Averages - how does this happen?

#1
I’d appreciate some help please with a research finding. I’ve surveyed a school where there are 48 teachers. Within the school there are 6 different departments and teachers can work across More than one department. Some indicated working for up to 4 departments but all only scored their career satisfaction once. I asked all teachers to score their satisfaction with their career, and an average of 4.15 out of 5 emerged. However, when I cross tab this career satisfaction score by department, the average for EACH of the 6 departments is slightly LESS than this. Can some please explain how this happens as I’ve never seen this before and I need to explain it to the school. THANKS VERY MUCH
 

Dason

Ambassador to the humans
#3
Do some of the lowest scoring individuals work for multiple departments. They could be bringing multiple departments averages down but they still only contribute a single value to the overall mean.

For example imagine there are four departments and one person that works for all four with a score of 1 and then let's say there are one other staff in each department each with a score of 5. So there are find staff total with an overall mean of (1+5+5+5+5)/5 = 4.2. In each department there are two scores - a 1 and a 5 so the average is 3.

So every department has a mean lower than the overall average because there is one person that isn't happy but they are represented in all the departments.

So it might be that satisfaction and the number of departments a person is counted in are negatively correlated? It's just a conjecture of course but it would be a plausible explanation.