John Walkup

I recently reviewed the draft for the national common core mathematics standards.  Precision and accuracy form concepts that interest me on a regular basis, so the common core standards lit me up when I read:

Above all, examples of precision need to clarify the distinction between precision and accuracy.  In this view, the standards writers unwisely selected the U.S. Census for an example.

First of all, we know the measurement is not accurate to within a single count because the census failed to count many individuals that comprise part of the world population.  But this error is not inherent in any given instance of a measurement: When I see an individual, I mark him or her down in my log with infinite precision.  The grand sum off all data points (that is, when I add all the counts to arrive at the total population) remains infinitely precise. On this count, if the census measurements reflect raw counts rather than interpolations and extrapolations, then even the last digit is precise.

Now if we dive deeper we realize that census takers don’t just make head counts. They ask heads of households the number of people living in the home, which we could cast as an imprecise measurement if the responses reflected a non-systematic error. But the common core standards are only correct on a technicality: we cannot expect students to know this. A better example of imprecision in measurement would be the width of the universe expressed to the nearest inch, an obvious overestimation of the precision capabilities of even our most sophisticated equipment.

This misunderstanding between precision and accuracy is pervasive, but also damaging. Failing to report raw counts in their entirety implies the use of an ill-defined round-off procedure. If I count the number of cars in my city and arrive at 23,492, do I report the result as 23,000, 23,500, or 23,490?  How could one justify one number over the other?

So let’s revisit the population census mentioned in the common core standards.  What does the number 304,059,724 represent? Answer: A raw count of people encountered by the census takers. The fact that this measurement does not correlate to the term population (that is, the actual number of people living in the U.S.) is a matter of accuracy.  The number 304,059,724 itself is precise.

Speaking of interpolation and extrapolation, I saw nothing in the common core standards that addresses these two
topics on the conceptual level, but the fact that interpolating is typically safer than extrapolating is important to know in mathematical and scientific reasoning.