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?

2020

1616

22

11

1313

66

1010
 
2020

1414

1717

55

1313

2020

1616

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Part 1

The range of the sample data is 

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hurricanes squared .hurricanes2.

years.years.

years squared .years2.

hurricanes.hurricanes.

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


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.

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?

2020, 16, 22, 11, 13, 6, 10, 20, 14, 17, 5, 13, 20, 16

This page provides a step-by-step solution to find the range, variance, and standard deviation of hurricane data for various years. Learn how to apply statistical formulas, calculate the mean, and understand the spread of data. Suitable for Grades 10-12.

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