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
 

To calculate the population variance, we follow these steps:

1. **Calculate the mean (average) of the data set**:
   \[
   \text{Mean} = \frac{\sum X_i}{N}
   \]
   where \( X_i \) are the values in the data set and \( N \) is the number of values.

2. **Calculate the squared differences from the mean for each data point**:
   \[
   (X_i - \text{Mean})^2
   \]

3. **Find the average of these squared differences** (this is the population variance):
   \[
   \text{Variance} = \frac{\sum (X_i - \text{Mean})^2}{N}
   \]

Let's compute it step by step.

### Step 1: Calculate the mean
The values are: 102, 97, 70, 69, 68, 68, 60, 59.

\[
\text{Mean} = \frac{102 + 97 + 70 + 69 + 68 + 68 + 60 + 59}{8} = \frac{593}{8} = 74.125
\]

### Step 2: Calculate the squared differences from the mean
\[
\begin{align*}
(102 - 74.125)^2 & = 774.015625 \\
(97 - 74.125)^2 & = 522.015625 \\
(70 - 74.125)^2 & = 17.015625 \\
(69 - 74.125)^2 & = 26.015625 \\
(68 - 74.125)^2 & = 37.015625 \\
(68 - 74.125)^2 & = 37.015625 \\
(60 - 74.125)^2 & = 201.015625 \\
(59 - 74.125)^2 & = 229.015625 \\
\end{align*}
\]

### Step 3: Calculate the population variance
\[
\text{Variance} = \frac{774.015625 + 522.015625 + 17.015625 + 26.015625 + 37.015625 + 37.015625 + 201.015625 + 229.015625}{8} = \frac{1843.125}{8} = 230.4
\]

So, the population variance is **230.4**.

Would you like more details on any part of this process? Here are some related questions that might interest you:

1. How is population variance different from sample variance?
2. What are some real-world applications of variance in financial analysis?
3. How does standard deviation relate to variance?
4. What is the significance of variance in data analysis?
5. Can you explain how variance is used in the context of probability distributions?

**Tip:** Variance provides a measure of how much the values in a data set differ from the mean. It is particularly useful in finance for assessing risk.

Learn how to calculate the population variance for Forbes' top 8 richest Americans, including Warren Buffett, Steve Ballmer, Michael Bloomberg, and others. Understand the steps involving mean calculation and squared differences from the mean. Round the answer to the nearest tenth, if necessary.

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