An Average Post…(harmonic, that is)
Sometimes I sits and thinks…and sometimes I just sits…
A.A.Milne
Sometimes I sits and thinks…and sometimes I sits and thinks about thinking…
Buford Lister, personal communication
I remember reading somewhere a long time ago, that what made humans unique in the animal world is that we can think about thinking. Not only do we think, but we can take the process to the next level and analyze the thinking itself. This is simply a guess, but I don’t believe any of the pets I have had during my life were capable of such a feat. Actually, I am not sure that most…well, you get the idea.
Over the decades, I have thought a lot about how thinking about thinking works. Back in the 80s, I tried, again and again, to sneak into a great course at Harvard. It was called Thinking about Thinking, and it was taught by Alan Dershowitz, the famous lawyer, Robert Noczik, one of the leading philosophers from the last century, and Stephen Jay Gould, the evolutionary biologist who has influenced me in more ways than I would care to admit. There you have it, three superstar professors from different departments coming together to dazzle a bunch of impressionable minds. The course was designed to show the students how different ways of thinking lead to different approaches to how we view the world and our place in it. At least, that is how I looked at it.
That course was highly popular, and it was near impossible to get anywhere near the classroom. When I was there, the course was taught in the Science Center, a building that resembles an old Polariod camera on its side. There were guards checking registration slips at each entrance. If you were a registered student and happened to forget your slip, you were out of luck, you missed that day’s class. I often thought of trying to crawl through the heating ducts to get in there. After a few minutes, I thought about my thinking and reconsidered, I thought better not to try it.
One day, I was sitting in the small cafe right inside the main entrance to the Science Center. Who do you think sat down near me? It was the three professors. I had no choice; I had to eavesdrop on their conversation. I fell into a bit of a pattern, I made sure I arrived at the cafe around the same time on the days when the class was meeting. I got to hear lots of conversations. I can only recall one topic, it was the same topic the three of them talked about every week. They talked about baseball, baseball, and then more baseball.
Why all this stuff about thinking about thinking, also known as meta-thinking? Easy, we are going to talk about how to take averages, really simple averages. How about this one: if Sally has 40 apples and Billy has 20 apples, what is the average number of apples that the kids have? If I am writing a post about such a thing, you should immediately start doing some meta-thinking, right? If it really was so straightforward, why would I be writing about it? And that is a very good point, and of course, it is true. I wouldn’t be writing about averages if I didn’t have something a little unusual and surprising to say about them.
The average of a and b are calculated in this familiar way:
So, if Sally has 40 apples and Billy has 20 apples, the average number of apples = 30. No problem. The answer is simple and straightforward. Now consider this:
Joe’s car gets 40 miles per gallon, and Steve’s gets 30 miles per gallon. What are the average miles per gallon of the two vehicles?
And, no surprise, that answer is wrong. Why? Let’s suppose that both Joe and Steve drive for 120 miles. Joe would use 3 gallons of gas, and Steve would use 4 gallons. Now, we can add everything up. A total of 7 gallons of gas was used to travel 240 miles. Therefore, the average is 34.28, and that is the correct answer. 34.28 is the harmonic mean or harmonic average, it is quite different from the simple averages we are used to calculating.
This is the equation for harmonic averages:
That equation can be reduced to the following:
The important thing to think about is that you want to get a common denominator, not a common numerator. That creates a lot of confusion when it comes to computing harmonic averages.
So, where do we now stand? We all now know that some averages are more simple to compute than others. As always, the trick is to know when to use a harmonic average instead of a simple one. I will be posting more about this topic in the future. Simple averages and harmonic averages are not alone in their “average” universe, check back in and you will see what I “mean.”
