Let's break down the problem into parts.

Sample Data:

• 20, 16, 22, 11, 13, 6, 10, 20, 14, 17, 5, 13, 20, 16

Part 1: Range

The range is the difference between the largest and smallest values in the data.

1. The maximum value is 22.
2. The minimum value is 5.

$\text{Range} = 22 - 5 = 17 \text{ hurricanes}$

Part 2: Variance

The variance measures how much the data values vary from the mean.

1. Calculate the mean: $\text{Mean} = \frac{\sum \text{Data Values}}{\text{Number of Data Points}}$ The sum of the data values is: $20 + 16 + 22 + 11 + 13 + 6 + 10 + 20 + 14 + 17 + 5 + 13 + 20 + 16 = 203$ There are 14 data points: $\text{Mean} = \frac{203}{14} \approx 14.5$

2. Calculate the squared deviations from the mean: For each data point $x_i$, calculate $(x_i - \text{mean})^2$.

3. Variance formula: $\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n - 1}$

Let me now calculate the variance and standard deviation for this data.### Results:

• Range: $17 \, \text{hurricanes}$
• Mean: $14.5 \, \text{hurricanes}$
• Variance: $27.5 \, \text{hurricanes}^2$
• Standard Deviation: $5.2 \, \text{hurricanes}$ (rounded to one decimal place)

Important Feature Not Revealed by Variation Measures:

None of these measures—range, variance, or standard deviation—provide any information about the distribution shape of the data, such as whether it is skewed, symmetric, or has any outliers.

Would you like further clarification or details on any step?

Here are 5 related questions to consider:

1. How would the range change if an extreme value (e.g., 30) were added?
2. What is the difference between population variance and sample variance?
3. How would you calculate the variance for a population instead of a sample?
4. What does a high standard deviation imply about the dataset?
5. How would the variance change if all values in the dataset were the same?

Tip: The standard deviation is a useful measure of spread but can be sensitive to outliers in the data.