The following data set includes the ages of the last six U.S. presidents at their inauguration. What is the variance for this data set? 

Round your final answer to one decimal place, if necessary.

**Age of Last Six U.S. Presidents at Inauguration**PresidentAge

George H.W. Bush 

64  

Bill Clinton

46

George W. Bush

54

Barack Obama

47

Donald Trump

70

Joe Biden

78

Provide your answer below:

The population variance, $\sigma^2$σ2​ , is  years.

To calculate the population variance $ \sigma^2 $ for the given data set, we use the formula:

\[
\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
\]

Where:
- $ N $ is the number of observations (in this case, 6).
- $ x_i $ represents each individual value in the data set.
- $ \mu $ is the population mean.

### 1. Find the Population Mean $ \mu $
\[
\mu = \frac{\sum x_i}{N} = \frac{64 + 46 + 54 + 47 + 70 + 78}{6}
\]

\[
\mu = \frac{359}{6} = 59.8333\ldots \approx 59.8
\]

### 2. Find Each $ (x_i - \mu)^2 $
Now, subtract the mean from each value, square the result, and then sum these squared differences.

\[
(x_1 - \mu)^2 = (64 - 59.8)^2 = (4.2)^2 = 17.64
\]
\[
(x_2 - \mu)^2 = (46 - 59.8)^2 = (-13.8)^2 = 190.44
\]
\[
(x_3 - \mu)^2 = (54 - 59.8)^2 = (-5.8)^2 = 33.64
\]
\[
(x_4 - \mu)^2 = (47 - 59.8)^2 = (-12.8)^2 = 163.84
\]
\[
(x_5 - \mu)^2 = (70 - 59.8)^2 = (10.2)^2 = 104.04
\]
\[
(x_6 - \mu)^2 = (78 - 59.8)^2 = (18.2)^2 = 331.24
\]

### 3. Calculate the Sum of Squared Differences
\[
\sum (x_i - \mu)^2 = 17.64 + 190.44 + 33.64 + 163.84 + 104.04 + 331.24 = 840.84
\]

### 4. Calculate the Population Variance $ \sigma^2 $
Finally, divide the sum of squared differences by $ N = 6 $:

\[
\sigma^2 = \frac{840.84}{6} = 140.14
\]

### Final Answer
The population variance $ \sigma^2 $ is **140.1** years (rounded to one decimal place).

---

Would you like to explore more details or have any questions?

Here are some related questions:
1. How does population variance differ from sample variance?
2. What is the standard deviation for this data set?
3. How would the variance change if the age of another president were added?
4. Why is the population variance calculated differently from the sample variance?
5. How does the age distribution of U.S. presidents compare to other world leaders?

**Tip:** When calculating population variance, remember to divide by $ N $ instead of $ N-1 $, which is used for sample variance.

Learn how to calculate the population variance for the ages of the last six U.S. presidents at their inauguration. Detailed step-by-step solution using statistical concepts of variance and mean. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation