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.

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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!