To compute the measure of dispersion for the given dataset of hours spent studying, we'll calculate the standard deviation step by step. Standard deviation is a common measure that tells us how spread out the data points are relative to the mean. Here’s the step-by-step process:

1. Data Set:

The study hours per week are:
$5, 7, 10, 8, 6, 9, 4, 12, 8, 6$

2. Calculate the Mean:

First, we need to compute the mean ($\mu$) of the data.

$\mu = \frac{\sum \text{Data Values}}{\text{Number of Values}} = \frac{5 + 7 + 10 + 8 + 6 + 9 + 4 + 12 + 8 + 6}{10} = \frac{75}{10} = 7.5$

The mean study hours per week is 7.5 hours.

3. Find the Deviations:

Next, calculate how far each value is from the mean (deviation):

• $5 - 7.5 = -2.5$
• $7 - 7.5 = -0.5$
• $10 - 7.5 = 2.5$
• $8 - 7.5 = 0.5$
• $6 - 7.5 = -1.5$
• $9 - 7.5 = 1.5$
• $4 - 7.5 = -3.5$
• $12 - 7.5 = 4.5$
• $8 - 7.5 = 0.5$
• $6 - 7.5 = -1.5$

4. Square Each Deviation:

Now, square each of the deviations:

• $(-2.5)^2 = 6.25$
• $(-0.5)^2 = 0.25$
• $(2.5)^2 = 6.25$
• $(0.5)^2 = 0.25$
• $(-1.5)^2 = 2.25$
• $(1.5)^2 = 2.25$
• $(-3.5)^2 = 12.25$
• $(4.5)^2 = 20.25$
• $(0.5)^2 = 0.25$
• $(-1.5)^2 = 2.25$

5. Sum the Squared Deviations:

Add up all the squared deviations:

$6.25 + 0.25 + 6.25 + 0.25 + 2.25 + 2.25 + 12.25 + 20.25 + 0.25 + 2.25 = 52.5$

6. Calculate the Variance:

The variance ($\sigma^2$) is the average of these squared deviations. Since we are calculating for the population (not a sample), we divide by the number of data points:

$\sigma^2 = \frac{52.5}{10} = 5.25$

7. Calculate the Standard Deviation:

Finally, the standard deviation ($\sigma$) is the square root of the variance:

$\sigma = \sqrt{5.25} \approx 2.29$

Conclusion:

The standard deviation of the study hours is approximately 2.29 hours, indicating that the study times typically vary about 2.29 hours from the mean of 7.5 hours.

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Would you like more details or have any questions about this process?

Here are five related questions to consider:

1. How would the process differ if we calculated the standard deviation for a sample instead of a population?
2. What are other measures of dispersion besides standard deviation?
3. How does the variance change if one of the study hours is significantly increased?
4. Can you interpret what a higher or lower standard deviation would imply in this scenario?
5. How would the median be affected by extreme values compared to the mean?

Tip: Always check whether you're working with a sample or a population, as it affects the divisor in variance and standard deviation calculations.