Let's calculate the required statistical measures step by step.

Given Data:

$2, 3, 1, 0, 4$

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Part 1: Range

The range is calculated as:
$\text{Range} = \text{Maximum value} - \text{Minimum value}$

$\text{Range} = 4 - 0 = 4$

Answer: $\mathbf{4}$

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Part 2: Variance ($s^2$)

Sample variance is calculated using the formula:

$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

Step 1: Compute the Mean ($\bar{x}$)

$\bar{x} = \frac{2 + 3 + 1 + 0 + 4}{5} = \frac{10}{5} = 2$

Step 2: Compute Each Deviation Squared

$(2 - 2)^2 = 0$ $(3 - 2)^2 = 1$ $(1 - 2)^2 = 1$ $(0 - 2)^2 = 4$ $(4 - 2)^2 = 4$

Step 3: Compute Sample Variance

$s^2 = \frac{0 + 1 + 1 + 4 + 4}{5 - 1} = \frac{10}{4} = 2.5$

Answer: $\mathbf{2.5}$ (rounded to one decimal place)

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Part 3: Standard Deviation ($s$)

Standard deviation is the square root of variance:

$s = \sqrt{s^2} = \sqrt{2.5}$

$s \approx 1.58$

Answer: $\mathbf{1.58}$ (rounded to two decimal places)

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Final Answers:

• Range: $4$
• Variance ($s^2$): $2.5$
• Standard Deviation ($s$): $1.58$

Would you like a further breakdown or another example?

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Related Questions:

1. What is the difference between sample variance and population variance?
2. How does increasing the sample size affect the standard deviation?
3. Can the standard deviation ever be negative? Why or why not?
4. How do outliers impact the range, variance, and standard deviation?
5. What are the applications of standard deviation in real-world scenarios?

Tip: The standard deviation is always in the same unit as the data, while the variance is in squared units.