  We analyze the form of the probability distribution function
P_{n}^{(\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n
x^{2}_j/n), where x_j are the eigenvalues of a given n \times n \beta-Gaussian
random matrix, \beta being the Dyson symmetry index. This variable, by
definition, can be considered as a measure of how any individual eigenvalue
deviates from the arithmetic mean value of all eigenvalues of a given random
matrix, and its distribution is calculated with respect to the ensemble of such
\beta-Gaussian random matrices. We show that in the asymptotic limit n \to
\infty and for arbitrary \beta the distribution P_{n}^{(\beta)}(w) converges to
the Mar\v{c}enko-Pastur form, i.e., is defined as P_{n}^{(\beta)}(w) \sim
\sqrt{(4 - w)/w} for w \in [0,4] and equals zero outside of the support.
Furthermore, for Gaussian unitary (\beta = 2) ensembles we present exact
explicit expressions for P_{n}^{(\beta=2)}(w) which are valid for arbitrary n
and analyze their behavior.
