Let's go step by step to calculate the range, variance, and standard deviation for the given sample data. Here is the list of the number of hurricanes per year:

Data: 2, 17, 6, 4, 17, 15, 7, 11, 12, 8, 3, 7, 10, 7

Step 1: Calculate the Range

The range is the difference between the maximum and minimum values in the dataset.

• Maximum value = 17
• Minimum value = 2

Range = Maximum - Minimum = 17 - 2 = 15 hurricanes.

Step 2: Calculate the Variance

The variance is the average of the squared differences from the mean. First, calculate the mean:

$\text{Mean} = \frac{\sum X}{n} = \frac{2 + 17 + 6 + 4 + 17 + 15 + 7 + 11 + 12 + 8 + 3 + 7 + 10 + 7}{14} = \frac{126}{14} = 9$

Now, calculate the squared differences from the mean for each value:

$\text{Squared differences} = (2-9)^2, (17-9)^2, \ldots, (7-9)^2$

$(2 - 9)^2 = 49, \quad (17 - 9)^2 = 64, \quad (6 - 9)^2 = 9, \quad (4 - 9)^2 = 25, \quad (17 - 9)^2 = 64$ $(15 - 9)^2 = 36, \quad (7 - 9)^2 = 4, \quad (11 - 9)^2 = 4, \quad (12 - 9)^2 = 9, \quad (8 - 9)^2 = 1$ $(3 - 9)^2 = 36, \quad (7 - 9)^2 = 4, \quad (10 - 9)^2 = 1, \quad (7 - 9)^2 = 4$

Now sum these squared differences:

$49 + 64 + 9 + 25 + 64 + 36 + 4 + 4 + 9 + 1 + 36 + 4 + 1 + 4 = 349$

Finally, divide by $n - 1$ (since this is a sample):

$\text{Variance} = \frac{349}{13} \approx 26.8 \text{ hurricanes}^2$

Step 3: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

$\text{Standard Deviation} = \sqrt{26.8} \approx 5.2 \text{ hurricanes}$

Summary of Results

• Range: 15 hurricanes
• Variance: 26.8 hurricanes²
• Standard Deviation: 5.2 hurricanes

Part 4: What important feature of the data is not revealed by these measures?

The measures of variation (range, variance, and standard deviation) do not reveal the distribution shape or any potential patterns in the data, such as trends over time. For example, these statistics do not indicate if the data is skewed, clustered, or contains any outliers.

Would you like further details or explanations on any part of this?

Here are some related questions:

1. What is the significance of the mean in this dataset?
2. How do outliers affect the standard deviation?
3. What does a high variance imply about the dataset?
4. Can the range give an indication of how spread out the data is?
5. How would the data distribution change if we added more data points?

Tip: The standard deviation gives a good sense of how much the values deviate from the mean, but it’s crucial to inspect data visually to understand its full nature.