**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.

Given that these are the top 10 salaries, do we know anything about the variation of salaries of TV personalities in general? Answer 4 Question 8 Yes.No.

Let's calculate the range, variance, and standard deviation for the given sample data. The data set consists of the following values (in millions of dollars):

\[ 38, 36, 35, 27, 15, 13, 12, 10, 9.6, 8.4 \]

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

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

### 2. **Variance**
Variance is calculated using the following formula for a sample:

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

Where:
- \( x_i \) represents each data point.
- \( \bar{x} \) is the sample mean.
- \( n \) is the number of data points.

First, calculate the sample mean:

\[
\bar{x} = \frac{38 + 36 + 35 + 27 + 15 + 13 + 12 + 10 + 9.6 + 8.4}{10} = \frac{203}{10} = 20.3 \text{ million dollars}
\]

Now, calculate each squared deviation from the mean:

\[
\text{Deviations}^2 = (38-20.3)^2, (36-20.3)^2, \ldots, (8.4-20.3)^2
\]

Summing these squared deviations and dividing by \( n-1 \):

\[
\sum (x_i - \bar{x})^2 = (17.7)^2 + (15.7)^2 + (14.7)^2 + \ldots + (-11.9)^2 = 1655.4
\]

\[
s^2 = \frac{1655.4}{9} \approx 183.93 \text{ million dollars}^2
\]

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

\[
s = \sqrt{183.93} \approx 13.56 \text{ million dollars}
\]

### Summary of Answers:
1. **Range:** 29.6 million dollars
2. **Variance:** 183.93 million dollars²
3. **Standard Deviation:** 13.56 million dollars

### 4. **Variation of Salaries of TV Personalities in General**
Since these are the top 10 salaries, they represent the higher end of the salary distribution. The large range and high variance indicate substantial variation among the top earners, suggesting that there's likely even more variation when considering all TV personalities. Thus, the answer is **Yes**.

Would you like further details or explanations?

Here are five related questions to explore:

1. How does changing one value in the sample affect the standard deviation?
2. What is the significance of variance in understanding data dispersion?
3. How would the range, variance, and standard deviation change if we included median salaries instead of just the top 10?
4. What other measures of dispersion could be useful in analyzing salary data?
5. How does the mean compare to the median in this dataset?

**Tip:** Variance and standard deviation are key indicators of how spread out the data is; the larger they are, the more dispersed the data points are around the mean.

Learn how to calculate the range, variance, and standard deviation for a sample of TV personalities' annual salaries, including detailed step-by-step solutions and interpretations. Understand the dispersion of data and its implications.

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