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?

1111

11

1212

77

1717

77

55
 
99

1212

1818

1010

33

1919

1111

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Part 1

The range of the sample data is 

1818 

hurricanes.hurricanes.

​(Round to one decimal place as​ needed.)

Part 2

The standard deviation of the sample data is 

enter your response here 

▼
 
years.years.

hurricanes.hurricanes.

years squared .years2.

hurricanes squared .hurricanes2.

​(Round to one decimal place as​ needed.)

Let's solve this problem step by step.

### 1. Range
The range is calculated as the difference between the maximum and minimum values in the dataset.

Given dataLet's solve this problem step by step.

### 1. Range
The range is calculated as the difference between the maximum and minimum values in the dataset.

Given   
\[11, 11, 12, 12, 7, 17, 7, 5, 9, 12, 18, 10, 3, 19, 11\]

- **Maximum value**: 19 (hurricanes)
- **Minimum value**: 3 (hurricanes)

\[
\text{Range} = 19 - 3 = 16 \text{ hurricanes}
\]

The correct range is 16 hurricanes, not 18.

### 2. Variance and Standard Deviation
The variance and standard deviation for the sample are calculated as follows:

#### Step 1: Calculate the mean (average) of the data.
\[
\text{Mean} = \frac{11 + 11 + 12 + 12 + 7 + 17 + 7 + 5 + 9 + 12 + 18 + 10 + 3 + 19 + 11}{15}
\]

#### Step 2: Calculate each deviation from the mean and square it.

#### Step 3: Calculate the sample variance.
\[
\text{Sample Variance} = \frac{\sum (x_i - \text{Mean})^2}{n-1}
\]
Where \( n \) is the number of observations (15 in this case).

#### Step 4: Calculate the sample standard deviation.
\[
\text{Sample Standard Deviation} = \sqrt{\text{Sample Variance}}
\]

Let's calculate these values.### Results:

- **Mean**: 10.9 hurricanes (rounded to one decimal place)
- **Sample Variance**: 20.6 hurricanes² (rounded to one decimal place)
- **Sample Standard Deviation**: 4.5 hurricanes (rounded to one decimal place)

### Interpretation:
- **Range**: The range of 16 hurricanes indicates a considerable spread in the number of hurricanes per year.
- **Variance and Standard Deviation**: The standard deviation of 4.5 hurricanes shows that the number of hurricanes varies moderately around the mean of 10.9 hurricanes per year.

### Important Feature Not Revealed by These Measures:
None of these measures of variation reveal any patterns, trends, or cycles in the data over time. For example, if there is an increasing or decreasing trend in the number of hurricanes over the years, the range, variance, and standard deviation alone would not capture that information.

Would you like more details or have any questions?

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Here are five related questions to deepen your understanding:
1. How would the inclusion of an outlier (e.g., a year with 30 hurricanes) affect the variance and standard deviation?
2. How can a time series analysis reveal patterns in the data that these measures do not?
3. What is the significance of a small vs. large standard deviation in this context?
4. How would the range change if you removed the maximum or minimum value from the dataset?
5. Can you interpret the variance in real-world terms, and why might it be useful?

**Tip:** Always look beyond basic statistics like range, variance, and standard deviation to explore the data's trends and patterns over time.

Learn how to calculate the range, variance, and standard deviation for a dataset of hurricane occurrences over multiple years. Understand the spread and variability of hurricane counts using statistical measures. Suitable for high school students studying statistics.

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