The Average Speed on the Highway

The following problem has appeared in elementary statistics books: You are driving on the highway and adjust your speed until the number of cars that you pass is equal to the number of cars that pass you. Is your speed the median speed or the mean speed of the cars on the highway? Students are expected to answer that since the number of cars going slower than you are is equal to the number of cars going faster than you are, your speed is the median speed. This certainly is true of the cars that you see, but that isn't what the problem asks, and it isn't the correct answer. The surprising results that we will derive illustrate the importance of careful and precise thinking when sampling. Here's an example. Imagine a highway with five lanes. In one lane, cars travel 20 miles per hour with two cars per mile. The other lanes have cars traveling 45 miles per hour with seven cars per mile, 55 miles per hour with two cars per mile, 65 miles per hour with three cars per mile, and 70 miles per hour with six cars per mile. The cars' median speed is 55 miles per hour since there are nine cars per mile going faster than 55 miles per hour and nine cars per mile going slower. When driving you cannot directly observe the number of cars moving at each speed. What will you see if you drive at the median speed of 55 mph? The number of cars going, say, 45 mph that you pass in an hour is equal to (55 ? 45) ? 7 = 70. The com? putations for all of the speeds are given below: