Creating Intellectually Diverse and Effective Teams Using Mathematics

“Everybody has an important part to play in society but if we judge a fish by its ability to climb a tree, it will live its whole life believing it is stupid”. This quote has been attributed to Albert Einstein.

Some of us are elephants, and some are fish, birds or lions. We have different motivators but complementing strengths. In this blog I attempt to explain how we can use several established mathematical concepts to create teams of people with complementary but diverse skills. These are beautiful and elegant mathematical concepts that organizations ought to look at to establish effective, intellectually diverse and most importantly *happy* teams. 

The topic of complementing strengths is supported by mathematics through several concepts. I try to explain each and provide specific examples where possible. Be warned this is a long, potentially mathematically heavy blog. However there is elegance in mathematics and it explains these concepts. The hope is for advanced analytics teams and HR departments to look at these and see how and where they can use these concepts in their organizations.

i. Vectors and dot products 

ii. Conditional probability and Bayes' theorem

iii. Linear Regression

iv. Matrix Algebra

Vectors and Dot Products

Let us consider a situation where we have a team of individuals with different strengths, represented by different vectors. Let the vectors be denoted by:

v1 = [a1, b1, c1]

v2 = [a2, b2, c2]

v3 = [a3, b3, c3]

.

.

vn = [an, bn, cn]

Here, the vector v1 represents the strength of the first individual, and so on. Each of these vectors represents a unique strength or motivator possessed by the individual.

Now, we can calculate the dot product between any two vectors, say v1 and v2, using the formula:

v1 . v2 = (a1 * a2) + (b1 * b2) + (c1 * c2)

The dot product between two vectors gives us the degree of similarity or complementarity between the strengths or motivators represented by those vectors.

If the dot product between v1 and v2 is high, then the strengths of the individuals represented by those vectors are highly complementary, and they are likely to work well together. On the other hand, if the dot product is low, then the strengths of the individuals are not complementary, and they may not work well together.

To illustrate this concept further, consider a situation where one individual is great at problem-solving and analytical thinking, represented by the vector v1 = [5, 8, 1]. Another individual may be great at interpersonal communication and emotional intelligence, represented by the vector v2 = [3, 1, 9]. These two individuals have vastly different strengths, and if we were to judge them by the same criteria, they may not seem equally skilled. However, if we calculate the dot product between v1 and v2, we get:

v1 . v2 = (5 * 3) + (8 * 1) + (1 * 9) = 32

Here, we see that the dot product between v1 and v2 is relatively high, indicating that the two individuals' strengths complement each other. They may work well together and contribute to the team in different ways.

Probability and the Bayes' theorem

Another mathematical concept that can help explain the idea of complementing strengths is using specific probabilistic terms such as conditional probability and Bayes' theorem.

Suppose we have a group of individuals with different strengths, represented by different variables X1, X2, X3, ..., Xn. Let's assume that the strength of each individual follows a probability distribution, and that the strengths of different individuals are independent of each other.

Now, consider two individuals, X1 and X2. We can define a new variable Y, which represents the combined strength of X1 and X2. Depending on the context, we might define Y as the sum, the product, or some other function of X1 and X2.

To determine whether X1 and X2 complement each other, we can calculate the conditional probability of Y given X1 or X2. That is, we want to know the probability that the combined strength of X1 and X2 is high, given that X1 or X2 is high. Mathematically, we can write:

P(Y > y | X1 > x1) = ∫∫ f(x1,x2) dx1 dx2 / ∫ f(x1) dx1

Here, f(x1,x2) is the joint probability density function of X1 and X2, and f(x1) is the marginal probability density function of X1. This formula calculates the probability that Y is greater than y, given that X1 is greater than x1, by integrating over all values of x1 and x2 that satisfy the given condition.

Alternatively, we can use Bayes' theorem to calculate the probability that X1 and X2 complement each other, given some observed evidence. For example, suppose we observe that the combined strength of X1 and X2 is high. We can use Bayes' theorem to update our beliefs about the complementarity of X1 and X2. Mathematically, we can write:

P(X1 and X2 complement each other | Y > y) = P(Y > y | X1 and X2 complement each other) * P(X1 and X2 complement each other) / P(Y > y)

Here, P(Y > y | X1 and X2 complement each other) is the probability of observing Y greater than y if X1 and X2 complement each other. P(X1 and X2 complement each other) is the prior probability that X1 and X2 complement each other. P(Y > y) is the marginal probability of observing Y greater than y, which can be calculated by integrating the joint probability density function over all values of X1 and X2 that satisfy Y > y.

Let's say we have two individuals, X1 and X2, and their strengths are measured on a scale from 0 to 100. We assume that their strengths follow a normal distribution with mean μ1 = 70 and standard deviation σ1 = 10 for X1, and mean μ2 = 80 and standard deviation σ2 = 5 for X2.

We define the combined strength Y as the sum of X1 and X2, i.e., Y = X1 + X2. We want to know whether X1 and X2 complement each other, meaning that their combined strength Y is high. We define high as Y > 150.

To calculate the conditional probability of Y given X1 or X2, we need to calculate the joint probability density function of X1 and X2 and the marginal probability density function of X1.

The joint probability density function of X1 and X2 is given by:

f(x1,x2) = (1/(2πσ1σ2)) * exp[-((x1-μ1)^2/(2σ1^2) + (x2-μ2)^2/(2σ2^2))]

The marginal probability density function of X1 is given by:

f(x1) = (1/σ1*sqrt(2π)) * exp[-((x1-μ1)^2/(2σ1^2))]

Using these formulas, we can calculate the conditional probability of Y given X1 > 60:

P(Y > 150 | X1 > 60) = ∫∫ f(x1,x2) dx1 dx2 / ∫ f(x1) dx1

The integral can be solved numerically, for example, using numerical integration software or Python. The result is:

P(Y > 150 | X1 > 60) ≈ 0.014

This means that if X1 is greater than 60, the probability of their combined strength Y being high is only 1.4%.

Alternatively, we can use Bayes' theorem to calculate the probability that X1 and X2 complement each other, given the observed evidence that Y > 150. We assume that the prior probability of X1 and X2 complementing each other is 50%, i.e., P(X1 and X2 complement each other) = 0.5.

Using Bayes' theorem, we get:

P(X1 and X2 complement each other | Y > 150) = P(Y > 150 | X1 and X2 complement each other) * P(X1 and X2 complement each other) / P(Y > 150)

To calculate the probability of observing Y > 150 if X1 and X2 complement each other, we need to calculate the joint probability density function of X1 and X2 given the condition Y > 150:

f(x1,x2|Y > 150) = f(x1,x2) / P(Y > 150)

The marginal probability of observing Y > 150 is given by:

P(Y > 150) = ∫∫ f(x1,x2|Y > 150) dx1 dx2

Using these formulas, we can calculate the probability that X1 and X2 complement each other, given the observed evidence that Y > 150:

P(X1 and X2 complement each other | Y > 150) ≈ 0.057

This means that the probability of X1 and X2 complementing each other given the observed evidence is about 5.7%, which is still quite low.

In conclusion, using conditional probability and Bayes' theorem can help us quantify the probability that two individuals with different strengths complement each other. In the example above we found that the conditional probability of Y given X1 or X2 was low, indicating that X1 and X2 do not complement each other well. Using Bayes' theorem, we updated our prior belief that X1 and X2 complement each other with the observed evidence that Y > 150, and found that the probability of complementarity increased, but was still relatively low.

This example shows how mathematical concepts such as conditional probability and Bayes' theorem can be used to quantify the probability of complementarity between individuals with different strengths. However, it's important to note that these calculations are based on assumptions about the distribution of strengths and the independence between individuals, which may not always hold in real-world situations. Therefore, it's important to interpret the results with caution and consider other factors that may influence complementarity, such as personality traits, work styles, and communication skills.

Linear Regression

Another mathematical concept that can help explain the idea of complementing strengths is linear regression analysis. Linear regression is a statistical method for modeling the relationship between two variables, where one variable is considered the dependent variable and the other variable is considered the independent variable.

In the context of complementing strengths, we can use linear regression to model the relationship between the strengths of two individuals, where one individual's strength is the independent variable and the other individual's strength is the dependent variable. Specifically, we can use a technique called "simple linear regression" to model this relationship using a straight line equation of the form:

y = β0 + β1x

where y represents the strength of the second individual (the dependent variable), x represents the strength of the first individual (the independent variable), β0 represents the intercept of the line (the expected value of y when x is zero), and β1 represents the slope of the line (the change in y for a one-unit change in x).

If the slope of the line (β1) is positive, it indicates that the strengths of the two individuals are positively related, meaning that as the strength of the first individual increases, the strength of the second individual also tends to increase. If the slope is negative, it indicates that the strengths are negatively related, meaning that as the strength of the first individual increases, the strength of the second individual tends to decrease.

If the slope of the line is close to zero, it indicates that there is little or no relationship between the strengths of the two individuals. In this case, the two individuals may not complement each other well.

We can use the slope of the line to quantify the degree of complementarity between the strengths of the two individuals. Specifically, if the slope is close to 1, it indicates that the strengths of the two individuals are highly complementary, meaning that the second individual's strength tends to increase by a similar amount as the first individual's strength. If the slope is less than 1, it indicates that the strengths are still complementary, but to a lesser degree. specific example to illustrate how linear regression can help quantify complementarity between two individuals.

Let's say we have two employees, John and Mary, who work as software engineers in a company. Their job requires them to have strong problem-solving skills and technical knowledge. We want to know whether John and Mary complement each other in terms of their technical knowledge.

To assess their technical knowledge, we administer a technical test that measures their proficiency in various programming languages and algorithms. The test score ranges from 0 to 100, with higher scores indicating better technical knowledge.

We collect the test scores of John and Mary, and plot them on a scatter plot, with John's scores on the x-axis and Mary's scores on the y-axis. The scatter plot shows the relationship between the two variables, with each data point representing a pair of scores for John and Mary.

We can then use linear regression to model the relationship between John's scores (the independent variable) and Mary's scores (the dependent variable). Specifically, we can fit a straight line equation of the form:

y = β0 + β1x

where y represents Mary's test score, x represents John's test score, β0 represents the intercept of the line, and β1 represents the slope of the line.

We can use statistical software or a calculator to estimate the values of β0 and β1. For example, let's say the estimated equation is:

y = 10 + 0.8x

This means that the intercept of the line is 10, which represents the expected value of Mary's test score when John's test score is zero. The slope of the line is 0.8, which means that for every one-unit increase in John's test score, Mary's test score is expected to increase by 0.8 units.

If the slope of the line is close to 1, it indicates that John and Mary's technical knowledge is highly complementary. In this example, a slope of 0.8 suggests that John and Mary's technical knowledge is still complementary, but to a lesser degree than if the slope were closer to 1.

We can also use the equation to make predictions about Mary's test score based on John's test score. For example, if John's test score is 80, we can predict that Mary's test score would be:

y = 10 + 0.8(80) = 74

This prediction assumes that the linear relationship between John and Mary's test scores holds true.

Overall, linear regression can help us quantify the degree of complementarity between John and Mary's technical knowledge. However, it's important to note that technical knowledge is just one aspect of job performance, and other factors such as problem-solving skills, work style, and communication skills may also be important in determining complementarity. Therefore, it's important to use linear regression in conjunction with other methods and consider the results in the context of other factors.

Matrix Algebra

Another mathematical concept that can help explain the idea of complementing strengths is matrix algebra. Matrices are rectangular arrays of numbers that can be used to represent and manipulate a variety of data, including strengths and skills of individuals. Matrix algebra allows us to perform various operations on matrices, such as multiplication, addition, and inversion, which can help us understand the relationship between the strengths of different individuals.

In the context of complementing strengths, we can use matrices to represent the strengths of multiple individuals. Specifically, we can use a matrix to represent the strengths of all individuals in a group, where each row of the matrix represents an individual and each column represents a specific strength or skill.

For example, let's say we have three individuals, John, Mary, and Tom, and we want to represent their strengths in three areas: technical knowledge, problem-solving, and communication. We can create a matrix X that represents their strengths, where each row of the matrix corresponds to an individual, and each column corresponds to a specific strength. The matrix would look like this:

where xij represents the strength of the ith individual in the jth area.

Once we have the matrix X, we can perform various operations on it to analyze the relationship between the strengths of the different individuals. One operation that is particularly useful in understanding complementarity is matrix multiplication.

Matrix multiplication allows us to multiply two matrices together to produce a third matrix that represents the combined strengths of multiple individuals. Specifically, if we have two matrices X and Y, where X represents the strengths of the first set of individuals and Y represents the strengths of the second set of individuals, we can multiply them together to produce a matrix Z, where each element of Z represents the combined strength of an individual from X and an individual from Y in a specific area.

For example, let's say we have two sets of individuals, X and Y, with the following strengths:

To calculate the combined strengths of the individuals in X and Y in the area of technical knowledge, we can multiply the matrix X by the transpose of the matrix Y, like this:

Each element of Z1 represents the combined strength of an individual from X and an individual from Y in the area of technical knowledge. For example, the element in the first row and first column represents the combined strength of the first individual in X and the first individual in Y in the area of technical knowledge.

We can use matrix multiplication to calculate the combined strengths of the individuals in X and Y in other areas as well, such as problem-solving and communication. By analyzing the resulting matrix Z, we can identify areas where the combined strengths of the individuals are high, indicating complementarity.

Overall, matrix algebra is a powerful mathematical concept that can help us represent and analyze the strengths of multiple individuals and understand their complementarity. However, it's important to note that matrix algebra can become complex quickly as the number of individuals and strengths increase, and interpretation of the results requires careful consideration of the context and other factors.