Use the following distribution to complete parts (a) through (d) below. $$ 3,5,9,12,11 $$, a) Compute the mean and standard deviation of the distribution. The mean is 8 . The standard deviation is $$ \square $$ Round to the nearest hundredth as needed.)

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$$
\textbf{Step 1: Calculate the Mean}
$$

The given distribution is: $$3, 5, 9, 12, 11$$.

The mean $$\overline{x}$$ is calculated as:
$$
\overline{x}=\frac{3+5+9+12+11}{5}=\frac{40}{5}=8.
$$

$$
\textbf{Step 2: Calculate Each Deviation and Its Square}
$$

Subtract the mean from each data value and square the result:
$$
(3-8)^2= (-5)^2=25,
$$
$$
(5-8)^2= (-3)^2=9,
$$
$$
(9-8)^2= 1^2=1,
$$
$$
(12-8)^2= 4^2=16,
$$
$$
(11-8)^2= 3^2=9.
$$

$$
\textbf{Step 3: Sum the Squared Deviations}
$$

Summing these squared deviations gives:
$$
25+9+1+16+9=60.
$$

$$
\textbf{Step 4: Compute the Standard Deviation}
$$

Since the problem refers to the distribution as a whole, we use the population standard deviation formula:
$$
\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i-\overline{x})^2}.
$$

Substitute the values:
$$
\sigma = \sqrt{\frac{60}{5}}=\sqrt{12}.
$$

Evaluating the square root, we have:
$$
\sqrt{12}\approx 3.4641.
$$

Rounded to the nearest hundredth:
$$
\sigma \approx 3.46.
$$

$$
\boxed{3.46}
$$
The standard deviation is approximately 3.46.

Use the following distribution to complete parts (a) through (d) below. \( 3,5,9,12,11 \), a) Compute the mean and standard deviation of the distribution. The mean is 8 . The standard deviation is \( \square \) Round to the nearest hundredth as needed.)

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