In this paper, we use the dimensional reduction technique to study the central limit theory (CLT) random quadratic forms based on sample means and sample covariance matrices. Specifically, we use a matrix denoted by $U_{p\times q}$, to map $q$-dimensional sample vectors to a $p$ dimensional subspace, where $q\geq p$ or $q\gg p$. Under the condition of $p/n\rightarrow 0$ as $(p,n)\rightarrow \infty$, we obtain the CLT of random quadratic forms for the sample means and sample covariance matrices.