Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year: George Lucas ​(5500​), Michael Jordan ​(1700​), Paul McCartney ​(1200​), J. K. Rowling ​(1000​), David Copperfield ​(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?

To find the **range**, **variance**, and **standard deviation** for the given sample data of celebrity net worths, we'll go through the following steps:

### Sample Data
The net worths of the six celebrities (in millions of dollars) are:
- George Lucas: 5500
- Michael Jordan: 1700
- Paul McCartney: 1200
- J. K. Rowling: 1000
- David Copperfield: 1000
- Jerry Seinfeld: 950

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

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

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

#### Step 2: Calculate the Variance
The **variance** measures how far the data points are spread out from the mean. The formula for the sample variance \( s^2 \) is:

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

where:
- \( x_i \) are the data points,
- \( \bar{x} \) is the mean,
- \( n \) is the sample size.

1. **Find the mean \( \bar{x} \):**

\[
\bar{x} = \frac{5500 + 1700 + 1200 + 1000 + 1000 + 950}{6} = \frac{11350}{6} \approx 1891.67
\]

2. **Calculate the squared differences \( (x_i - \bar{x})^2 \) for each data point:**

\[
(5500 - 1891.67)^2 \approx 13027361.11
\]
\[
(1700 - 1891.67)^2 \approx 3672.22
\]
\[
(1200 - 1891.67)^2 \approx 479340.28
\]
\[
(1000 - 1891.67)^2 \approx 794234.72
\]
\[
(1000 - 1891.67)^2 \approx 794234.72
\]
\[
(950 - 1891.67)^2 \approx 884006.94
\]

3. **Sum of squared differences:**

\[
13027361.11 + 3672.22 + 479340.28 + 794234.72 + 794234.72 + 884006.94 \approx 16095849.99
\]

4. **Find the variance:**

\[
s^2 = \frac{16095849.99}{6 - 1} = \frac{16095849.99}{5} \approx 3219169.998
\]

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

\[
s = \sqrt{3219169.998} \approx 1794.21
\]

### Results Summary:
- **Range**: 4550 million dollars
- **Variance**: 3,219,170 (million dollars)^2
- **Standard Deviation**: 1794.21 million dollars

### Interpretation of Results
- **Range**: The large range (4550 million dollars) shows a significant difference between the highest and lowest net worths in this sample.
- **Variance and Standard Deviation**: The high variance and standard deviation reflect a wide spread in the data, suggesting that the wealth of these celebrities varies significantly from the mean.
  
### Precision of Data
- **Precision of Net Worth**: Given that these amounts are in millions of dollars, there could be some level of estimation. The large values imply that the exact figures might not be known with perfect accuracy and could be rounded or estimated. Hence, the data's precision may not be exact to the dollar level but is likely accurate to within a reasonable range.

Do you want more details or have any questions?

### Related Questions:
1. What is the significance of using sample variance instead of population variance here?
2. How would the inclusion of more celebrities impact the standard deviation?
3. Could we use a different measure of central tendency, like the median, and why might it be useful here?
4. What conclusions can we draw about the distribution of celebrity wealth from these measures?
5. How might outliers, like George Lucas's wealth, influence these calculations?

### Tip:
When analyzing data with high variance, consider how extreme values (outliers) can distort measures like the mean and standard deviation.

Here are 6 celebrities with some of the highest net worths​ (in millions of​ dollars) in a recent​ year: George Lucas ​(5500​), Michael Jordan ​(1700​), Paul McCartney ​(1200​), J. K. Rowling ​(1000​), David Copperfield ​(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 how to calculate the range, variance, and standard deviation for the net worths of 6 top celebrities. Learn key concepts in descriptive statistics, such as range (4550 million dollars), variance (3,219,170 million dollars²), and standard deviation (1794.21 million dollars), and what they reveal about the population of wealthy individuals.

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