  In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness
of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically
$p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n:
|\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A
function $f$ is called statistically $p$-ward continuous on a subset $A$ of the
set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy
sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically
$p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a statistically
$p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued
function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if
there exists a positive integer $p$ such that $f$ preserves statistically
$p$-quasi-Cauchy sequences of points in $A$.
