Which series is growing more?

Question

I am a c# .NET programmer with a math question

I have a list of numbers. Imagine they were test-scores of a set of students in different schools.

Is there a formula to will tell me which of these students is improving the most, the "most improved student"? I'd like to rank the students best to worst.

 John: 45, 65, 67, 65, 56, 80
 Sall: 78, 88, 89, 89, 90, 78, 88, 89
 Samy: 56, 78, 89, 87, 87

Answer 1

A simple and efficient way is computing for each student the average of the successive "improvements" to get an idea of how much the score of the student has improved so far.

Formally, if $\langle x_0,\dots,x_{n-1} \rangle$ are the scores of the student, you compute: $$\frac{1}{n-1}\sum_{i = 1}^{n-1} (x_i - x_{i-1}).$$ (Note that, of course, you need at least two scores).

From your example you would get:

John: ((65-45) + (67-65) + (65-67) + (56-65) + (80-56)) / 5 = 7
Sall: ((88-78) + (89-88) + (89-89) + (90-89) + (78-90) + (88-78) + (89-88)) / 7 = 1.58
Samy: ((78-56) + (89-78) + (87-89) + (87-87)) / 4 = 7.75

So Sall has not improved much (the scores are high but quite similar), whereas Samy and John have improved much more.