  We have discussed the validity of the factorization approximation (FA) and
nonextensivity-induced correlation, by using the multivariate $q$-Gaussian
probability distribution function (PDF) for $N$-unit independent nonextensive
systems. The Tsallis entropy is shown to be expressed by $S_q^{(N)} =
S_{q,FA}^{(N)}+ \Delta S_q^{(N)}$ where $q$ denotes the entropic index,
$S_{q,FA}^{(N)}$ a contribution in the FA, and $\Delta S_q^{(N)}$ a correction
term. It is pointed out that the correction term of $\Delta S_q^{(N)}$ is
considerable for large $| q-1 |$ and/or large $N$ because the multivariate PDF
cannot be expressed by the factorized form which is assumed in the FA. This
implies that the pseudoadditivity of the Tsallis entropy, which is obtained
with PDFs in the FA, does not hold although it is commonly postulated in the
literatures. We have calculated correlations defined by $C_m= < (\delta x_i
\:\delta x_j)^m >_q -< (\delta x_i)^m >_q\: < (\delta x_j)^m >_q$ for $i \neq
j$, where $\delta x_i=x_i -< x_i >_q$ and $<\cdot >_q$ stands for $q$-average
over the escort PDF. It has been shown that $C_1$ expresses the intrinsic
correlation and that $C_m$ with $m \geq 2$ signifies correlation induced by
nonextensivity whose physical origin is elucidated within the superstatistics.
PDFs calculated for the classical ideal gas and harmonic oscillator are
compared with the $q$-Gaussian PDF. A discussion on the $q$-product PDF is
presented also.
