We know that mathematics can solve problems in the classroom, but what can it do for your business? **Oscar Fernandez**, author of Everyday Calculus, takes a look at how knowledge of numbers can help your bottom line.

**Why You Should Be Learning ****Math Even If You Don’t Need It for Your Job**

I want to tell you a short story about epic triumph in the midst of adversity. Okay, I’m exaggerating a bit, but hear me out.

A couple of years ago, I approached Boston Scientific—an S&P 500 component—with a crazy idea: let me and a team of students from Wellesley College (a liberal arts college for women) and Babson College (a business school) do consulting work for you. It was a crazy idea because what could I—a mathematician who knew nothing about their business—and some students—who hadn’t even graduated yet—possibly offer the company? Plenty, it turns out, all thanks to our common expertise: mathematics.

Mathematics, often depicted in movies as something pocket-protector-carrying people with less than stellar social skills do, is actually quite ubiquitous. I’d even say that mathematicians are the unsung heroes of the world. Alright, that’s a bit of hyperbole. But think about it. Deep in the catacombs of just about every company, there are mathematicians. They work in low light conditions, hunched over pages of calculations stained with days-old coffee, and think up ways to save the company money, optimize their revenue streams, and make their products more desired. You may never notice their efforts, but you’ll surely notice their effects. That recent change in the cost of your flight? Yep, it was one of us trying to maximize revenue. The reason that UPS truck is now waking you up at 6 a.m.? One of us figured out that the minimum cost route passes through your street.

But we’re do-good people too. We help optimize bus routes to get children to school faster and safer. We’ve spent centuries modelling the spread of disease. More recently, we’ve even reduced crime by understanding how it spreads. That’s why I was confident that my team and I could do something useful for Boston Scientific. Simply put, we knew math.

We spent several weeks pouring over data the company gave us. We tried everything we could think of to raise their revenues from certain products. Collectively, we were trained in mathematics, economics, computer science, and psychology. But nothing worked. It seemed that we—and math—had failed.

Then, with about three weeks left, I chanced upon an article from the *MIT Technology Review* titled “Turning Math Into Cash.” It describes how IBM’s 200 mathematicians reconfigured their 40,000 salespeople over a period of two years and generated $1 *billion* in additional revenue. Wow. The mathematicians analyzed the company’s price-sales data using “high-quantile modeling” to predict the maximum amount each customer was willing to spend, and then compared that to the actual revenue generated by the sales teams. IBM then let these mathematicians shuffle around salespeople to help smaller teams reach the theoretical maximum budget of each customer. Genius, really.

I had never heard of quantile regression before, and neither had my students, but one thing math does well is to train you to make sense of things. So we did some digging. We ran across a common example of quantile modelling: food expenditure vs. household income. There’s clearly a relationship, and in 1857 researchers quantified the relationship for Belgian households. They produced this graph:

That red line is the linear regression line—the “best fit to the data.” It’s useful because the slope of the line predicts a 50 cent increase in food expenditure for a $1 increase in household income. But what if you want information about the food expenditure of the top 5% of households, or the bottom 20%? Linear regression can’t give you that information, but quantile regression can. Here’s what you get with quantile regression:

The red line is the linear regression line, but now we also have various quantile regression lines. To understand what they mean let’s focus on the top-most dashed line, which is the 95^{th} percentile line. Households above this line are in the 95^{th} percentile (or 0.95 quantile) of food expenditure. Similarly, households below the bottom-most line are in the 5^{th} percentile (or 0.05 quantile) of food expenditure. Now, if we graph the slopes of the lines as a function of the percentile (also called “quantile”), we get:

(The red line is the slope of the linear regression line; it doesn’t depend on the quantile, which is why it’s a straight line.) Notice that the 0.95 quantile (95^{th} percentile) slope is about 0.7, whereas the 0.05 quantile (5^{th} percentile) slope is about 0.35. This means that for every $1 increase in household income, this analysis predicts that households in the 95^{th} percentile of food expenditure will spend 70 cents more, whereas households in the 5^{th} percentile will spend only 35 cents more.

Clearly quantile regression is powerful stuff. So, my team and I went back and used quantile regression on the Boston Scientific data. We came up with theoretical maximum prices that customers could pay based on the region the product was sold in. As with IBM, we identified lots of potential areas for improvement. When my students presented their findings to Boston Scientific, the company took the work seriously and was very impressed with what a few students and one professor could do. I can’t say we generated $1 billion in new revenue for Boston Scientific, but what I can say is that we were able to make serious, credible recommendations, all because we understood mathematics. (And we were just a team of 5 working over a period of 12 weeks!)

April is Mathematics Awareness Month, and this year’s theme is “math drives careers.” After my Boston Scientific experience and after reading about IBM’s success, I now have a greater appreciation of this theme. Not only can mathematics be found in just about any career, but if you happen to be the one to find it (and use it), you could quickly be on the fast track to success. So in between celebrating March Madness, Easter, Earth Day, and April 15^{th} (I guess you’d only celebrate if you’re due a tax refund), make some time for math. It just might change your career.

Oscar Fernandez is the author of Everyday Calculus. He is assistant professor of mathematics at Wellesley College.