The Role of Estimates and Expectations

A pop quiz question in my college physics class asked:

How many piano tuners are there in New York City? Show your work.

We had 5 minutes. This was pre-internet.

Not being from New York City, nor having ever played much less tuned a piano, I had little to go on. We didn’t have access to a phone book or any suitable reference.

We did have the power of logic though.

Just Guess

I could have simply guessed, say 123 piano tuners. The work to show was a bit scant, yet I could write down it was simply a guess.

Create Set of Possible Bounds

Given NYC is a big city, with about 5 million people (even this was a guess for someone from Northern Wisconsin), and it stood to reason that not everyone in the city was a piano tuner, therefore I could set the upper bound at 5 million.

For the lower bound, it would be possible, though not likely there were no tuners in the city, thus a lower bound of zero.

This didn’t help much in creating an estimate.

No Everyone has a Piano, nor Tunes it Regularly

Then I struck on the path of creating an estimate of the number of pianos that may require tuning. Let’s assume that 1 in 1,000 people have a piano, that suggest there are 5,000 pianos in the city.

Let’s assume each of these piano’s are tuned at least once a year on average. Some more and some never. That means there likely is enough tuners to tune each piano in the city once a year.

How long does it take to tune a piano? Here again I have no clue and suspect some jobs are quick while others may take all day or longer. Let’s say with travel, setup, and actual tuning it takes an average of 4 hours. That means in a year there is enough work to fill about 20,000 hours.

A full time job is 40 hours a week for 50 weeks, or 2,000 hours. Thus it would take 10 full time tuners to tune 5,000 pianos each year in NYC.

My answer is 10.

You’re estimate may differ.

Why Were We Asked This Question?

Upon reading the question I suspect all of my fellow students wondered if we were in the right classroom. What was the point of knowing the number of piano tuners in NYC?

Of course, the lesson that day went on to extoll the necessary step of guessing. Of creating a reasonable estimate before setting out to run an experiment or run the calculations.

The idea was the need to set an expectation of what the answer or results should be before looking for the results.

Preparing to Be Surprised

The idea of creating a quick, and reasonable, estimate is an acquired skill. It’s a bit of experience, a bit of logic, and a bit of guessing. The idea is to create a future state that permits you to compare the results obtained with the expected results.

If the results match the estimates, you have confirmed with data what you previously only guessed. If the result does not match, you have been surprised.

Being surprised leads to learning something new. To scientific discovery. To the ability to uncover previously unknown truths.

Follow Up on Being Surprised

There are a number of reasons your estimate is not matching the results.

I may have made one or more faulty assumptions. For example, what is it takes a full day to tune a piano and full time work only includes 4 piano tunings per week. Instead of 10 tuners, we would need more to cover the work.

With more than 5 minutes, and now with Google’s search engine, we can refine our many assumptions to improve our estimate.

Yet even with reasonable assumptions, we may not have considered all the salient factors that influence the result. For example, how about the number of part time tuners, or those dedicated to a specific property, like Carnegie Hall. Again including these considerations would have increased my estimate.

Also, when setting up an experiment or doing mathematical calculations, the difference may reveal a basic mistake in either the estimate or the calculation. During the class discussion, we talked about estimating the mass of an object. If the result of our measurements resulting in a negative number, we would have a difference from our initial guess worth exploring.

Finally, the difference may suggest something unexpected occurred. Something that may lead to a new material, novel solution, or fundamental principle.

By not being prepared to be surprised, we may view the result of the experiment or calculation as simply a result. We may miss the significance of the result (or the possibility of an error that tainted the result).

When you next sit down to design an experiment or grab your calculator (or spreadsheet) stop for a moment and set your expectations. Write it down. Then be prepared to be surprised.

PS: if anyone knows how many piano tuners there actually are in NYC I would be interested in using that number to compare to my estimate.