  We investigate spacing statistics $p(s)$ and distribution of eigenvalues
$D(\epsilon)$ for ensembles of various real random matrices (of order $n \times
n, n=2$ and $n>>2$) where the matrix-elements have various Probability
Distribution Function (PDF: $f(x)$) including Gaussian. We construct ensembles
of $1000$, $100 \times 100$ real random matrices $R$, $C$ (cyclic) and $T$
(tridiagonal) and real symmetric matrices: ${\cal R}'$, ${\cal R}=R+R^t$,
${\cal Q}=RR^t$, ${\cal C}$ (cyclic), ${\cal T}$ (tridiagonal), $T'$
(pseudo-symmetric Tridiagonal), $\Theta$ (Toeplitz) , ${\cal D}=CC^t$ and
${\cal S}=TT^t$. We find that the spacing distribution of the adjacent levels
of matrices ${\cal R}$ and ${\cal R}'$ under any symmetric PDF of matrix
elements is $p_{AB}(s)=A s e^{-Bs^2}$ which approximately conforms to the
Wigner surmise as $A/2 \approx B \approx \pi/4$. But under asymmetric PDFs we
observe $A/2 \approx B >>\pi/4$, where $A,B$ are also sensitive to the choice
of the matrix and the PDF. More interestingly, the real symmetric matrices
${\cal C}, {\cal T}, {\cal Q}$, $\Theta$ (excepting ${\cal D}$ and ${\cal S}$)
and $T'$ (pseudo-symmetric tridiagonal) all conform to the Poisson distribution
$p_{\mu}(s) =\mu e^{-\mu s}$, where $\mu$ depends upon the choice of the matrix
and PDF. Let complex eigenvalues of $R$, $C$ and $T$ be $E^c_n$. We show that
all $p(s)$ arising due to $\Re(E^c_n)$, $\Im(E^c_n)$ and $|E^c_n|$ of $R$, $C$
and $T$ are also of Poisson type: $\mu e^{-\mu s}$. We observe $p(s)$ as
half-Gaussian for two real eigenvalues of $C$. For real matrices $R, C, T$, we
associate new types of $p(s)$ with them. Lastly, we study the distribution
$D(\epsilon)$ of eigenvalues of symmetric matrices (of large order) discussed
above.
