Let $X$ be an $N$-dimensional random vector which describes the ordered eigenvalues of a $β$-Hermite ensemble, and let $z$ the vector containing the ordered zeros of the Hermite poynomial $H_N$. We present an explicit estimate for $P(\|X-z\|_2\geε)$ for small $ε>0$ and large parameters $β$. The proof is based on a central limit theorem for these ensembles for $β\to\infty$ with explicit eigenvalues of the covariance matrices of the limit. The estimate is similar to previous estimates of Dette and Imhof (2009).