To find the standard deviation of a data set, follow these steps:

### Step-by-Step Process:

1. **Find the mean (average)**:
   - Add up all the numbers in your data set.
   - Divide the sum by the number of data points (n).
   \[
   \text{Mean} (\mu) = \frac{\sum x_i}{n}
   \]
   where \( x_i \) represents each value in the data set.

2. **Calculate the squared differences from the mean**:
   - For each data point \( x_i \), subtract the mean \( \mu \) and then square the result.
   \[
   (x_i - \mu)^2
   \]

3. **Find the average of the squared differences** (this is the variance):
   - Add up all the squared differences and divide by the number of data points. 
   - If you’re calculating for a **sample**, divide by \( n - 1 \) instead of \( n \).
   \[
   \text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n} \quad \text{for population}
   \]
   or
   \[
   \text{Variance} (s^2) = \frac{\sum (x_i - \mu)^2}{n-1} \quad \text{for sample}
   \]

4. **Take the square root of the variance**:
   - This gives you the standard deviation.
   \[
   \text{Standard Deviation} (\sigma) = \sqrt{\sigma^2}
   \]
   or
   \[
   \text{Standard Deviation} (s) = \sqrt{s^2}
   \]

### Example:
Given the data set: 2, 4, 4, 4, 5, 5, 7, 9

1. **Mean**:
   \[
   \mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5
   \]

2. **Squared differences**:
   \[
   (2 - 5)^2 = 9, \quad (4 - 5)^2 = 1, \quad (4 - 5)^2 = 1, \quad (4 - 5)^2 = 1
   \]
   \[
   (5 - 5)^2 = 0, \quad (5 - 5)^2 = 0, \quad (7 - 5)^2 = 4, \quad (9 - 5)^2 = 16
   \]
   
3. **Variance**:
   \[
   \sigma^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = \frac{32}{8} = 4
   \]

4. **Standard Deviation**:
   \[
   \sigma = \sqrt{4} = 2
   \]

Would you like more details on any part of this process?

### Related Questions:
1. What is the difference between population and sample standard deviation?
2. How do you interpret standard deviation in a data set?
3. Can standard deviation ever be negative?
4. How does standard deviation relate to variance?
5. What is the formula for standard deviation in grouped data?

### Tip:
For large data sets, consider using a calculator or software (Excel, Python, etc.) to quickly compute the standard deviation.

Learn how to calculate the standard deviation of a data set with a detailed step-by-step guide. This tutorial covers key statistical concepts, including mean, variance, and standard deviation, suitable for students in Grades 9-12.

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