I is not true. The mean of $S_1$
$\begin{array}{cl}
= & \dfrac{d-6+d-2+d-1+d+3+d+5+d+7}{6} \\
= & d +1
\end{array}$
The mean of $S_2$
$\begin{array}{cl}
= & \dfrac{d-7+d-5+d-3+d+1+d+2+d+6}{6} \\
= & d -1
\end{array}$
Therefore, the means of $S_1$ and $S_2$ are not equal.
II is true. The standard deviation of $S_1$
$\begin{array}{cl}
= & \sqrt{\dfrac{(d-6-(d+1))^2 +(d-2-(d+1))^2+\ldots +(d+7-(d+1))^2}{6}} \\
= & 4.434\ 711\ 565
\end{array}$
The standard deviation of $S_2$
$\begin{array}{cl}
= & \sqrt{\dfrac{(d-7-(d-1))^2 +(d-5-(d-1))^2+\ldots +(d+6-(d-1))^2}{6}} \\
= & 4.434\ 711\ 565
\end{array}$
Therefore, the standard deviations of $S_1$ and $S_2$ are equal.
III is true. The inter-quartile range of $S_1$
$\begin{array}{cl}
= & (d+5)-(d-2) \\
= & 7
\end{array}$
The inter-quartile range of $S_2$
$\begin{array}{cl}
= & (d+2)-(d-5) \\
= & 7
\end{array}$
Therefore, the inter-quartile ranges of $S_1$ and $S_2$ are equal.
