Since square $PQRS$ has side length 2, then $PQ=QR=RS=SP=2$. Since $W, X, Y, Z$ are the midpoints of the sides of $PQRS$, then $PW=PZ=1$. Since $\angle ZPW=90^{\circ}$, then $WZ=\sqrt{PW^{2}+PZ^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$. Therefore, square $WXYZ$ has side length $\sqrt{2}$. The area of square $WXYZ$ is $(\sqrt{2})^{2}=2$ and the area of square $PQRS$ is $2^{2}=4$. The ratio of these areas is $2: 4$ or $1: 2$.