A real valued function $f$ defined on a subset $E$ of $\mathbb R$, the set of real numbers, is statistically upward (resp. downward) continuous if it preserves statistically upward (resp. downward) half quasi-Cauchy sequences; A subset $E$ of $\mathbb R$, is statistically upward (resp. downward) compact if any sequence of points in $E$ has a statistically upward (resp. downward) half quasi-Cauchy subsequence, where a sequence ($x_n$) of points in $\mathbb R$ is called statistically upward half quasi-Cauchy if \[ im_{noıfty}\frac1n|\{keq n:x_k-x_{k+1}\geqǎrepsilon\}|=0, \] and statistically downward half quasi-Cauchy if \[ im_{noıfty}\frac1n|\{keq n:x_{k+1}-x_k\geqǎrepsilon\}|=0 \] for every $\varepsilon>0$. We investigate statistically upward and downward continuity, statistically upward and downward half compactness and prove interesting theorems. It turns out that any statistically upward continuous function on a below bounded subset of $\mathbb R$ is uniformly continuous, and any statistically downward continuous function on an above bounded subset of $\mathbb R$ is uniformly continuous.