Are the sample range and sample variance independent when population is normally distributed? If a population is normally distributed, the sample mean and sample variance are independent.
What about the sample range and sample variance? Are they independent too?
I am trying to derive Studentized Range Distribution,
and it might be simpler if it is... or not...
 A: Here's a sample of 10000 normal sample range-variance pairs for $n$=30.

Plainly, those aren't independent.
A: If the sample variance and sample range is calculated from the same sample, it's highly correlated. From the simulation I conducted, I realized that the sample size needs to be over 1000 so that the sample variance and the sample range is independent. 
Given that, it's quite interesting that even the range of sample mean of sample size 3 is almost independent from sample variance(spooled sample variance, the estimated population variance from different sample variances assuming each sample is sampled from the same population).
Here's the result from R simulation.
par(mfcol=c(2,3))

sim=1000 # number of simulations
g=3  # number of groups
n=2 # sample size of a group

x<-array(rnorm(sim*g*n), dim=c(g,sim,n))
x2<-array(rnorm(sim*g*n), dim=c(g,sim,n))


# a for sample sd for each grouop
# b for sample mean
if (n>=2) a<-apply(x, c(1,2), sd)
if (n==1) a<-apply(x,c(1,2), c)

if (n>=2) a2<-apply(x2, c(1,2), sd)
if (n==1) a2<-apply(x2,c(1,2), c)

b<-apply(x,c(1,2),mean)

if (n>=2) aa<-apply(a, 2, function(y) {sqrt((n-1)*sum(y^2)/((n-1)*g))})
if (n==1) aa<-apply(a, 2, sd)

if (n>=2) aa2<-apply(a2, 2, function(y) {sqrt((n-1)*sum(y^2)/((n-1)*g))})
if (n==1) aa2<-apply(a2, 2, sd)

if (g>=2) bb<-apply(b, 2, function(y) {range(y)[2]-range(y)[1]})
if (g==1) bb<-apply(b, 2, c)

r<-bb/aa*sqrt(n)
r2<-bb/aa2*sqrt(n)

hist(r, xlim=c(0,10), breaks="Scott", prob=T)

dx=0.1
alpha=sqrt(n)
if (n>=2) curve((ptukey((x+dx),g,g*(n-1))-ptukey(x,g,g*(n-1)))/(dx), xlim=c(0,10))
if (n==1) curve((ptukey((x+dx),g,g-1)-ptukey(x,g,g-1))/(dx*alpha), xlim=c(0,10))
if (n>=2) curve((ptukey((x+dx),g,g*(n-1))-ptukey(x,g,g*(n-1)))/(dx), xlim=c(0,10))
if (n==1) curve((ptukey((x+dx),g,g-1)-ptukey(x,g,g-1))/(dx), xlim=c(0,10))

hist(r2, xlim=c(0,10), breaks="Scott", prob=T)

plot(aa,bb)
plot(aa2,bb)

This is when the sample standard deviation and sample range is from the same sample with size 3

This is when the number of groups is 3, sample size for each group is 2,
Notice the range of sample mean and estimated population s.d is almost uncorrelated!

