Listed below are the numbers of hurricanes that occurred in each year in a certain region. The data are listed in order by year. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. What important feature of the data is not revealed by any of the measures of​ variation?
2
17
6
4
17
15
7

11
12
8
3
7
10
7
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Part 1
The range of the sample data is

enter your response here
▼
hurricanes squared .
years.
years squared .
hurricanes.
​(Round to one decimal place as​ needed.)

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.

Listed below are the numbers of hurricanes that occurred in each year in a certain region. The data are listed in order by year. Find the range, variance, and standard deviation for the given sample data. Include appropriate units in the results. What important feature of the data is not revealed by any of the measures of variation? Data: 2, 17, 6, 4, 17, 15, 7, 11, 12, 8, 3, 7, 10, 7

This problem involves calculating the range, variance, and standard deviation for hurricane data. The dataset includes hurricane counts from multiple years, and the solution shows step-by-step calculations for each measure of variation. The result provides a deeper understanding of hurricane frequency and variability.

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