To calculate the standard deviation of the data set $\{6, 2, 8, 1, 1, 7, 3, 4\}$, follow these steps:

Step 1: Find the Mean

The mean $\mu$ of the data set is given by: $\mu = \frac{\sum x_i}{n}$ where $x_i$ are the data points and $n$ is the number of data points.

$\mu = \frac{6 + 2 + 8 + 1 + 1 + 7 + 3 + 4}{8} = \frac{32}{8} = 4$

Step 2: Calculate the Variance

The variance $\sigma^2$ is the average of the squared differences between each data point and the mean: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$

(6 - 4)^2 &= 4, \\ (2 - 4)^2 &= 4, \\ (8 - 4)^2 &= 16, \\ (1 - 4)^2 &= 9, \\ (1 - 4)^2 &= 9, \\ (7 - 4)^2 &= 9, \\ (3 - 4)^2 &= 1, \\ (4 - 4)^2 &= 0. \end{align*}$$ Now, sum these squared differences: $$4 + 4 + 16 + 9 + 9 + 9 + 1 + 0 = 52$$ Then, divide by the number of data points $$n = 8$$: $$\sigma^2 = \frac{52}{8} = 6.5$$ ### Step 3: Calculate the Standard Deviation The standard deviation $$\sigma$$ is the square root of the variance: $$\sigma = \sqrt{6.5} \approx 2.55$$ Thus, the standard deviation of the data set is approximately $$2.55$$. --- Do you want more details or have any questions? Here are 5 related questions for practice: 1. What is the variance of the set $$\{3, 5, 7, 9, 11\}$$? 2. How does changing one data point affect the standard deviation? 3. Find the standard deviation for the set $$\{10, 15, 20, 25, 30\}$$. 4. How is population standard deviation different from sample standard deviation? 5. How do outliers affect the standard deviation of a data set? **Tip:** Standard deviation measures the spread of a data set; larger values mean more spread out data!