodel can also be viewed as a point-in-time snapshot of the compositions ofmany different departments. In particular, if we have a large collection of de-partments of size N, and if we randomly distribute minorities among thesedepartments such that the average number of minorities per department is Nf,


then the probability that any given department contains exactly i minorities isgiven by equation 2. As an example, if we have 200 departments of size 20,and if the minority fraction averages 10%, then at any given time we shouldexpect 12.16% (24 departments) to contain no minorities; 27.02% (54 depart-ments) to contain 1 minority; 28.52% (57 departments) to contain 2 minorities,and so on. As indicated earlier, more departments will have less than the averagenumber than will have more than the average. In this example, 39.17% willhave one or less; 28.52% will have exactly the average number of 2; and only32.31% will have 3 or more minorities. Once again, the intuitive expectationthat randomness should, somehow, lead to uniformity is incorrect.3 Most depart-ments, at most times, will struggle to reach the expected average, and theirstruggles will be of long duration. These are necessary consequences of anyminority-blind process.

With these findings in mind, we offer the following suggestions for depart-ments wishing to provide visible and convincing evidence of the fairness oftheir hiring processes. All of our suggestions are designed to enrich the avail-ability pool and/or reduce the otherwise inevitable fluctuations without introduc-ing any intrusive or manipulative elements of bias:

1. Recognize that the recruitment process is, in many respects, more importantthan the final hiring process, and take affirmative steps to assure a diversepool of candidates. Simply relying on the uncontrolled accumulation of re-sponses to advertisements of position vacancies will not guarantee either adiverse pool of candidates or a pool that truly contains “the best” potentiallyavailable candidates.

2. Recognize that the recruitment networks of the present faculty members arelikely to be richer in candidates who are “like themselves,” and seek assis-tance in tapping networks that reach candidate pools that are more diversethan the department itself. Early success in diversifying the faculty will natu-rally broaden and deepen the department’s own recruitment networks forfuture hires.

3. Define the vacant positions in ways that do not unnecessarily restrict the sizeof the potential availability pool. Smaller numbers are inevitably subject tolarger fluctuations, and work against both diversification and quality. Everyrestrictive qualifier should be discussed and debated before deciding that itis essential to the department’s needs.

4. Define the vacant positions in ways that attempt to assure reasonable unifor-mity of credentials among the candidates. Advertising at all professorial lev-els in a single search will generate a candidate pool that includes both inexpe-rienced recent graduates and experienced senior faculty. Comparisons of theirrecords of accomplishments will generally favor the more experienced candi-dates. During periods of demographic change, most of the experienced candi-


dates will be majority candidates, leading to an unreasonably small minorityfraction among the finalists.

5. Create your own availability pool. We have argued that a “true” value for fmust exist in principle, but that it would be very difficult to determine anumerical value for this hypothetical “true f.” Offices of affirmative actionuse various methodologies in attempting to estimate reasonable availabilitypools for purposes of judging compliance and setting goals. Advocacy groupspushing for greater diversity tend to use comparisons with large-scale demo-graphics, and often imply f values that are considered by the departments tobe unrealistically large. In the end, though, it is the recruitment and advertis-ing practices of a department that define some effective availability poolfrom which the department will hire; the f value of that pool has no predeter-mined relationship to any of the others. Departments should spend sometime studying the issue of availability, benchmarking with peer departments(particularly with departments they view as being leaders in the field) andsetting goals that can be achieved in realistic time scales. If, for example, adepartment uses the sort of affirmative recruitment suggested here to assurethat at least one or two highly competitive minority candidates are containedin each short-list of five, then they will be recruiting as if f = 0.2 or 0.4,independently of whatever passive value f may have assumed in the absenceof recruitment.

6. When a single search identifies more than one candidate who, in the depart-ment’s experience, is not only the best of the current candidates, but wouldhave been among the best in all recent searches, explore the possibility ofhiring all such candidates. Temporarily increasing the size of the departmentto take advantage of the unexpected opportunity to hire unusually good can-didates can speed the approach to equilibrium if the candidate pools arediverse.

These suggestions can reduce the time required to achieve a particular goalfor departmental diversity without simultaneously increasing the magnitudes ofthe fluctuations. This reduces the time-averaged magnitude of the fluctuationsso that departments are near their target goals more frequently during any periodof observation.

Most importantly, however, none of our suggestions implies either quotas orpreferential hiring criteria. They simply make use of known properties—includ-ing those described in this article—of search and hiring processes so as to makethe selection process as fair, in the broadest sense of the term, as possible.

Acknowledgments. The authors wish to express their gratitude to the following col-leagues for many helpful discussions and suggestions: Bruce Beck, Martha Casey, PhillipCertain, Bernice Durand, Loyal Durand, Brad Franklin, Margaret Harrigan, Olga Holtz,


Michael Iltis, Thomas Kurtz, Philip Miles, Charlene Tortorice, and Laura Wright. Wealso thank anonymous reviewers for many helpful comments and suggestions.

NOTES

1. We express gratitude to an anonymous reviewer for pointing out Hopkins’s prior use and publica-tion of this model, and for making us aware of the related work in Becker and Williams (1986)and Wise (1975).

2. We wish to thank Olga Holtz for proving that the eigenvalues of the canonical department transi-tion matrix T(N,f) are λ0 = 1, λ1 = 1 − 1/N, . . . , λN−1 = 1/N, λN = 0.

3. The reader who doubts our assertions about the failure of the human mind to develop accurateintuition about truly random processes is invited to try the following experiment. Draw a squarebox on a piece of paper or a chalkboard, and invite someone to place ten points randomly withinthe box. Inevitably, the first few points are placed quickly, in a seemingly random fashion, afterwhich the subject begins searching for large “holes” as potential sites for the remaining “random”placements. This is, of course, not a process of random placement. On the contrary, it is a processthat simulates some repulsive interaction that tends to maximize the distances between points,and leads to a more-or-less uniform coverage with no other obvious pattern. Our flawed intuitiontells us that if a region is already occupied by one point, then placing another point in closeproximity would confer some “special attractiveness” to that region and would, therefore, violatethe condition of randomness. The absence of any obvious pattern or regularity is a necessary,but not a sufficient, condition for randomness. Truly random placement will populate the spacewith many clusters of various sizes and many large empty regions, as can easily be confirmedby simulating this experiment using the random number generator and graphing features of aspreadsheet.

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