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Scenario: Imagine tracking the number of hours spent studying per week by students in a class: 5, 7, 10, 8, 6, 9, 4, 12, 8, 6. Now, Identify and compute the appropriate measure of dispersion using the step-by-step process.

To analyze the dispersion of the data (the spread of the number of hours students spent studying), we can use measures like **range, variance, and standard deviation**. Here's a step-by-step breakdown:

### Data:  
5, 7, 10, 8, 6, 9, 4, 12, 8, 6

### 1. **Range**  
The range is the difference between the maximum and minimum values.
- **Maximum**: 12
- **Minimum**: 4  
\[
\text{Range} = 12 - 4 = 8
\]
So, the range is **8 hours**.

### 2. **Variance**  
Variance measures how far each number in the data set is from the mean (the average).

#### Step 1: Find the Mean
Sum of the values:  
\[
5 + 7 + 10 + 8 + 6 + 9 + 4 + 12 + 8 + 6 = 75
\]
Number of values: 10  
\[
\text{Mean} = \frac{75}{10} = 7.5
\]

#### Step 2: Find the squared differences from the mean  
For each value, subtract the mean (7.5) and square the result:
\[
(5 - 7.5)^2 = (-2.5)^2 = 6.25
\]
\[
(7 - 7.5)^2 = (-0.5)^2 = 0.25
\]
\[
(10 - 7.5)^2 = (2.5)^2 = 6.25
\]
\[
(8 - 7.5)^2 = (0.5)^2 = 0.25
\]
\[
(6 - 7.5)^2 = (-1.5)^2 = 2.25
\]
\[
(9 - 7.5)^2 = (1.5)^2 = 2.25
\]
\[
(4 - 7.5)^2 = (-3.5)^2 = 12.25
\]
\[
(12 - 7.5)^2 = (4.5)^2 = 20.25
\]
\[
(8 - 7.5)^2 = (0.5)^2 = 0.25
\]
\[
(6 - 7.5)^2 = (-1.5)^2 = 2.25
\]

#### Step 3: Calculate the average of these squared differences  
Sum of the squared differences:
\[
6.25 + 0.25 + 6.25 + 0.25 + 2.25 + 2.25 + 12.25 + 20.25 + 0.25 + 2.25 = 52.5
\]
Since this is a sample, divide by \(n - 1 = 10 - 1 = 9\):
\[
\text{Variance} = \frac{52.5}{9} \approx 5.83
\]

So, the variance is approximately **5.83 hours²**.

### 3. **Standard Deviation**  
The standard deviation is the square root of the variance:
\[
\text{Standard Deviation} = \sqrt{5.83} \approx 2.41
\]

Thus, the standard deviation is approximately **2.41 hours**.

---

### Summary of Measures of Dispersion:
- **Range**: 8 hours
- **Variance**: 5.83 hours²
- **Standard Deviation**: 2.41 hours

Would you like more details on any step or further clarifications?

### Related Questions:
1. What is the importance of using the standard deviation over the range?
2. How does sample size affect the variance calculation?
3. Can you calculate the median for the given data and how does it relate to dispersion?
4. What happens to the standard deviation if one outlier is significantly larger or smaller?
5. How do you calculate the coefficient of variation and when would it be useful?

### Tip:
The **standard deviation** is sensitive to outliers, so always inspect your data to see if extreme values are affecting the spread!

Imagine tracking the number of hours spent studying per week by students in a class: 5, 7, 10, 8, 6, 9, 4, 12, 8, 6. Now, identify and compute the appropriate measure of dispersion using the step-by-step process.

Learn how to calculate the measures of dispersion, including range, variance, and standard deviation, using the weekly study hours of students: 5, 7, 10, 8, 6, 9, 4, 12, 8, 6. This step-by-step guide explains the formulas and process, suitable for students in Grades 10-12.

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