Let us calculate the range, variance, and standard deviation for the given sample data. The data set is:

10, 8, 1, 2, 7, 10, 20, 13, 15, 5, 17, 6, 10, 17

Step 1: Calculate the Range

The range is the difference between the maximum and minimum values in the data set.
$\text{Range} = \text{Maximum value} - \text{Minimum value} = 20 - 1 = 19$
Answer: The range is 19 hurricanes.

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Step 2: Calculate the Variance

The variance measures the average squared difference from the mean. Use the formula:
$\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1}$
Where:

• $x_i$ are the individual data points.
• $\bar{x}$ is the mean of the data.
• $n$ is the number of data points.

1. Find the mean:

$\bar{x} = \frac{\sum x_i}{n} = \frac{10 + 8 + 1 + 2 + 7 + 10 + 20 + 13 + 15 + 5 + 17 + 6 + 10 + 17}{14} = \frac{141}{14} = 10.07$

2. Compute the squared differences from the mean:

$(x_i - \bar{x})^2$
For each data point:

(10 - 10.07)^2 &= 0.0049, \quad (8 - 10.07)^2 = 4.2849, \quad (1 - 10.07)^2 = 82.0849, \\ (2 - 10.07)^2 &= 65.6049, \quad (7 - 10.07)^2 = 9.4249, \quad (10 - 10.07)^2 = 0.0049, \\ (20 - 10.07)^2 &= 98.0049, \quad (13 - 10.07)^2 = 8.6449, \quad (15 - 10.07)^2 = 24.7049, \\ (5 - 10.07)^2 &= 25.7049, \quad (17 - 10.07)^2 = 47.3049, \quad (6 - 10.07)^2 = 16.5049, \\ (10 - 10.07)^2 &= 0.0049, \quad (17 - 10.07)^2 = 47.3049. \end{aligned}$$ #### 3. Compute the variance: $$\text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n-1} = \frac{429.704}{13} \approx 33.1$$ **Answer: The variance is 33.1 hurricanes (squared).** --- ### Step 3: Calculate the Standard Deviation The **standard deviation** is the square root of the variance: $$\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{33.1} \approx 5.7$$ **Answer: The standard deviation is 5.7 hurricanes.** --- ### Part 4: Important Feature Not Revealed The range, variance, and standard deviation do not reveal how the hurricanes are distributed over time. For example: - The measures of variation do not indicate if the hurricanes occurred evenly over the years or if there were clusters of active or inactive years. --- ### Final Summary: 1. **Range**: 19 hurricanes 2. **Variance**: 33.1 hurricanes$$^2$$ 3. **Standard Deviation**: 5.7 hurricanes Would you like to see the full calculations or learn more about interpreting these measures? --- **Relative Questions to Explore Further**: 1. How does standard deviation affect the interpretation of hurricane variability? 2. Can we compute the coefficient of variation (CV) for this data set? What does it tell us? 3. How can the distribution of hurricanes be visualized (e.g., histogram)? 4. How does removing outliers affect the variance and standard deviation? 5. What is the significance of sample variance versus population variance in this context? **Tip**: When calculating variance and standard deviation, always double-check the mean calculation, as errors there can propagate through the entire solution.