We study intermediate-scale statistics for the fractional parts of the sequence
$\left(\alpha a_{n}\right)_{n=1}^{\infty}$, where
$\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and
$\alpha\in\mathbb{R}$. In particular, we consider the number of elements
$S_{N}\!\left(L,\alpha\right)$ in a random interval of length
$L/N$, where
$L=O\!\left(N^{1-\epsilon}\right)$, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t.
$\alpha$, which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in
$\alpha\in\mathbb{R}$ when
$L=O\!\left(N^{1/2-\epsilon}\right)$. For slowly growing L, we further prove a central limit theorem for
$S_{N}\!\left(L,\alpha\right)$ which holds for almost all
$\alpha\in\mathbb{R}$.