Divergence from uniform distribution (continuous): dispersion measure?

Question

I have data of a continuous random variable within the range [-1,1], which sometimes is concentrated around 0, while other times is concentrated toward -1 and 1, while zero is relatively underpopulated.

What measure can I use for both these cases to measure the divergence from a uniform distribution?

In other terms: I am looking for a measure of how evenly spread out the data is within the range, but standard dispersion measures (like variance) don't seem to work, since they favor distributions in which the tails are higher than the 'peak', e.g., when the region around zero is relatively underpopulated.

Answer 1

Expanding on @Glen_b 's comment:

KS works quite well here. In R, first create distributions like those mentioned

set.seed(123)
library(moments)
library(lattice)

UniDist <- runif(10000,-1,1)
NormDist <- rnorm(10000,0,1)
TruncNormDist <- NormDist[NormDist > -1 & NormDist < 1]
LowDist <- rnorm(5000,-.5,.1)
HighDist <- rnorm(5000,.5,.1)
TruncLowDist <- LowDist[LowDist < 0 & LowDist > -.5]
TruncHighDist <- HighDist[HighDist > 0 & HighDist < .5]
InvDist <- c(TruncLowDist, TruncHighDist)

Then test with ks.test:

ks.test(TruncNormDist, UniDist)
ks.test(InvDist, UniDist)

However, the D values are not intuitive (at least, not to me); ordinarily, I would suggest also using quantile plots:

qqplot(TruncNormDist, UniDist)

But that does not work well here (there is almost no divergence). Nor is a basic scatterplot much help:

xyplot(TruncNormDist~UniDist)

But Tukey mean difference plots are another matter

tmd(xyplot(TruncNormDist~UniDist))

Shows that the difference between the two distributions is near 0 at -1 and 1, but larger in the middle.

Comparing the inverse distribution to the truncated normal is much easier; both qqplot and xyplot show huge differences, and tmdplot is fascinating.