Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year:

George LucasGeorge Lucas

​(55005500​),

Steven SpielbergSteven Spielberg

​(37003700​),

Oprah WinfreyOprah Winfrey

​(32003200​),

Paul McCartneyPaul McCartney

​(12001200​),

J. K. RowlingJ. K. Rowling

​(10001000​),

and

Jerry SeinfeldJerry Seinfeld

​(950950​)

.

Find the​ range, variance, and standard deviation for the sample data. What do the results tell us about the population of all​ celebrities? Based on the nature of the​ amounts, what can be inferred about their​ precision?

Let's calculate the **range**, **variance**, and **standard deviation** for the given data.

### Step 1: List the data
The net worths of the celebrities in millions of dollars are:

- George Lucas: 5500
- Steven Spielberg: 3700
- Oprah Winfrey: 3200
- Paul McCartney: 1200
- J.K. Rowling: 1000
- Jerry Seinfeld: 950

### Step 2: Calculate the **range**
The range is the difference between the maximum and minimum values.

\[
\text{Range} = \text{Max value} - \text{Min value}
\]
\[
\text{Range} = 5500 - 950 = 4550
\]

### Step 3: Calculate the **variance** (sample variance)
The formula for the sample variance \(s^2\) is:

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

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

#### Mean:
\[
\bar{x} = \frac{5500 + 3700 + 3200 + 1200 + 1000 + 950}{6} = \frac{15550}{6} \approx 2591.67
\]

#### Deviations from the mean and squared deviations:
- \((5500 - 2591.67)^2 = 8470173.61\)
- \((3700 - 2591.67)^2 = 1237340.28\)
- \((3200 - 2591.67)^2 = 369506.94\)
- \((1200 - 2591.67)^2 = 1935067.28\)
- \((1000 - 2591.67)^2 = 2567340.28\)
- \((950 - 2591.67)^2 = 2696734.03\)

#### Sum of squared deviations:
\[
8470173.61 + 1237340.28 + 369506.94 + 1935067.28 + 2567340.28 + 2696734.03 = 17294162.42
\]

#### Sample variance:
\[
s^2 = \frac{17294162.42}{6-1} = \frac{17294162.42}{5} = 3458832.48
\]

### Step 4: Calculate the **standard deviation**
The standard deviation \(s\) is the square root of the variance:

\[
s = \sqrt{3458832.48} \approx 1860.37
\]

### Summary of results:
- **Range**: 4550 million dollars
- **Variance**: 3,458,832.48 (million dollars squared)
- **Standard Deviation**: 1,860.37 million dollars

### Interpretation:
1. **Range**: The large range of 4550 million dollars indicates a significant disparity in the net worths among these celebrities, with George Lucas being much wealthier than Jerry Seinfeld.
2. **Variance and Standard Deviation**: Both measures of variability are quite large. The high standard deviation of 1860.37 million suggests that the net worth values are widely spread out from the mean.
3. **Precision**: Given that these amounts are estimates, the precision of these figures might not be exact down to the dollar. These numbers are typically rounded, so the exact precision is difficult to guarantee, especially for extremely high net worths like these.

### Relative Questions:
1. What are the potential outliers in this dataset?
2. How would the results change if we included additional celebrities?
3. How does the variance help in understanding the spread of data in this case?
4. What is the significance of having such a large standard deviation in net worth?
5. Can the range be misleading without looking at other measures of spread like standard deviation?

### Tip:
When interpreting financial data, both the range and standard deviation give a clearer picture of wealth distribution, but remember that high outliers can heavily influence these values.

Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year:

George Lucas (5500), Steven Spielberg (3700), Oprah Winfrey (3200), Paul McCartney (1200), J. K. Rowling (1000), and Jerry Seinfeld (950). Find the​ range, variance, and standard deviation for the sample data. What do the results tell us about the population of all​ celebrities? Based on the nature of the​ amounts, what can be inferred about their​ precision?

Explore the analysis of celebrity net worths including George Lucas, Oprah Winfrey, and others. Learn how to calculate the range (4550 million dollars), variance (3,458,832.48), and standard deviation (1,860.37 million dollars) for sample data. Understand wealth distribution and variability in high-net-worth populations.

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