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The question in the image shows a back-to-back stem-and-leaf diagram comparing the data of two different groups: men and women. The goal is to determine which of the provided answers best describes the data, with a focus on comparing the mean and range of the two distributions.

Let me first extract the relevant data from the stem-and-leaf diagram:

### Stem-and-Leaf Diagram
For women:
- Stems: \(4, 5, 6, 7, 8\)
- Leaves: 2, 4, 5, 8, 0, 1, 5, 6, 6, 3

For men:
- Stems: \(4, 5, 6, 7, 8\)
- Leaves: 5, 8, 2, 7, 2, 4, 4, 7, 7, 4, 1

### Steps to Compare the Distributions:
1. **Calculate the Mean for Women and Men:** The mean is the sum of all the values divided by the number of values.
2. **Find the Range for Women and Men:** The range is the difference between the maximum and minimum values for each group.

Let me compute the mean and range for both groups now.import numpy as np

# Data from the stem-and-leaf diagram
women_data = [42, 44, 45, 48, 50, 51, 55, 56, 56, 63]
men_data = [45, 48, 42, 47, 52, 54, 54, 57, 57, 54, 51]

# Calculate the mean for women and men
mean_women = np.mean(women_data)
mean_men = np.mean(men_data)

# Calculate the range for women and men
range_women = np.max(women_data) - np.min(women_data)
range_men = np.max(men_data) - np.min(men_data)

mean_women, mean_men, range_women, range_menThe analysis of the data yields the following:

- **Mean for Women:** 51.0
- **Mean for Men:** 51.0
- **Range for Women:** 21
- **Range for Men:** 15

### Comparison:
- Both groups have the same mean of 51.0.
- However, the range for women is larger (21) than the range for men (15). This suggests that the women’s data is more spread out compared to the men’s data.

Based on this, we can conclude that the best answer should reflect the fact that the two groups have the same average (mean) but different spreads (range).

Would you like to go over any further details or calculations? Here are some related questions to explore:

1. What conclusions can be drawn from the same mean but different ranges?
2. How does the shape of the distribution impact interpretation in real-world scenarios?
3. How do outliers influence the mean and range in a dataset?
4. Can a larger range indicate more variability in a dataset?
5. How would the median of the datasets compare to the mean?

**Tip:** The mean gives a central tendency, but the range shows how varied the data can be. Always consider both when analyzing data distributions.

The question in the image shows a back-to-back stem-and-leaf diagram comparing the data of two different groups: men and women. The task is to determine which of the provided answers best describes the data, with a focus on comparing the mean and range of the two distributions.

This problem involves comparing the distributions of two datasets—men and women—using a back-to-back stem-and-leaf plot. The task includes calculating and analyzing the mean and range for both groups to determine the best descriptive conclusion. Ideal for students in Grades 6-8.

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