How to measure how well a list of numbers is uniformly decreasing? I have a list of numbers that I want to measure how well are sorted/decrease. I want something more detailed than simply inversion, I want to know how uniformly they decrease.
For examples, the list (10,8,6,4,2,0) should have a higher score than (10,9,8,,3,2,1).
(10,8,6,4,2,0) and (10, 8, 6, 7, 4, 2) should both have a higher score than (10.5,10.4,10.3, 10.1, 8, 6).
In other words, I want to measure how well the numbers descend, and also I want to favour the magnitude of descending.
My idea is to assign each number an index, and then to measure the correlation. However, what kind of correlation would I want? Should it be Pearson since I am looking for a linear correlation. Or is there a better method?
More info:
In some tests I have done Pearson gave the list [10.91, 4.84, 4.75, 4.75, 4.397, 4.37, 3.85, 3.05, 3.05] a lower correlation than the list [14.71, 14.71, 14.71, 10.15, 10.15, 10.17, 10.22, 10.22, 10.22].
The first list is more desirable to me since the range is bigger - we can see the list is descending more strongly. Should I use spearmans rank instead? If so, why?
 A: Convert to differences, so (10, 8, 6, 4, 2, 0) = (-2, -2, -2, -2, -2) and (10, 8, 6, 7, 4, 2) = (-2, -2, 1, -3, -2). Find mean, variance, and standard deviation, and relative standard deviation (aka coefficient of variation) for each list. Lower relative standard deviation means more uniform descent, higher relative standard deviation means less uniform descent.
If you want to also penalize lists having individually smaller decreases, construct a composite score of these two important aspects. Use mean difference of each list as a weight in the final score, scaling it however you need, e.g., final score = mean difference - 2 * relative standard deviation, in which case score of (10,8,6,4,2,0) = 2, (10,8,6,7,4,2) = -1.11, (10.5,10.4,10.3,10.1,8,6)= -0.98, and as desired (10,8,6,4,2,0) is better than (10,8,6,7,4,2), both of which are better than (10.5,10.4,10.3,10.1,8,6).
A: As @C8H10N4O2 noticed, you are interested in differences between consecutive numbers. You want the differences to be small and you want them to be "uniform", not to vary too much. It seems like you want to measure two things. The first thing can be easily measured by looking at the mean of the differences. The second thing, by looking at the standard deviation of the differences.
> x1 <- c(10,8,6,4,2,0)
> x2 <- c(10,9,8,3,2,1)
> x3 <- c(10, 8, 6, 7, 4, 2)
> x4 <- c(10.5,10.4,10.3, 10.1, 8, 6)
> s <- function (x) c(mean=mean(diff(x)), sd=sd(diff(x)))
> s(x1)
mean   sd 
  -2    0 
> s(x2)
     mean        sd 
-1.800000  1.788854 
> s(x3)
     mean        sd 
-1.600000  1.516575 
> s(x4)
    mean       sd 
-0.90000  1.05119 

Of course, you can combine the two statistics, for example, use something as coefficient of variation or calculate the weighted sum of them. The question to ask yourself is how important are both criteria? Would you apply the same weights to both? Even if you choose some statistic that measures both things at the same time, it likely will penalize one of them stronger than another, so you need to ask yourself this question.
