Each year, Forbes, magazine compiles a lists of the 400 richest Americans. The table below includes the top 8 on the list. Determine the population variance. Round your answer to the nearest tenth, if necessary.

Billionaires' Club
Person	
Wealth

(in billions of dollars)

Warren Buffett	 
102
 
Steve Ballmer	 
97
 
Michael Bloomberg 	 
70
 
Jim Walton	 
69
 
Alice Walton	 
68
 
Rob Walton	 
68
 
Phil Knight & Family	 
60
 
McKenzie Scott	 
59
 
Provide your answer below:

The population variance,  $\sigma^2$ , is 
 billion dollars.

To determine the population variance ($\sigma^2$) of the wealth of the 8 billionaires listed, we can follow these steps:

1. **Calculate the mean ($\mu$)** of the wealth values.
2. **Subtract the mean** from each wealth value and **square the result**.
3. **Sum** all the squared differences.
4. **Divide** by the number of data points (since it's the population variance).

Let's go through each step.

### Step 1: Calculate the Mean ($\mu$)
The wealth values in billions of dollars are: 102, 97, 70, 69, 68, 68, 60, 59.

\[
\mu = \frac{102 + 97 + 70 + 69 + 68 + 68 + 60 + 59}{8} 
\]

\[
\mu = \frac{593}{8} = 74.125
\]

### Step 2: Calculate the Squared Differences
Next, we'll subtract the mean from each value, square the result, and sum these squared differences.

\[
\text{Squared Differences} = (102 - 74.125)^2, (97 - 74.125)^2, (70 - 74.125)^2, \ldots, (59 - 74.125)^2
\]

\[
(102 - 74.125)^2 = (27.875)^2 = 777.140625
\]

\[
(97 - 74.125)^2 = (22.875)^2 = 523.140625
\]

\[
(70 - 74.125)^2 = (-4.125)^2 = 17.015625
\]

\[
(69 - 74.125)^2 = (-5.125)^2 = 26.265625
\]

\[
(68 - 74.125)^2 = (-6.125)^2 = 37.515625
\]

\[
(68 - 74.125)^2 = (-6.125)^2 = 37.515625
\]

\[
(60 - 74.125)^2 = (-14.125)^2 = 199.515625
\]

\[
(59 - 74.125)^2 = (-15.125)^2 = 228.765625
\]

### Step 3: Sum the Squared Differences

\[
\text{Sum} = 777.140625 + 523.140625 + 17.015625 + 26.265625 + 37.515625 + 37.515625 + 199.515625 + 228.765625 = 1846.875
\]

### Step 4: Calculate the Population Variance

\[
\sigma^2 = \frac{\text{Sum of Squared Differences}}{n} = \frac{1846.875}{8} = 230.859375
\]

Rounded to the nearest tenth:

\[
\sigma^2 \approx 230.9 \text{ billion dollars}^2
\]

So, the population variance, $\sigma^2$, is **230.9 billion dollars squared**.

Would you like any further details or explanations?

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Here are five related questions to expand your understanding:
1. How is population variance different from sample variance?
2. What would the standard deviation be in this scenario?
3. How does variance help in understanding the distribution of data?
4. If another billionaire were added to this list, how would the variance change?
5. What are some potential outliers in this dataset?

**Tip:** Always ensure that you clearly distinguish between population variance and sample variance when working with data, as the formulas differ slightly.

Learn how to calculate the population variance of the wealth of the top 8 billionaires listed by Forbes. Follow a step-by-step solution to find $ \sigma^2 $, rounding to the nearest tenth where necessary. Understand the statistical concepts of variance and its application in data analysis.

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