The space shuttle has a very low aspect ratio, a design which makes it a very poor glider. What is aspect ratio? And what does it have to do with statistics and the law of averages?
Both the delta wing design of the shuttle and the low aspect ratio move the wings as far back as practical, so that the leading edge of the wing is not in the shock wave formed by the nose of the aircraft as it travels at supersonic speeds. This geometric information about the wings is summarized by aspect ratio.
With that in mind, here are 3 ways to summarize information:
1. Aspect ratio summarizes two dimensional, geometric characteristics of a wing. For a given aspect ratio, there are an infinite number of ways to design an airplane’s wing, so some information is lost when presenting this summary.
2. Statistics are also summaries of specific characteristics of data sets. And similarly, some information is lost by summarizing data in a statistic.
3. An aircraft photo captures a single moment in space and time. Just like the two statements above, a photo is an attempt to describe a highly complex and dynamic phenomena graphically. In the photo above, we have not even done a good job illustrating aspect ratio.
We can assess, at least qualitatively, the aspect ratio of the space shuttle with a very brief glimpse at the photograph, and furthermore, we can make this assessment from almost any camera angle. In that sense, we have an incredible ability to interpret data graphically.
We think that we understand statistics. We think that we know the inherent strengths and weaknesses of a particular statistic, like the average. But sometimes our intuition fails us…
If I had told you the aspect ratio of this aircraft, would you have envisioned it? (For more information about this aircraft, come back this next Wednesday for the Airplanes by Design column.)
I think, we understand the average of particular set of data, but we do not understand the relationship of the average to any set of data. Let me say it another way.
We can look at a set of data and understand how the average summarizes the information contained therein.
But we cannot, at least not effectively and consistently, look at an average and understand the uncountable infinitude of information that it represents.
The moral of the story is this: don’t make the flaw of averages. The average does not accurately represent every collection of data.
How do we find our way then, when we are exploring the unknown, blazing a trail into uncharted territory? How do we apply elementary statistical principles to transform uncertainty into decisive action? What is to prevent us from making a preposterous application of ATOMs when we deal with very complex situations, those in which our intuition fails?
These questions are not much different from those faced by Chuck Yeager before he ever broke the sound barrier or Neil Armstrong as he took that first step on the moon. Neither of these men, nor anyone around them–with hundreds or thousands of highly educated, very scientific people on these teams–knew what to expect. Or did they…?
ATOMs is a monthly column that introduces analytical tools of mathematics and statistics and illustrates their application. To read more about ATOMs, you can read Where Do We Go From Here, or view the online workbook here.