# A mathematical proof that almost all drivers speed almost all the time.

Until recently when I started paying more attention to my surroundings from the left side of the car, I assumed naively that most drivers obeyed speed limits.

The title is a little contentious since there are some restrictions. Obviously cars can’t speed when other cars or traffic lights get in the way. However, when there are no restrictions on speed other than the speed limit itself, it seems most drivers speed most of the time.

The proof involves sampling from a population to estimate the proportion of cars that are speeding. At each point in time each car on the road is either speeding or not speeding. I have assumed that the proportion of cars which are speeding does not change over time. This is a necessary assumption as the sample is collected at different points in time so we want to assume we are still sampling from the same population.

The sampling itself is fairly straight forward. Just jump in the car and drive (and count). Drive just above the speed limit on a road where there are no traffic lights and traffic conditions do not impede driver’s speeds. Count the number of cars you pass (non-speeders) and the number of cars that pass you (non-non-speeders). Collect enough sample points as per standard statistical theory and use the results to estimate the proportion of drivers who speed.

I have done the experiment and the result is summarised by the title of the story. It’s clear now why my dad used to say that most drivers consider the speed limit as a minimum rather than a maximum.

One of the strange characteristics of the driver population is that individual cars can change their state from one time to another by either slowing down or speeding up. This is different from the usual scenario where for example the proportion of defectives on a production line is estimated. An item that is defective stays defective. The car is like Schrödinger’s cat in the famous thought experiment.

As an addendum, I note that The Age newspaper recently (2 June 2017) published an article bearing out the above observation using a survey of drivers. See the story Motorists behaving badly

Motorists behaving badly … and they know it Speeding, texting while driving and answering hand-held mobile phones. These dangerous driving practices are rife on Victorian roads.