# PMean: A biased sample of car speeds

Dear Professor Mean, I read a newspaper report about speed limits and how few people obeyed them. A reporter decided to collect some hard data and drove exactly at the speed limit (55 mph in this particular setting). The reporter noticed that nine cars passed his car for every car that he passed, and concluded that most people are breaking the speed limit. I’m wondering if this is really a valid way to collect data.

Your worries are on target. To assess the appropriateness of any sampling procedure, you need to ask yourself “Who was left out?” In this sampling procedure,  the cars left out are the cars that neither passed you nor were passed by you, because they were travelling at exactly the same speed as you.

Suppose that 90% of the cars drive at 55mph and 9% drive faster than 55mph and 1% drive slower than 55mph, then 9 cars will pass you for every 1 that you pass. But 91% of the cars obey the speed limit. You never see most of them because if they are 1,000 feet in front of you or behind you when you start your data collection, those cars will remain 1,000 feet away when you are done with your data collection. Now, realistically, I don’t believe that 90% of the cars drive at exactly 55mph, but the data collection described above is clearly biased because it is unable to sample the fraction who travel exactly at 55mph and it undersamples those cars that drive only a tiny bit faster or slower than 55 mph.

What’s worse is that the cars that drive slower are very unlikely to drive a lot slower than 55 mph. It would suicide to drive more than 5 mph below the speed limit. But among those who drive faster than the speed limit, the sky’s the limit. Some may only drive 2 mph above the speed limit and some may drive only 5 mph above the speed limit, but there are still more than enough drivers out there who are 10 to 20 mph above the speed limit and these are the drivers who are most likely to appear in your sample.

So, it’s clearly a biased sample. It would be better to sit by the side of the road at 0 mph and point your radar gun at every car that passes by. There may still be some biases, especially if there is a relationship between the speed at which you drive and the frequency and duration of your trips. But if there are biases, they would be a lot less than the biases inherent in the reporter’s sampling method.