The nose wheel on my plane has 8 numbers on it, and being degenerate gamblers we bet on which number will end up on the bottom each time we land. Also, being nerds, we keep track of the results. I now have 67 data points from various landings.

The results in sequence: 7,3,6,5,5,6,3,8,6,1,5,6,3,8,4,8,5,3,2,1,3,3,1,4,2,8,7,6,7,4,3,6,5,6,3,2,8,7,3,

8,3,2,5,7,7,2,3,2,3,8,4,5,1,5,4,2,2,1,5,3,1,1,5,3,4,3,5

The numbers are written on the wheel in chalk and are periodically redrawn. The chalk stays on the wheel for a long time, but any rain washes the numbers pretty much completely off. This data has been gathered over several years, and many rechalkings of the wheel. If there is bias in a particular chalking (i.e., if they didn’t get the wheel exactly divided into 8ths) we are presuming that over time (over many chalkings) this gets randomized and/or is not statistically significant. In any case the chi-square test will help ferret out such bias, if any.

Sometimes the wheel lands “too close to call” (right on a line) and we call that a push. That happened 7 times out of 74 landings; on the assumption that the pushes are random events like everything else I have simply taken them out of this data for this analysis (leaving the 67 data points shown).

Now the interesting (to geeks) question is this one: are these data points consistent with the notion that the wheel is random?

There are a variety of ways to test for this. Today we start with the chi-square test.

Our Null Hypothesis: the wheel is random and each number should show up N/8 = 8.375 times.

The actual counts are:

1: 7

2: 8

3: 15

4: 6

5: 11

6: 7

7: 6

8: 7

The number 5 is heavy, 3 even more, and the others are light. Is this random variation or is there some systematic reason 3 is coming up? Let’s see what chi-square says.

To compute chi-square, we take the sum of ((O-E)**2 / E) … “O” being the observed count, “E” being expected (and “**2” meaning “squared”). This calculation comes out to 8.10.

From that, and the seven degrees of freedom (with eight possible values there are N-1 = 7 degrees of freedom), we can look up “p” which is, roughly, the probability that these variations came from randomness rather than some statistically significant factor. There are also online calculators for this, such as http://www.graphpad.com/quickcalcs/chisquared2.cfm

The calculated “p” value for this data is 0.32. When you hear statisticians say “blah blah blah is not statistically significant” they are usually talking about a chi-square test, and the threshold for significance is having a “p” value of 0.05 or less. Larger than that and we say that it is reasonably likely that the variations come from random factors rather than some specific cause. Our “p” here is quite a bit above 0.05, so the data is consistent with the wheel being random.

Stated the other way around, while the numbers obviously vary from the exact 8.375 count for each value, the amount of the variation is well within the boundaries one would expect from normal randomness (just as when you are in Vegas and see the roulette wheel come up red four times in a row on occasion – it doesn’t mean the wheel is rigged, it’s just one of those streaky things that happens from time to time).

Next scheduled trip is back to Canada in February (I pick the best spots to go in the winter, don’t I?) … action on the wheel can be booked by simply sending me a message – pick your numbers and your amounts!

Neil,

Great post!! I don’t suspect you recall meeting me circa Fall 2006 when I was an aviation advisor and finance specialist with JPMorgan. Anyways, I still recall a few great aviation stories you shared back then, and really enjoyed this post. Best wishes for continued safe, fun travels and please advise if I can ever assist with aviation advisory and/or jet financing. Best, Joe