Why would you expect the mean of a set of numbers to be the reciprocal of the mean of reciprocals of those numbers? It seems pretty obvious that those two operations (taking the reciprocal of the numbers, taking the mean) can’t be reordered.
To take a trivial example, the mean of 1 and 2 is 3/2, but the mean of 1 and 1/2 is 3/4. It’s easy to see that 3/2 ≠ 4/3. Now let’s try something that pumps the difference up. The mean of 1/1000 and 1 is 1001/2000, about half. On the other hand, the mean of 1000 and 1 is 1001/2, about 500. Clearly 500 is not the reciprocal of 1/2.
In the case where we have the two sticks, the ratio of small to big is going to have to be somewhere between 0 and 1, because we’re taking the mean of a set of numbers between 0 and 1. On the other hand, in the ratio of big to small, we’re taking the mean of a set of numbers between 1 and ∞.
To take a trivial example, the mean of 1 and 2 is 3/2, but the mean of 1 and 1/2 is 3/4. It’s easy to see that 3/2 ≠ 4/3. Now let’s try something that pumps the difference up. The mean of 1/1000 and 1 is 1001/2000, about half. On the other hand, the mean of 1000 and 1 is 1001/2, about 500. Clearly 500 is not the reciprocal of 1/2.
In the case where we have the two sticks, the ratio of small to big is going to have to be somewhere between 0 and 1, because we’re taking the mean of a set of numbers between 0 and 1. On the other hand, in the ratio of big to small, we’re taking the mean of a set of numbers between 1 and ∞.