I need to develop a grading system for a course. The first assignment is evaluated based on only one criteria, a. The second assignment based on the first criteria, a, plus one new criteria, b, the third assignment based on three criteria, a, b, and c, etc. The 16th assignment is graded based on 16th criteria, a through p.
Assignment 1: a
Assignment 2: a+b
Assignment 3: a+b+c
…
I initially devised this distribution:
Assignment 1: 1
Assignment 2: 1/2+1/2
Assignment 3: 1/3+1/3+1/3
…
Assignment 16: 1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16, 1/16+1/16+1/16
This does not suit the problem well though, because the first criteria introduced are the most important, so the earlier criteria should always take more weight than later criteria. I need a system where a is always greater than b and b is always greater than c, meanwhile, making sure that p never gets to be too much smaller than ~1/50 or so.
Assignment 16: a > b > c > d > e > f > g > h > i > j > k > l > m > n > o > p
How can I distribute the value of each criteria for the assignments?
Do you really need fractions? You can make them an arithmetic progression. For assignment $k$, the average weight is $\frac 1k$. Let the difference between terms be $d$. The weights range from $\frac 1k+\frac{(k-1)d}2$ to $\frac 1k-\frac {(k-1)d}2$. So if $k=15, d=0.005, \frac 1k \approx 0.066667$ and the weights range from $0.1016667$ down to $0.0316667$. Now you just have to pick a $d$ for each $k$ that makes you happy. Then you can convert these back to fractions if you want.