Solution
\[
\begin{aligned}
& \text { Mean }=\frac{\sum x_{i}}{8}=\frac{-10-7-1+x+y+9+2+16}{8}=\frac{7}{2} \\
& \Rightarrow x+y=19 \quad \ldots(1) \\
& \text { Variance }=\frac{1}{n} \sum x_{i}^{2}-\mu^{2}=\frac{293}{4} \\
& =\frac{1}{8}\left(100+49+1+x^{2}+y^{2}+81+4+256\right)-\left(\frac{7}{2}\right)^{2}=\frac{293}{4}
\end{aligned}
\]
\[
\begin{aligned}
& \Rightarrow x^{2}+y^{2}+406+85=8\left[\frac{49+293}{4}\right]=684 \\
& \Rightarrow x^{2}+y^{2}=684-491=193 \text { from }(1) \\
& \Rightarrow x^{2}+(19-x)^{2}=193 \\
& \Rightarrow(x-12)(x-7)=0 \\
& x=12 \mid x=7 \\
& y=7 \mid y=12
\end{aligned}
\]
required numbers $\mathrm{x}, \mathrm{y}, \mathrm{x}+\mathrm{y}+1,|\mathrm{x}-\mathrm{y}|$ are 12, 7, 20, 5
\[
\text { Mean }=\frac{12+7+20+5}{4}=11 \text { Answer }(1)
\]
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