To find which equation results in the greatest standard deviation of the $y$-values for $x$ from 1 to 20, we need to understand how linear transformations affect standard deviation.

Let $x$ be a variable with a standard deviation $S D_x$.
If $y=m x+b$, then the standard deviation of $y\left(S D_y\right)$ is given by:

$$
\(S D_y=|m| \times S D_x\)
$$


In this problem, the $x$-values are integers from 1 to 20 . This means the set of $x$-values is

$$
\(\{1,2,3, \ldots, 20\}\)
$$


The standard deviation of this set of $x$-values ( $S D_x$ ) is a fixed positive value. We don't need to calculate it, only recognize that it's constant for all options.

To maximize $S D_y$, we need to maximize $|m|$, the absolute value of the slope.
Let's find the slope $(m)$ for each equation:
(A) $\(y=\frac{x}{5}\)$

The slope $\(m=\frac{1}{5}=0.2\)$.

$$
\(|m|=0.2\)
$$

(B) $\(y=\frac{x}{2}+2\)$

The slope $\(m=\frac{1}{2}=0.5\)$.

$$
\(|m|=0.5\)
$$

(C) $y=x$

The slope $m=1$.

$$
\(|m|=1\)
$$

(D) $\(y=-2 x+7\)$

The slope $\(m=-2\)$.

(E) $\(y=4 x-9$\)

The slope $\(m=4\)$.

$$
\(|m|=|4|=4\)
$$


Now, let's compare the absolute values of the slopes:

$$
\(0.2,0.5,1,2,4\)
$$


The greatest absolute value of the slope is 4 , which corresponds to equation (E).
Therefore, the standard deviation of the $y$-values will be the greatest for equation (E).
The final answer is $\(\mathrm{y}=4 \mathrm{x}-9\)$.