Let $x_{1}, x_{2}, x_{3}, \ldots, x_{20}$ be in geometric progression with $x_{1}=3$ and the common ratio $\frac{1}{2}$. A new data is constructed replacing each $x_{i}$ by $\left(x_{i}-i\right)^{2}$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is ____________.