For normal data the sample mean A and standard deviation S are
stochastically independent, even though not functionally independent
(because A appears in the formula for S). Accordingly, if one takes a
large number m of samples of size n, and plots the m pairs (A, S),
there is no hint of association and the sample correlation of A and S
is very near 0. This independence of A and S holds only for normal
data.
One of my favorite examples of zero correlation *with* association
is to do the same procedure as above, but with data from the bathtub
shaped distribution BETA(.3, .3). The symmetry implies 0 correlation,
so one wonders how the inevitable association of A and S will
manifest itself. Here is a plot of S against A for m = 50,0000 and
n = 5. In effect, the transformation from 5-D data space to (A, S)
has 'squashed' the 5-D unit hypercube into 2 dimensions. Because
BETA(.3, .3) places much probability near faces, edges, and corners
of the hypercube, vestiges of them are visible in the plot of S vs A.
The sample correlation of A and S for the picture shown is
r = -0.0002795444.
R code:
m = 50000; n = 5; x = rbeta(m*n, .3, .3)
DTA = matrix(x, nrow=m)
a = rowMeans(DTA)
s = apply(DTA, 1, sd)
plot(a, s, pch="."); cor(a,s)
------------------------------
Bruce Trumbo
Original Message:
Sent: 08-03-2016 06:14
From: Sergio Yanez-Canal
Subject: Example zero correlation implies independence
Please tell me an example different that the bivariate normal case
Sergio Yáñez Canal
Profesor
Universidad Nacional
Medellín