The U.S. Census Bureau frequently conducts nationwide surveys on the characteristics of U.S. households. Suppose the following data are on the number of people per household for a sample of 20 households. Find the five-number summary for this data set by hand. Report the five-number summary in the following order: **Min, Q1, Median (Q2), Q3, Max**

Number of

People

Frequency

1

4

2

7

3

4

4

2

5

1

6

1

7

1

To find the five-number summary (Min, Q1, Median, Q3, Max) for this dataset, we'll go through the following steps:

1. **Organize the data** by listing each number of people with its corresponding frequency.
2. **Find the five-number summary**, including the minimum, the quartiles (Q1, Median, Q3), and the maximum.

### Step 1: Organizing the Data
The frequency table gives us the number of households corresponding to each number of people. We'll list the numbers accordingly:

- 1 person: 4 households → \( 1, 1, 1, 1 \)
- 2 people: 7 households → \( 2, 2, 2, 2, 2, 2, 2 \)
- 3 people: 4 households → \( 3, 3, 3, 3 \)
- 4 people: 2 households → \( 4, 4 \)
- 5 people: 1 household → \( 5 \)
- 6 people: 1 household → \( 6 \)
- 7 people: 1 household → \( 7 \)

Now, combining them into one ordered list:
\[
1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 7
\]

### Step 2: Finding the Five-Number Summary
- **Minimum (Min)**: The smallest number in the dataset is \( \mathbf{1} \).
  
- **Maximum (Max)**: The largest number in the dataset is \( \mathbf{7} \).

- **Median (Q2)**: Since we have 20 data points, the median is the average of the 10th and 11th values in the ordered list:
  \[
  Q2 = \frac{2 + 2}{2} = 2
  \]

- **First Quartile (Q1)**: The first quartile is the median of the lower half of the data (the first 10 values). The 5th and 6th values are both 2, so:
  \[
  Q1 = \frac{2 + 2}{2} = 2
  \]

- **Third Quartile (Q3)**: The third quartile is the median of the upper half of the data (the last 10 values). The 15th and 16th values are 3 and 4, so:
  \[
  Q3 = \frac{3 + 4}{2} = 3.5
  \]

### Final Five-Number Summary
- **Min**: 1
- **Q1**: 2
- **Median (Q2)**: 2
- **Q3**: 3.5
- **Max**: 7

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Would you like more details or have any questions? 

Here are 5 related questions for practice:
1. How do you calculate the interquartile range (IQR) using the five-number summary?
2. If the dataset had an extra household with 8 people, how would the five-number summary change?
3. How would you interpret the third quartile (Q3) in this context?
4. What does the five-number summary tell us about the spread of household sizes in this sample?
5. How could you use this five-number summary to identify potential outliers?

**Tip**: Quartiles divide your data into four equal parts. Q1 is the 25th percentile, the median (Q2) is the 50th percentile, and Q3 is the 75th percentile.

The U.S. Census Bureau frequently conducts nationwide surveys on the characteristics of U.S. households. Suppose the following data are on the number of people per household for a sample of 20 households. Find the five-number summary for this data set by hand. Report the five-number summary in the following order: Min, Q1, Median (Q2), Q3, Max. Number of People: 1 (4 households), 2 (7 households), 3 (4 households), 4 (2 households), 5 (1 household), 6 (1 household), 7 (1 household).

Learn how to calculate the five-number summary for a U.S. household data sample. The data includes the number of people per household from a sample of 20 households. This problem covers key concepts in descriptive statistics and is suitable for students in Grades 6-8.

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