Go figure!

Back when I was in my final year of high school, I was given leave to attend a lecture at the mathematics department of the local university. The title of the lecture was “Figures never lie, but liars often figure”, and its precis was how statistics, whilst a powerful tool to understanding the world, are also a powerful tool to sow confusion, falsehoods and deceit.

The lecture room was the biggest they had at the University, and when full seated over 300 students. The day of the lecture it was indeed pretty full, with representatives from maths classes from every high school in the county.

After everyone had piled into the room and found a seat and settled down, the lecturer stood up at the lecturn to address the participants.

“I’d like to start by making a prediction. The next person who enters through that door”

he said, gesturing theatrically towards the room’s main Entry/Exit point

“will have more than the average number of legs”.

Having given this, slightly odd statement, a few moments to digest in the bowels of the audience, he continued with the lecture proper.

He hadn’t been going long when the main lecture door opened, and in sidled a late-comer to the lecture (as, I imagine, the lecturer anticipated). This chap started tiptoe-ing towards an empty seat a few rows up from the front, when he suddenly noticed that the speaker had stopped his lecture, and that everyone in the auditorium was staring at him intently. After a few moments a few people sniggered, then everyone started laughing. The guy, now totally confused and embarrassed, went a shade of red that was visible even under the subdued lighting of the theatre, then slowly backed back out towards the door, then turned and fled, not to be seen again.

TLDR; Note: the point of the lecturers assertion that the next person to walk through the room having more than the average number of legs was that, even for something as mundane as the notion of “average” needs careful evaluation. The most common use of average (mean) would put the average number of legs of a human as just below 2, something like 1.98, as whilst some people might have only one or no legs at all, the incidence of three legged (or more) people is pretty much unknown. As such, the lecturer could be pretty sure he’d win his bet using the arithmetic mean. A better measure of average in this case would be the one known as the “mode”. This represents the quantity which occurs the most frequently in a set of numbers. In the situation of most reasonably representing the “average” number of legs a person has, the mode would be 2.