This post is about an elementary fact of probability applied to University policy. The policy challenge is what many universities face, namely how to bring the racial and ethnic composition of the faculty in line with the pool of qualified candidates. One policy idea that gets voiced repeatedly is “When you conduct a faculty search, just make sure you invite the top few URM applicants [1] for an interview.” This sounds reasonable at first blush, but under current California law such a policy is illegal. Here is a way to understand why:

Suppose for the sake of argument that the targeted identity group are redheads. Policy proposal (A) is “Make sure you interview the top few redheads”. Because academic merit has nothing to do with hair color, the average redhead is just as good as the average blond, right? Yes, but not so for the top-ranked redhead. If redheads make up 5% of the applicant pool [2] then on a merit-ranked list of all applicants the top-ranked redhead will typically be at position #20.

Now consider this policy (B): “First rank all the applicants by merit. Then pull out the redheads and improve their rank by a factor of 20; for example a redhead ranked #60 on the list gets promoted to #3. Then invite people for interviews from the top of the list.”

Most people will look at policy (B) and say “that’s ridiculous, you can’t justify that”. But (A) and (B) are (on average) the same policy. The proof – as probability books love to say – is left to the reader.

[1] For non-US readers: URM = under-represented minority.
[2] I’m pulling that number out of thin air.

8 thoughts on “Minority hiring and the geometric distribution”

Wow, thank you so much for your VERY meaningful contribution to this complicated topic. Turns out, it’s all just simple math! Wow!! 10/10 would read again

I sense a hint of sarcasm in this comment, but at the risk of belaboring the obvious: It *is* a complicated topic, and no, it’s not just simple math. The post merely points out that two policies that sound different are actually the same. They are not pulled out of thin air: Policy A is discussed frequently by people at my university whereas Policy B would be rejected out of hand by those same people. I don’t pretend that the issues surrounding minorities in academia boil down to a math problem, nor that I have a ready solution.

First of all, I sincerely apologize for my sarcasm. It’s not a productive way to have this conversation and I regret coming in hot like that. I appreciate your grace in moving beyond my tone to hear my message and have a dialogue. This is a challenging conversation and we all need to have patience with each other, so thank you for setting the tone in that respect.

That said, allow me to clarify my original message…

I think you’ve taken a very clear political position here. I think it’s a position that supports the racism that’s baked into the ivory tower. And I think you’re hiding behind math to abdicate your responsibility, as one who holds power in that institution, to do better. In your hyper-reductionist hypothetical scenario, you’ve excluded every variable that matters most and only included those variables that make the ongoing racial awfulness of academia seem reasonable. You are in a position of power and influence. You can do so much better than this. You could be working actively to increase access to people who’ve been historically locked out of higher education. Beyond this blog post, you might be doing that, I don’t know. But this blog post works in the opposite direction by gaslighting those who know for a fact that this issue goes so much deeper than the math you’ve laid out.

Why do you think there is a single, knowable, measurable axis along which candidates can be objectively ranked? Why do you ignore the fact that URM candidates with identical measurables consistently get ranked lower? Why do you assume that candidates are even evaluated in a way that can be described as objective? Why do you assume that a hiring committee has the ability to discriminate the quality of candidates with any level precision?

I request that you either delete this blog post or, even better, make a follow up in which your hypothetical scenario includes the possibility of any/all of the following: a multidimensional space of candidate ranking (e.g. an estimate of candidates’ ability to cope with and overcome adversity; an estimate of candidates’ ability to make students feel welcome; an estimate of how likely a candidate’s work is to meaningfully address an unmet need in society); bias on the part of the hiring committee; noise on the part of the hiring committee; an incentive for the hiring body to hire people with a range of diverse backgrounds; a threshold above which all candidates are considered “qualified” above a given ranking; … etc etc etc

Your blog post also harkens to a very common implicit idea not only academia but in any hyper-competitive space within the umbrella of western capitalism: “Our institution won’t be able to compete if we let the URMs in.” You will be hard pressed to convince me that you don’t believe that sentiment nor will you be able to easily convince me that that sentiment isn’t fundamental to your motivation to make this blog post to begin with. I do have an open mind and an open heart. But you’ve got a long row to hoe if you want to change my mind in this regard.

The ivory tower does not need to be protected from being overrun by “underqualified” URM outsiders. Quite the opposite: insofar as the culture of academia mimics the hyper-competitive capitalist culture of western society at large, it must seek healing (and progress) by making space for those who have traditionally been locked out. Or else it will fall victim to the same sickness afflicting the rest of our institutions.

Thanks for taking the time to consider these issues.

Markus, your point that humans interpret the same statistics differently, depending on how they are presented, is well-established in psychology, and important to keep in mind. It is a cognitive bias that yields cognitive errors and irrational behavior, just as our other cognitive biases do.

I personally do not approach the complex issue of diversity from a purely statistical perspective, but I agree that some statistical thinking is valuable, so it is worth developing the hair color argument a bit more…

If academic merit is unrelated to hair color, and 5% of the population is redhead, then the expected rank of the top redhead would be 10, rather than 20 (there is likely to be one redhead in the top twenty, with equal likelihood of ranking anywhere within that 20); nevertheless the probability of having at least one redhead in the top 6 faculty candidates would still be pretty low, 26% = 95% chance that each of the top 6 is non-redhead = 0.95 * 0.95 * 0.95 * 0.95 * 0.95 * 0.95.

That said, when considering the rubric to “invite the top few URM applicants” (as opposed to the top few redheads) one must include applicants from other underrepresented hair color groups, so let’s assume the following:
50% brunette
5% redhead, 5% blond, 5% black, 5% grey, 5% white, 5% bald, 5% blue, 5% purple, 5% green, 5% pink
Even with the assumption that half of the candidates with the most academic merit will be brunette, the probability that the 6 candidates with most academic merit are all brunette is exceedingly low = ½ * ½ * ½ * ½ * ½ * ½ = 1.6%. Said another way, there is a 98.4% chance that at least one of the top 6 candidates will be from an underrepresented hair color group (“URM”). There is an 89% chance that two or more of the top six candidates will be URM, and a 66% chance that three or more will be URM (1.6% chance of zero, 9.4% chance of exactly one, 23.4% chance of exactly two, 31.3% chance of exactly three, 23.4% chance of exactly four, 9.4% chance of exactly five, 1.6% chance of exactly six).

Given such statistics, if a process for evaluating merit repeatedly yielded all-brunette pools, I would start to suspect that the evaluation process was highly flawed. If, on average, only 3 of the top candidates should be brunette, but my evaluation process kept giving me 6 out of 6 brunettes, I would worry that 3 of those candidates are not truly in the top 6, but are erroneously being pulled from the #7-12 ranks. I would become highly distrustful of my evaluation process and consider a complete overhaul, as suggested in the comment by Josh Downer. And until a more optimal evaluation process for identifying merit was implemented, a reasonable rubric might be to take the top 3 brunettes, and the top 3 non-brunettes. To be sure, such a rubric would be imperfect, but it would seem more likely to identify more of the very top candidates.

Since we are actually talking about race rather than hair color, let’s also do the calculation assuming 60% of the candidates with the most academic merit will be brunette, since roughly 60% of Americans are white. The probability that 6 of 6 candidates will be from the majority group is 0.6*0.6*0.6*0.6*06*0.6 = 4.7%, a highly unlikely outcome. In science, Asians are usually not considered underrepresented, so we can recalculate based on 66% of the population being from the majority group (white or Asian), and there is still only an 8% chance that all 6 of 6 candidates should be from that majority group.

One might note that in a pool of applications for a faculty position, the percentage of applicants from underrepresented groups is often lower than one might expect from the population at large. This is to be expected if each stage of the lifelong academic evaluation process is more likely to identify merit in the majority group, and to overlook merit in other groups. The merit of any candidate who survives such a serial, skewed statistical process is likely to be truly exceptional (i.e, better, on average, than the majority candidates who make it to that final stage of selection), and hence worthy of careful consideration.

In my opinion, the above statistical arguments, as well as the extensive literature demonstrating bias based on race, skin color and other personal characteristics, should make us highly distrustful of any system for identifying and developing the very best academic merit that consistently fails to yield members of underrepresented groups in the top candidate pools.

Jennifer, I agree with the spirit of what you write. It’s a good exercise to keep track of the odds of various outcomes in the course of a faculty search, as you do in your calculations. It takes minimal effort and lets you check whether what happened is surprising or not, given the distribution of the applicant pool. Of course it’s not the only criterion by which to evaluate your search process.

One quibble with the numbers in your paragraph 3: The expected rank is 20, not 10 as you write. The rank of the top redhead follows the geometric distribution (see title of my post). For the top redhead to appear at position n, the sequence must start with n-1 non-redheads followed by 1 redhead. The probability of that happening is P(n)=(1-p)^(n-1)*p. The expectation value of that distribution is 1/p=20.

We’ve already had an email exchange about this topic, but I’ve been feeling the need to post a public comment.

I believe that, coming from your position of power and expertise, it is detrimental to our community to post this substantial oversimplification of affirmative action. By framing this problem as an ‘elementary fact of probability’ you mislead the reader to believe that affirmative action is unjustifiable and ridiculous. Your substitution of redheads for Black people (or other URMs) allows you to skirt important historical and contemporary context.

“The Supreme Court has consistently recognized…that an applicant’s race can be considered as one of an array of factors in assessing the entirety of a student’s application.” [1-3] For example, if two applicants are considered to be fairly equivalent, it is legal to consider representation at that point. This makes a lot of sense to me given the noisy process of evaluating applicants and the known biases that are in play. [4] I believe, given the known unequal distribution of resources and unequal treatment, we should assume that URMs in the top pool of applicants have greater underlying aptitude than other applicants.

We must face the reality that we live in a racist society and have therefore inherited biases that we cannot control. While we cannot control our inherited biases, we can control how we respond to them. I encourage you to engage with material on this subject. By recruiting and supporting URMs in our community, we contribute to a better future.

In light of these concerns, I ask you to please add a comment on your original post that this mathematical puzzle is a considerable oversimplification of affirmative action that does not provide any motivating historical or contemporary context.

Laura Driscoll and Lea Duncker

[1] https://admissionscase.harvard.edu/key-points
[2] University of California Regents v. Bakke, 438 U.S. 265, 98 S. Ct. 2733, 57 L. Ed. 2d 750 (1978).
[3] Grutter v. Bollinger, 539 U.S. 306, 123 S. Ct. 2325, 156 L. Ed. 2d 304 (2003).
[4] Eaton, A.A., Saunders, J.F., Jacobson, R.K. et al. How Gender and Race Stereotypes Impact the Advancement of Scholars in STEM: Professors’ Biased Evaluations of Physics and Biology Post-Doctoral Candidates. Sex Roles 82, 127–141 (2020). https://doi.org/10.1007/s11199-019-01052-w

My post contributed 4 paragraphs to a topic about which many thousands of paragraphs have been written. It should hardly be necessary to warn the reader that these 2 minutes of reading will not fully inform about all aspects of Affirmative Action. But just to be safe, I adopt your comment as that disclaimer.

The post focuses on one tiny facet of the problem with which I have some personal experience: How should a faculty search committee proceed? Yes, the idea that “if two applicants are considered to be fairly equivalent, it is legal to consider representation at that point” is reasonable. But numbers matter. Using the sample numbers in my post, if you adopt a policy of “invite the top 3 URMs” that is equivalent to stating “we consider the top 60 applicants as equivalently qualified for the position”. The faculty search panels that I have seen from the inside would find that claim ridiculous and would refuse to work if that notion were enforced from on high. For better or worse, we typically spend most of our time debating the relative merits in the top 10 list. Finding the top 60 can be done by the Human Resources department.

However, this will depend on the type of position being filled, and probably the academic discipline. You could ask faculty at your University about their experience: “After ranking the candidates by merit, how far down the rank list are you happy to completely scramble the order?” Perhaps Professor Raymond, who has commented here as well, can help us with some personal insights.

“I’m sometimes asked when will there be enough [women on the Supreme Court] And I say when there are nine, people are shocked. But there’d been nine men, and nobody’s ever raised a question about that.”

Wow, thank you so much for your VERY meaningful contribution to this complicated topic. Turns out, it’s all just simple math! Wow!! 10/10 would read again

LikeLike

I sense a hint of sarcasm in this comment, but at the risk of belaboring the obvious: It *is* a complicated topic, and no, it’s not just simple math. The post merely points out that two policies that sound different are actually the same. They are not pulled out of thin air: Policy A is discussed frequently by people at my university whereas Policy B would be rejected out of hand by those same people. I don’t pretend that the issues surrounding minorities in academia boil down to a math problem, nor that I have a ready solution.

LikeLike

First of all, I sincerely apologize for my sarcasm. It’s not a productive way to have this conversation and I regret coming in hot like that. I appreciate your grace in moving beyond my tone to hear my message and have a dialogue. This is a challenging conversation and we all need to have patience with each other, so thank you for setting the tone in that respect.

That said, allow me to clarify my original message…

I think you’ve taken a very clear political position here. I think it’s a position that supports the racism that’s baked into the ivory tower. And I think you’re hiding behind math to abdicate your responsibility, as one who holds power in that institution, to do better. In your hyper-reductionist hypothetical scenario, you’ve excluded every variable that matters most and only included those variables that make the ongoing racial awfulness of academia seem reasonable. You are in a position of power and influence. You can do so much better than this. You could be working actively to increase access to people who’ve been historically locked out of higher education. Beyond this blog post, you might be doing that, I don’t know. But this blog post works in the opposite direction by gaslighting those who know for a fact that this issue goes so much deeper than the math you’ve laid out.

Why do you think there is a single, knowable, measurable axis along which candidates can be objectively ranked? Why do you ignore the fact that URM candidates with identical measurables consistently get ranked lower? Why do you assume that candidates are even evaluated in a way that can be described as objective? Why do you assume that a hiring committee has the ability to discriminate the quality of candidates with any level precision?

I request that you either delete this blog post or, even better, make a follow up in which your hypothetical scenario includes the possibility of any/all of the following: a multidimensional space of candidate ranking (e.g. an estimate of candidates’ ability to cope with and overcome adversity; an estimate of candidates’ ability to make students feel welcome; an estimate of how likely a candidate’s work is to meaningfully address an unmet need in society); bias on the part of the hiring committee; noise on the part of the hiring committee; an incentive for the hiring body to hire people with a range of diverse backgrounds; a threshold above which all candidates are considered “qualified” above a given ranking; … etc etc etc

Your blog post also harkens to a very common implicit idea not only academia but in any hyper-competitive space within the umbrella of western capitalism: “Our institution won’t be able to compete if we let the URMs in.” You will be hard pressed to convince me that you don’t believe that sentiment nor will you be able to easily convince me that that sentiment isn’t fundamental to your motivation to make this blog post to begin with. I do have an open mind and an open heart. But you’ve got a long row to hoe if you want to change my mind in this regard.

The ivory tower does not need to be protected from being overrun by “underqualified” URM outsiders. Quite the opposite: insofar as the culture of academia mimics the hyper-competitive capitalist culture of western society at large, it must seek healing (and progress) by making space for those who have traditionally been locked out. Or else it will fall victim to the same sickness afflicting the rest of our institutions.

Thanks for taking the time to consider these issues.

LikeLike

Markus, your point that humans interpret the same statistics differently, depending on how they are presented, is well-established in psychology, and important to keep in mind. It is a cognitive bias that yields cognitive errors and irrational behavior, just as our other cognitive biases do.

I personally do not approach the complex issue of diversity from a purely statistical perspective, but I agree that some statistical thinking is valuable, so it is worth developing the hair color argument a bit more…

If academic merit is unrelated to hair color, and 5% of the population is redhead, then the expected rank of the top redhead would be 10, rather than 20 (there is likely to be one redhead in the top twenty, with equal likelihood of ranking anywhere within that 20); nevertheless the probability of having at least one redhead in the top 6 faculty candidates would still be pretty low, 26% = 95% chance that each of the top 6 is non-redhead = 0.95 * 0.95 * 0.95 * 0.95 * 0.95 * 0.95.

That said, when considering the rubric to “invite the top few URM applicants” (as opposed to the top few redheads) one must include applicants from other underrepresented hair color groups, so let’s assume the following:

50% brunette

5% redhead, 5% blond, 5% black, 5% grey, 5% white, 5% bald, 5% blue, 5% purple, 5% green, 5% pink

Even with the assumption that half of the candidates with the most academic merit will be brunette, the probability that the 6 candidates with most academic merit are all brunette is exceedingly low = ½ * ½ * ½ * ½ * ½ * ½ = 1.6%. Said another way, there is a 98.4% chance that at least one of the top 6 candidates will be from an underrepresented hair color group (“URM”). There is an 89% chance that two or more of the top six candidates will be URM, and a 66% chance that three or more will be URM (1.6% chance of zero, 9.4% chance of exactly one, 23.4% chance of exactly two, 31.3% chance of exactly three, 23.4% chance of exactly four, 9.4% chance of exactly five, 1.6% chance of exactly six).

Given such statistics, if a process for evaluating merit repeatedly yielded all-brunette pools, I would start to suspect that the evaluation process was highly flawed. If, on average, only 3 of the top candidates should be brunette, but my evaluation process kept giving me 6 out of 6 brunettes, I would worry that 3 of those candidates are not truly in the top 6, but are erroneously being pulled from the #7-12 ranks. I would become highly distrustful of my evaluation process and consider a complete overhaul, as suggested in the comment by Josh Downer. And until a more optimal evaluation process for identifying merit was implemented, a reasonable rubric might be to take the top 3 brunettes, and the top 3 non-brunettes. To be sure, such a rubric would be imperfect, but it would seem more likely to identify more of the very top candidates.

Since we are actually talking about race rather than hair color, let’s also do the calculation assuming 60% of the candidates with the most academic merit will be brunette, since roughly 60% of Americans are white. The probability that 6 of 6 candidates will be from the majority group is 0.6*0.6*0.6*0.6*06*0.6 = 4.7%, a highly unlikely outcome. In science, Asians are usually not considered underrepresented, so we can recalculate based on 66% of the population being from the majority group (white or Asian), and there is still only an 8% chance that all 6 of 6 candidates should be from that majority group.

One might note that in a pool of applications for a faculty position, the percentage of applicants from underrepresented groups is often lower than one might expect from the population at large. This is to be expected if each stage of the lifelong academic evaluation process is more likely to identify merit in the majority group, and to overlook merit in other groups. The merit of any candidate who survives such a serial, skewed statistical process is likely to be truly exceptional (i.e, better, on average, than the majority candidates who make it to that final stage of selection), and hence worthy of careful consideration.

In my opinion, the above statistical arguments, as well as the extensive literature demonstrating bias based on race, skin color and other personal characteristics, should make us highly distrustful of any system for identifying and developing the very best academic merit that consistently fails to yield members of underrepresented groups in the top candidate pools.

LikeLike

Jennifer, I agree with the spirit of what you write. It’s a good exercise to keep track of the odds of various outcomes in the course of a faculty search, as you do in your calculations. It takes minimal effort and lets you check whether what happened is surprising or not, given the distribution of the applicant pool. Of course it’s not the only criterion by which to evaluate your search process.

One quibble with the numbers in your paragraph 3: The expected rank is 20, not 10 as you write. The rank of the top redhead follows the geometric distribution (see title of my post). For the top redhead to appear at position n, the sequence must start with n-1 non-redheads followed by 1 redhead. The probability of that happening is P(n)=(1-p)^(n-1)*p. The expectation value of that distribution is 1/p=20.

LikeLike

Hi Markus,

We’ve already had an email exchange about this topic, but I’ve been feeling the need to post a public comment.

I believe that, coming from your position of power and expertise, it is detrimental to our community to post this substantial oversimplification of affirmative action. By framing this problem as an ‘elementary fact of probability’ you mislead the reader to believe that affirmative action is unjustifiable and ridiculous. Your substitution of redheads for Black people (or other URMs) allows you to skirt important historical and contemporary context.

“The Supreme Court has consistently recognized…that an applicant’s race can be considered as one of an array of factors in assessing the entirety of a student’s application.” [1-3] For example, if two applicants are considered to be fairly equivalent, it is legal to consider representation at that point. This makes a lot of sense to me given the noisy process of evaluating applicants and the known biases that are in play. [4] I believe, given the known unequal distribution of resources and unequal treatment, we should assume that URMs in the top pool of applicants have greater underlying aptitude than other applicants.

We must face the reality that we live in a racist society and have therefore inherited biases that we cannot control. While we cannot control our inherited biases, we can control how we respond to them. I encourage you to engage with material on this subject. By recruiting and supporting URMs in our community, we contribute to a better future.

In light of these concerns, I ask you to please add a comment on your original post that this mathematical puzzle is a considerable oversimplification of affirmative action that does not provide any motivating historical or contemporary context.

Laura Driscoll and Lea Duncker

[1] https://admissionscase.harvard.edu/key-points

[2] University of California Regents v. Bakke, 438 U.S. 265, 98 S. Ct. 2733, 57 L. Ed. 2d 750 (1978).

[3] Grutter v. Bollinger, 539 U.S. 306, 123 S. Ct. 2325, 156 L. Ed. 2d 304 (2003).

[4] Eaton, A.A., Saunders, J.F., Jacobson, R.K. et al. How Gender and Race Stereotypes Impact the Advancement of Scholars in STEM: Professors’ Biased Evaluations of Physics and Biology Post-Doctoral Candidates. Sex Roles 82, 127–141 (2020). https://doi.org/10.1007/s11199-019-01052-w

LikeLike

Hi Laura and Lea:

My post contributed 4 paragraphs to a topic about which many thousands of paragraphs have been written. It should hardly be necessary to warn the reader that these 2 minutes of reading will not fully inform about all aspects of Affirmative Action. But just to be safe, I adopt your comment as that disclaimer.

The post focuses on one tiny facet of the problem with which I have some personal experience: How should a faculty search committee proceed? Yes, the idea that “if two applicants are considered to be fairly equivalent, it is legal to consider representation at that point” is reasonable. But numbers matter. Using the sample numbers in my post, if you adopt a policy of “invite the top 3 URMs” that is equivalent to stating “we consider the top 60 applicants as equivalently qualified for the position”. The faculty search panels that I have seen from the inside would find that claim ridiculous and would refuse to work if that notion were enforced from on high. For better or worse, we typically spend most of our time debating the relative merits in the top 10 list. Finding the top 60 can be done by the Human Resources department.

However, this will depend on the type of position being filled, and probably the academic discipline. You could ask faculty at your University about their experience: “After ranking the candidates by merit, how far down the rank list are you happy to completely scramble the order?” Perhaps Professor Raymond, who has commented here as well, can help us with some personal insights.

LikeLike

Ruth Bader Ginsburg said it best:

“I’m sometimes asked when will there be enough [women on the Supreme Court] And I say when there are nine, people are shocked. But there’d been nine men, and nobody’s ever raised a question about that.”

LikeLike