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?
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Part 1
The range is ​$
  
4550 million.
​(Round to the nearest integer as​ needed.) 
Part 2
The variance is 
  
enter your response here million dollars squared.
​(Round to the nearest integer as​ needed.) 

To solve the problem, let's go step by step to compute the **range**, **variance**, and **standard deviation** for the given sample data.

### Given Data:
Net worths of the celebrities (in millions):
- George Lucas: 5500
- Michael Jordan: 1700
- Paul McCartney: 1200
- J. K. Rowling: 1000
- David Copperfield: 1000
- Jerry Seinfeld: 950

### Part 1: **Range**
The range is calculated as the difference between the highest and lowest values in the dataset.

\[
\text{Range} = \text{Highest value} - \text{Lowest value}
\]

In this case, the highest value is George Lucas with 5500 million, and the lowest value is Jerry Seinfeld with 950 million.

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

So, the **range** is **4550 million**.

### Part 2: **Variance and Standard Deviation**

To calculate the variance, we follow these steps:

1. Find the mean (average) of the data.
2. Find the squared differences between each data point and the mean.
3. Calculate the average of those squared differences to get the variance.
4. Take the square root of the variance to get the standard deviation.

#### Step 1: Calculate the Mean

The formula for the mean $ \mu $ is:

\[
\mu = \frac{\text{Sum of all values}}{\text{Number of values}}
\]

Sum of all values:

\[
5500 + 1700 + 1200 + 1000 + 1000 + 950 = 11350 \, \text{million}
\]

Number of values: 6

\[
\mu = \frac{11350}{6} = 1891.67 \, \text{million}
\]

#### Step 2: Calculate the Squared Differences

For each data point, subtract the mean and then square the result:

\[
(5500 - 1891.67)^2 = 13007123.61
\]
\[
(1700 - 1891.67)^2 = 36722.22
\]
\[
(1200 - 1891.67)^2 = 477777.78
\]
\[
(1000 - 1891.67)^2 = 795694.44
\]
\[
(1000 - 1891.67)^2 = 795694.44
\]
\[
(950 - 1891.67)^2 = 885069.44
\]

#### Step 3: Calculate the Variance

The variance $ \sigma^2 $ for a sample is the average of these squared differences:

\[
\sigma^2 = \frac{13007123.61 + 36722.22 + 477777.78 + 795694.44 + 795694.44 + 885069.44}{6 - 1}
\]

Now let's compute this.

### Variance:
The sample variance is approximately **3,202,417 million dollars squared** (rounded to the nearest integer).

### Standard Deviation:
The standard deviation is approximately **1,790 million dollars** (rounded to the nearest integer).

### Interpretation of the Results:
- The **range** of the data, 4550 million, indicates a significant disparity between the highest and lowest net worths among these celebrities.
- The **variance** and **standard deviation** suggest there is a high level of variability in the net worths. Some celebrities have significantly higher net worths (like George Lucas), contributing to this wide spread.
- In terms of **precision**, the values reflect approximations, and given the vast differences in net worths, the precision might not need to be too fine-tuned. The wealth figures are likely rounded or estimates, especially when dealing with large amounts.

---

Would you like more details or have any questions?

Here are 5 relative questions:
1. How would the variance change if we removed George Lucas from the dataset?
2. Why is it important to distinguish between sample variance and population variance?
3. What is the impact of extreme values (like George Lucas) on standard deviation?
4. How would we interpret these statistics if we had data on all celebrities?
5. What other statistical measures could help better understand the distribution of net worths?

**Tip:** Always consider the impact of outliers when interpreting variance 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?

Discover how to calculate the range, variance, and standard deviation for the net worths of top celebrities like George Lucas and Michael Jordan. This step-by-step solution covers essential statistical concepts, including the range formula, variance, and standard deviation, suitable for students in Grades 10-12.

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