**Find the range, variance, and standard deviation for the given sample data.**

Listed below are the top 10 annual salaries (in millions of dollars) of TV personalities (based on data from OK! magazine).

38 36 35 27 15 13 12 10 9.6 8.4

Range: Answer 1 Question 8 millions of dollars.
Variance: Answer 2 Question 8 millions of dollars2.
Standard Deviation: Answer 3 Question 8 millions of dollars.

Let's calculate the range, variance, and standard deviation for the given sample data. The data set consists of the top 10 annual salaries (in millions of dollars) of TV personalities:

**Data Set:** 38, 36, 35, 27, 15, 13, 12, 10, 9.6, 8.4

### 1. Range
The range is the difference between the maximum and minimum values in the data set.

\[
\text{Range} = \text{Max} - \text{Min}
\]

Max = 38  
Min = 8.4

\[
\text{Range} = 38 - 8.4 = 29.6 \text{ million dollars}
\]

### 2. Variance
The variance is a measure of the spread of the data points around the mean. For a sample, the variance is calculated using the formula:

\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

Where:
- \(n\) is the number of data points
- \(x_i\) are the individual data points
- \(\bar{x}\) is the sample mean

#### Step 1: Calculate the mean (\(\bar{x}\))

\[
\bar{x} = \frac{38 + 36 + 35 + 27 + 15 + 13 + 12 + 10 + 9.6 + 8.4}{10} = 20.7 \text{ million dollars}
\]

#### Step 2: Calculate each \((x_i - \bar{x})^2\)

\[
(38 - 20.7)^2 = 299.29
\]
\[
(36 - 20.7)^2 = 236.49
\]
\[
(35 - 20.7)^2 = 204.49
\]
\[
(27 - 20.7)^2 = 39.69
\]
\[
(15 - 20.7)^2 = 32.49
\]
\[
(13 - 20.7)^2 = 59.29
\]
\[
(12 - 20.7)^2 = 73.29
\]
\[
(10 - 20.7)^2 = 114.49
\]
\[
(9.6 - 20.7)^2 = 123.21
\]
\[
(8.4 - 20.7)^2 = 150.69
\]

#### Step 3: Calculate the variance

\[
s^2 = \frac{299.29 + 236.49 + 204.49 + 39.69 + 32.49 + 59.29 + 73.29 + 114.49 + 123.21 + 150.69}{9} = 148.24 \text{ million dollars}^2
\]

### 3. Standard Deviation
The standard deviation is the square root of the variance:

\[
s = \sqrt{s^2} = \sqrt{148.24} \approx 12.17 \text{ million dollars}
\]

### Final Answers:
- **Range:** 29.6 million dollars
- **Variance:** 148.24 million dollars\(^2\)
- **Standard Deviation:** 12.17 million dollars

Would you like more details or explanations on these calculations?

### Related Questions:
1. What does a high variance indicate in a data set?
2. How would the range, variance, and standard deviation change if we added more data points?
3. Can standard deviation ever be negative?
4. What is the difference between population variance and sample variance?
5. How does the mean affect the calculation of variance?

### Tip:
Always double-check your data entry when calculating statistics to ensure accuracy in your results.

Learn how to calculate the range, variance, and standard deviation for a set of TV personalities' annual salaries. Includes step-by-step calculations and explanations. Suitable for advanced high school students and those interested in statistics.

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