Sunday, January 27, 2013

Math Literacy: So Vital in Day-to-Day Business

A close relative to science literacy is math literacy - since good science often relies on doing the math right.

In the Richard Feynman collection of essays "Surely, You're Joking, Mr. Feynman!" is the chapter "Lucky Numbers" (the story is also available here "Feynman vs. The Abacus")

Feynman tells the story of his encounter with a man trying to sell abacuses at a restaurant and challenging customers to competitions of computation speed.  The waiters suggest the salesman challenge Feynman with his mathematical problems, which he does and Feynman accepts. The first problems are simple, and the salesman wins handily.  But as the problems become more complex, the salesman gets slower.  Eventually, Feynman wins on a more complex problem, computing the cube root of a number.  As Feynman explains - the salesman knew his tool (the abacus) but Feynman knew numbers.  A variant of this story appears in the Feynman biopic, "Infinity" (Wikipedia).

I've had a number of interesting experiences over my professional life which further illustrate the importance of math literacy, or math competence.

One of my early jobs was doing bookkeeping at a large tomato packing firm.  I was employed there my last years of high school and the job funded my first round of college education.  This was back in the day when small business computers were just coming on the scene and personal computers were looming on the horizon.  One of the company's accountants was trying to estimate the total amount paid by the company in social security wages (i.e. wages subject to social security tax).  The numbers readily available from the accounting system of the day were the total amount paid in wages,  (which included wages with and without social security tax deducted), the total amount paid in social security taxes, and the tax rate.

For those readers outside the U.S. or unfamiliar with the Social Security system in the United States (Wikipedia), Social Security taxes are deducted from paychecks at a fixed percentage, say SSrate, of a persons gross salary, Salary, up to some limiting amount, SalaryLimit, above which Social Security tax is not deducted.  Back in the 1970s,  employer contribution was equal to the employee contribution.

Now at the company I worked for, most of the employees paid Social Security taxes on their entire paycheck.   Only a few employees had annual pay that accumulated above the limit, SalaryLimit, to be exempt from the tax.  The accountant wanted to avoid computing the quantity of interest by adding up all the weekly employee paychecks for the year.

Now to me, the solution was obvious.  Since

SStax = SSrate * Salary

The total amount paid in Social Security is just the sum of SStax for each employee.  The tricky part comes in including employees who had part of their wages above the Social Security tax limit, SalaryLimit.

For anyone familiar with algebra, the obvious solution was that the total amount paid in SS wages was

 total amount paid in SS wages = (total paid in SS taxes)/(SSrate)

I suggested this to the accountant and he looked at me surprised. Had this obvious (to me) solution actually not occurred to him?  He ran his numbers through my suggestion and seemed satisfied.

Now this dollar amount computed this way will probably not be exact to the penny.  Because each individual tax computation is rounded to the nearest penny, adding them up for the total amount of the payroll propagates the rounding errors into the total.  Ideally, for an organization with a large number of employees with different pay rates, you round up about as many times as you round down, so the total should be close, but odds are it will not be exact.  This gets to Feynman's next point of his story - the power of approximation methods.

Why Does It Matter?
The point is that mathematical skills are valuable.  While many of the cranks whose science I discuss on this blog often try to dismiss the mathematics, it nonetheless can accurately describe how things work in the real world when the understanding of the fundamentals is correct.

In addition, scientific and mathematical literacy have applications far beyond the ability to do research.  As I demonstrate above, it has applications in day-to-day business.  It has applications in business budgets and inventory control, which requires us to understand flows of material and money to produce and distribute a successful product.  I have known a lot of people in businesses who fail not because they don't have a good product, or can't sell their product, but because they can't keep track of these fundamentals.

Exercise for Readers:
  • Assuming the accounting software did proper rounding, how would you calculate the largest possible error?  
  • How about the most probable error?

Additional Resources:
  • Wikipedia: Approximation
  • Enrico Fermi had a reputation for estimating quantities to determine if they were important or relevant to experiments or even everyday life (Wikipedia).  A collection of these type of "Fermi Problems" is maintained at the University of Maryland

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