Can Average Test Scores Increase Without Students Scoring Any Higher?

You bet.

If you're in charge of the school district, you can increase average scores (and probably your salary) by simply shuffling students from one school to another. With absolutely no change in individual test scores, the averages will increase.

Illustration:

Let's say that there are two schools, one with low-scoring students, and another with high-scoring students. In fact, here are their grades:

School "A"
95
95
90
90
85
85
Average = 90.0

School "B"
90
85
85
80
75
65
Average = 80.0


What you need to do is make some morally superior platitude about how the School "B" students are suffering from segregation, underfunding, discrimination, etc., and then transfer the worst School "A" students to School "B".

In this example, let's transfer two School "A" students. The new distributions are:

School "A"
95
95
90
90
Average = 92.5, an increase of 2.5 points!

School "B"
90
85
85
85
85
80
75
65
Average = 81.25, an increase of 1.25 points!

Now you can report that the average scores in both schools have increased, and can look like the highly-respected public servant that you are.

This process has a name, the Will Rogers Phenomenon, and it has already been shown to reveal deceptive medical statistics -- specifically in cancer survival.

Must be careful with those numbers...