Describe the process of calculating a standard deviation. Give a simple example of its calculation (such as calculating the standard deviation of the numbers $$ 2,3,4,4 $$, and 6 ). What is the standard deviation if all of the sample values are the same? Fill in the blanks to complete the process of calculating a standard deviation. Compute the mean of the data set. Then find the deviation from the mean for every data value by subtracting the mean from the data value. Find the squares of all the deviations from the mean, and then add them together. Divide this sum by the total number of data values minus 1 . The standard deviation is the square root of this quotient. The standard deviation of the numbers $$ 2,3,4,4 $$, and 6 is approximately (Round to three decimal places as needed.)

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**1. Compute the mean of the data set:**  
For the numbers $$ 2, 3, 4, 4, 6 $$, the mean $$ \bar{x} $$ is calculated as:  
$$
\bar{x} = \frac{2+3+4+4+6}{5} = \frac{19}{5} = 3.8
$$

**2. Find the deviations from the mean:**  
Subtract the mean from each data value:  
$$
2 - 3.8 = -1.8,\quad 3 - 3.8 = -0.8,\quad 4 - 3.8 = 0.2,\quad 4 - 3.8 = 0.2,\quad 6 - 3.8 = 2.2
$$

**3. Square each deviation:**  
$$
(-1.8)^2 = 3.24,\quad (-0.8)^2 = 0.64,\quad (0.2)^2 = 0.04,\quad (0.2)^2 = 0.04,\quad (2.2)^2 = 4.84
$$

**4. Sum the squared deviations:**  
$$
\text{Sum} = 3.24 + 0.64 + 0.04 + 0.04 + 4.84 = 8.8
$$

**5. Divide by the number of data values minus 1:**  
Since there are 5 values, the divisor is $$ 5 - 1 = 4 $$. This gives the variance:  
$$
\text{Variance} = \frac{8.8}{4} = 2.2
$$

**6. Take the square root of the variance to find the standard deviation:**  
$$
\text{Standard Deviation} = \sqrt{2.2} \approx 1.483
$$

**What is the standard deviation if all sample values are the same?**  
If all values in the data set are identical, then every value equals the mean, making each deviation $$ 0 $$. Thus, the squared deviations and their sum will be $$ 0 $$. Consequently, the variance is $$ 0 $$ and the standard deviation is:  
$$
\sqrt{0} = 0
$$
**Calculating Standard Deviation:** 1. **Compute the mean:** For numbers $$ 2, 3, 4, 4, 6 $$, the mean is $$ 3.8 $$. 2. **Find deviations from the mean:** $$ 2 - 3.8 = -1.8 $$, $$ 3 - 3.8 = -0.8 $$, $$ 4 - 3.8 = 0.2 $$, $$ 4 - 3.8 = 0.2 $$, $$ 6 - 3.8 = 2.2 $$. 3. **Square each deviation:** $$ (-1.8)^2 = 3.24 $$, $$ (-0.8)^2 = 0.64 $$, $$ (0.2)^2 = 0.04 $$, $$ (0.2)^2 = 0.04 $$, $$ (2.2)^2 = 4.84 $$. 4. **Sum the squared deviations:** $$ 3.24 + 0.64 + 0.04 + 0.04 + 4.84 = 8.8 $$. 5. **Divide by $$ n-1 $$ (where $$ n = 5 $$):** $$ 8.8 \div 4 = 2.2 $$. 6. **Take the square root of the variance:** $$ \sqrt{2.2} \approx 1.483 $$. **Standard Deviation:** The standard deviation of $$ 2, 3, 4, 4, 6 $$ is approximately **1.483**. **If all sample values are the same:** The standard deviation is **0**.

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