Traveler Spending The data show the traveler spending in billions of dollars for a recent year for a sample of the states. Round your answers to two decimal places. 20.6, 33.4, 21.4, 58.1, 23.4, 110.3, 30.1, 24.9, 74, 60.6, 40.8, 45.4, 65.7 . find the range? find the variance?

Let's calculate the range and the variance step by step for the given dataLet's calculate the range and the variance step by step for the given 

### Data:
Traveler spending in billions of dollars:
20.6, 33.4, 21.4, 58.1, 23.4, 110.3, 30.1, 24.9, 74, 60.6, 40.8, 45.4, 65.7

### Step 1: Calculate the **Range**

The range is the difference between the largest and smallest values.

- Maximum value: \( 110.3 \)
- Minimum value: \( 20.6 \)

\[
\text{Range} = 110.3 - 20.6 = 89.7
\]

### Step 2: Calculate the **Variance**

Variance (\( \sigma^2 \)) is calculated using the following steps:

1. **Find the mean** (\( \mu \)) of the dataLet's calculate the range and the variance step by step for the given 

### Data:
Traveler spending in billions of dollars:
20.6, 33.4, 21.4, 58.1, 23.4, 110.3, 30.1, 24.9, 74, 60.6, 40.8, 45.4, 65.7

### Step 1: Calculate the **Range**

The range is the difference between the largest and smallest values.

- Maximum value: \( 110.3 \)
- Minimum value: \( 20.6 \)

\[
\text{Range} = 110.3 - 20.6 = 89.7
\]

### Step 2: Calculate the **Variance**

Variance (\( \sigma^2 \)) is calculated using the following steps:

1. **Find the mean** (\( \mu \)) of the 
   \[
   \mu = \frac{\text{Sum of all data points}}{\text{Number of data points}}
   \]

2. **Calculate each squared difference** from the mean:
   \[
   (x_i - \mu)^2
   \]

3. **Find the average of these squared differences**:
   \[
   \text{Variance} = \frac{\sum (x_i - \mu)^2}{n}
   \]

Let me compute this now.### Results:

- **Range**: \( 89.7 \)
- **Variance**: \( 638.66 \)

Would you like to see more details on these calculations, or do you have any other questions?

Here are some related questions:
1. How do you calculate the standard deviation from variance?
2. What is the effect of outliers on the range and variance?
3. How would the variance differ if we used sample variance instead of population variance?
4. How can traveler spending be modeled with other statistical measures (like median or mode)?
5. How does variance interpretation differ in the context of spending data?

**Tip**: Variance gives an idea of how spread out the data is, while the standard deviation, the square root of variance, is often more intuitive for interpreting variability.

Traveler Spending: The data show the traveler spending in billions of dollars for a recent year for a sample of the states. Round your answers to two decimal places. 20.6, 33.4, 21.4, 58.1, 23.4, 110.3, 30.1, 24.9, 74, 60.6, 40.8, 45.4, 65.7. Find the range? Find the variance?

This example demonstrates how to calculate the range and variance of traveler spending data for different states, with values given in billions of dollars. Explore the detailed steps, including finding the mean, squared differences, and variance calculation. Suitable for high school students studying statistics (Grades 10-12).

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