Given that $x_1, x_2, x_3, \dots, x_{10}$ are 10 unique numbers whose mean, $\bar{x}$, is 11.8 and standard deviation is 4.729. Find the value of

1. Use the mean formula to find the sum of the 10 numbers.

$\bar{x}=\frac{x_1+x_2+x_3+\cdots+x_{10}}{10}$
Given $\bar{x}=11.8$,
$x_1+x_2+x_3+\cdots+x_{10}=10(11.8)=118$

1. Write the standard deviation formula and substitute the given values.

For $n=10$, $\bar{x}=11.8$ and standard deviation $=4.729$,
$4.729=\sqrt{\frac{x_1^2+x_2^2+\cdots+x_{10}^2}{10}-11.8^2}$

2. Rearrange the formula and calculate the sum of squares.

Squaring both sides,
$4.729^2=\frac{x_1^2+x_2^2+\cdots+x_{10}^2}{10}-11.8^2$
Hence
$x_1^2+x_2^2+\cdots+x_{10}^2=10(4.729^2+11.8^2)$
$=10(22.363441+139.24)$
$=1616.03441$
To the nearest whole number,
$x_1^2+x_2^2+\cdots+x_{10}^2=1616.$

1. Identify how the transformation changes each value's distance from the mean.

For any value below the mean, $x_n<\bar{x}$, decreasing it by 2 makes it even further below $\bar{x}$.
For any value above the mean, $x_n>\bar{x}$, increasing it by 2 makes it even further above $\bar{x}$.

2. Conclude the effect on the standard deviation.

Since the data values become more spread out from the mean, the standard deviation becomes larger.