Mathematical Proof of the Value of Diversity

I recently came across a novel mathematical equation explaining the value of diversity in problem solving. It was presented by Professor Ned Smith during one of my MBA classes at Kellogg. He attributed this proof mostly to Scott Page, a social scientist at the University of Michigan.

It’s important to highlight two critical points. First, diversity is objectively a good thing for several moral and ethical reasons. In other words, we should not have to prove its value. Our world is incredibly diverse and, thus, we should expect organizations to be comprised of individuals who are representative of this diversity. If not, the natural conclusion is there is some sort of bias or discrimination at play. Second, diversity is valueless without inclusion. As diversity advocate Vernā Myers put it, “Diversity is being invited to the party. Inclusion is being asked to dance.”

However, diversity also has incredible power when it comes to problem solving. It has long been accepted that a team comprised of individuals who have a diverse set of perspectives will generate more innovative solutions. This relates to socio-cognitive diversity and not demographic diversity. Demographic diversity does not necessarily result in socio-cognitive diversity (or vice versa), but a correlation between the two often exists.

While I was working at Cummins, this idea was presented in terms of a business case for diversity. Firms can leverage diversity as a competitive advantage through the innovation and creative problem solving that a diverse workforce generates. In short, on top of it being the right thing to do, firms should invest in diversity and inclusion because it produces a positive return on investment.

While I understand this concept and believe in it fundamentally, I always have felt the argument was a bit too qualitative or abstract. This is where the diversity equation comes in, as it objectively and quantifiably captures the business case for diversity.

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It helps if you think of a group trying to predict a solution to a theoretical problem where there is some finite correct answer (the “truth” or 𝜃).

Let’s use an example of a group guessing the number of M&Ms in a glass jar. 

The “crowd” (a random and, by extension, diverse group) submits a prediction. The difference between the prediction and the correct answer (truth) is the crowd error. The equation demonstrates that the crowd error is the difference between the average of how much each individual guess deviates from the truth (average error) and the average of how much each individual guess deviates from the crowd prediction (diversity). Thus, to minimize the crowd error (how much the group prediction deviates from the truth), we want maximize the difference in our guesses — diversity!

If you are having a hard time following the math (or just don’t believe it), I have attached a simple example in Excel. Using the same example as above — imagine 30 people trying to guess the number of M&Ms in a jar. The correct answer (truth or 𝜃) is 529. In the first simulation, I used a random number generation (0 to 1000) to simulate 30 individual guesses. In the second simulation, I simply replaced every fifth prediction with the same guess (in red) to artificially create less diversity. As the proof suggests, the crowd error increases in the second simulation — the one with less diversity.

While this proof and example revolve around a problem where there is some finite truth, the logic can easily be applied to other problem types. Increasing diversity does improve a team’s ability to solve a problem, especially when the solution is unknown or the search costs associated with identifying the individuals who can solve the problem are too high.