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Let $M_n$ be a $n \times n$ Wigner or sample covariance random matrix, andlet $\mu_1(M_n), \mu_2(M_n), ..., \mu_n(M_n)$ denote the unordered eigenvaluesof $M_n$. We study the fluctuations of the partial linear eigenvalue statistics$$ \sum_{i=1}^{n-k} f(\mu_i(M_n)) $$ as $n \rightarrow \infty$ for sufficientlynice test functions $f$. We consider both the case when $k$ is fixed and when$\min{k,n-k}$ tends to infinity with $n$.

Partial Linear Eigenvalue Statistics for Wigner and Sample Covariance Random Matrices