Statistics: Health service waiting lists

I have just been catching up on Radio 4's excellent programme More or Less: Behind the statistics and this week one of their listeners raised this curious point:

If a hospital procedure is carried out on everyone in exactly 10 weeks, what is the average waiting time? I was particularly taken with this piece because I have been very wary of "averages" for a long time and never accept them in a business context unless I have seen the raw data. The reason for this is the average, and even above average, person is clueless as to what an "average" really tells us, especially with small data sets. (Pun intended)

Anyway, the answer to this question took me by surprise: 5 weeks.

It seems that we are really being told average waiting times of those in the queue and they get this by asking each person in the queue how long they have been there. So if the sample is large enough it is reasonable to assume that some will have been there for 1 day, others for 2 days etc all the way to 69 days. The average of these wait times is indeed 35 days.

To be fair the programme does go on to say that the Government is moving towards total treatment times, but not before misleading us as to waiting waiting times.