1. Let's start by stating the problem: We want to understand how to calculate the standard deviation of a data set. 2. The standard deviation measures how spread out the numbers in a data set are from the mean (average). 3. The formula for the standard deviation $\sigma$ of a population is: $$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}$$ where $N$ is the number of data points, $x_i$ are the data points, and $\mu$ is the mean. 4. For a sample standard deviation $s$, the formula is: $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ where $n$ is the sample size, $x_i$ are the sample points, and $\bar{x}$ is the sample mean. 5. Steps to calculate standard deviation: 1. Find the mean $\mu$ or $\bar{x}$ by summing all data points and dividing by the number of points. 2. Subtract the mean from each data point to find the deviation of each point. 3. Square each deviation. 4. Sum all squared deviations. 5. Divide by $N$ for population or $n-1$ for sample. 6. Take the square root of the result. 6. Example: Data set $\{2, 4, 4, 4, 5, 5, 7, 9\}$ - Mean $\bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = 5$ - Deviations: $\{-3, -1, -1, -1, 0, 0, 2, 4\}$ - Squared deviations: $\{9, 1, 1, 1, 0, 0, 4, 16\}$ - Sum of squares: $9+1+1+1+0+0+4+16=32$ - Sample variance: $\frac{32}{8-1} = \frac{32}{7} \approx 4.57$ - Sample standard deviation: $s = \sqrt{4.57} \approx 2.14$ Final answer: The sample standard deviation of the data set is approximately $2.14$.