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?

1212

66

88

55

2020

22

1010

1111

77

1111

2020

1818

33

1717

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Part 1

The range of the sample data is

enter your response here

▼

hurricanes squared .hurricanes2.

years.years.

years squared .years2.

hurricanes.hurricanes.

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

Let's begin by organizing and analyzing the given data to calculate the range, variance, and standard deviation.

### Given Data:
12, 6, 8, 5, 20, 2, 10, 11, 7, 11, 20, 18, 3, 17

#### Step 1: **Calculate the Range**
The range is calculated as the difference between the maximum and minimum values in the dataset.
- Maximum value = 20
- Minimum value = 2

\[
\text{Range} = \text{Max} - \text{Min} = 20 - 2 = 18 \text{ hurricanes}
\]

#### Step 2: **Calculate the Variance**
Variance is a measure of how much the data points deviate from the mean.

First, we need the **mean** of the data.
\[
\text{Mean} = \frac{12 + 6 + 8 + 5 + 20 + 2 + 10 + 11 + 7 + 11 + 20 + 18 + 3 + 17}{14} = \frac{150}{14} \approx 10.7 \text{ hurricanes}
\]

Now, for each data point, we compute the squared difference from the mean, then calculate the average of those squared differences.

\[
\text{Variance} (\sigma^2) = \frac{(12-10.7)^2 + (6-10.7)^2 + (8-10.7)^2 + \dots + (17-10.7)^2}{n - 1}
\]
Where \( n = 14 \) (the number of data points).

Let me compute the variance.### Results:

1. **Range**: The range of the sample data is 18 hurricanes.
2. **Variance**: The variance of the sample data is approximately 36.8 hurricanes².
3. **Standard Deviation**: The standard deviation of the sample data is approximately 6.1 hurricanes.

### Important Feature Not Revealed:
These measures of variation (range, variance, standard deviation) do not reveal the shape of the data distribution, such as whether the data is skewed or if there are clusters or gaps.

Do you want further details or clarifications on any part? 

Here are five follow-up questions you might find interesting:
1. How does skewness affect the interpretation of standard deviation?
2. Can you calculate the interquartile range (IQR) for this data?
3. What would be the effect of adding an extreme outlier to the data?
4. How do range and standard deviation complement each other in understanding data spread?
5. What are the differences between population and sample variance?

**Tip**: Always check for outliers in your data, as they can significantly impact the variance and standard deviation.

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: 12, 6, 8, 5, 20, 2, 10, 11, 7, 11, 20, 18, 3, 17.

Learn how to calculate the range, variance, and standard deviation for a set of hurricane data. This problem covers statistical concepts such as measures of variation and includes detailed formulas and step-by-step solutions. Suitable for students in Grades 10-12.

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