Alex's Adventures in Numberland - Alex Bellos [157]
Platykurtic and leptokurtic distributions.
The desire to see the bell curve in data is perhaps most strongly felt in education. The awarding of grades from A to E in end-of-year exams is based on where a pupil’s score falls on a bell curve to which the distribution of grades is expected to approximate. The curve is divided into sections, with A representing the top section, B the next section down, and so on. For the education system to run smoothly, it is important that the percentage of pupils getting grades A to E each year is comparable. If there are too many As, or too many Es, in one particular year the consequences – not enough, or too many, people on certain courses – would be a strain on resources. Exams are specifically designed in the hope that the distribution of results replicates the bell curve as much as possible – irrespective of whether or not this is an accurate reflection of real intelligence. (It might be as a whole, but is probably not in all cases.)
It has even been argued that the reverence some scientists have for the bell curve actively encourages sloppy practices. We saw from the quincunx that random errors are distributed normally. So, the more random errors we can introduce into measurement, the more likely it is that we will get a bell curve from the data – even if the phenomenon being measured is not normally distributed. When the normal distribution is found in a set of data, this could simply be because the measurements have been gathered too shambolically.
Which brings me back to my baguettes. Were their weights really normally distributed? Was the tail thin or fat? First, a recap. I weighen argued00 baguettes. The distribution of their weights was chapter 10. The graph showed some hopeful trends – there was a mean of somewhere around 400g, and a more or less symmetrical spread between 380 and 420g. If I had been as indefatigable as Henri Poincaré, I would have continued the experiment for a year and had 365 (give or take days of bakery closure) weights to compare. With more data, the distribution would have been clearer. Still, my smaller sample was enough to get an idea of the pattern forming. I used a trick, compressing my results by redrawing the graph with a scale that grouped baguette weights in bounds of 8g rather than 1g. This created the following graph:
When I first drew this out I felt relief, as it really looked like my baguette experiment was producing a bell curve. My facts appeared to be fitting the theory. A triumph for applied science! But when I looked closer, the graph wasn’t really like the bell curve at all. Yes, the weights were clustered around a mean, but the curve was clearly not symmetrical. The left side of the curve was not as steep as the right side. It was as if there was an invisible magnet stretching the curve a little to the left.
I could therefore conclude one of two things. Either the weights of Greggs’ baguettes were not normally distributed, or they were normally distributed but some bias had crept in to my experimentation process. I had an idea of what the bias might be. I had been storing the uneaten baguettes in my kitchen, and I decided to weigh one that was a few days old. To my surprise it was only 321g – significantly lower than the lowest weight I had measured. It dawned on me then that baguette weight was not fixed because bread gets lighter as it dries out. I bought another loaf and discovered that a baguette loses about 15g between 8 a.m. and noon.
It was now clear that my experiment was flawed. I had not taken into account the hour of the day when I took my measurements. It was almost certain that this variation was providing a bias to the distribution of weights. Most of the time I was the first person in the shop, and weighed my loaf at about 8.10 a.m., but sometimes I got up late. This random variable