Summary: Let \(\beta>0\) and considering an \(n\)-point process \(\lambda_1,\lambda_2,\ldots, \lambda_n\) from Hermite \(\beta\) ensemble on the real line \(\mathbb{R}\), some previous authors discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper, we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than \(\sqrt{\log n}\), i.e., local semicircle law holds. And the number of positive states in \((0, \infty)\) is proved to fluctuate normally around its mean \(n/2\) with variance like \(\log n/\pi^2\beta\). The proof of the results relies mainly on the counting states in any interval and the classical martingale method.