Question

Using the following symmetric box plots, compare the distributions. (Explain what the data is showing)
[25, 40, 55, 70, 85]
[60, 75, 80, 85, 100]

Answer 1

The first is grades with no homework, and the second is grades with homework

Answer 2

To compare the distributions represented by the two sets of symmetric box plots, let's break down the data points and what they each signify.

### Dataset 1: [25, 40, 55, 70, 85]
- **Minimum**: 25
- **1st Quartile (Q1)**: 40
- **Median (Q2)**: 55
- **3rd Quartile (Q3)**: 70
- **Maximum**: 85

### Dataset 2: [60, 75, 80, 85, 100]
- **Minimum**: 60
- **1st Quartile (Q1)**: 75
- **Median (Q2)**: 80
- **3rd Quartile (Q3)**: 85
- **Maximum**: 100

### Analysis of the Box Plots

1. **Range**:
- The first dataset has a range of 60 (85 - 25).
- The second dataset has a range of 40 (100 - 60).

This shows that the first dataset has a wider spread of values.

2. **Interquartile Range (IQR)**:
- The IQR for the first dataset is Q3 - Q1 = 70 - 40 = 30.
- The IQR for the second dataset is Q3 - Q1 = 85 - 75 = 10.

This indicates that the central 50% of the data in the first dataset is more dispersed compared to the second dataset, which is more concentrated.

3. **Median Comparison**:
- The median of the first dataset is 55, while the median of the second dataset is 80.

This suggests that, on average, the values in the second dataset are higher than those in the first dataset.

4. **Overall Distribution**:
- The first dataset (from 25 to 85) includes lower values, indicating that it may represent a population with lower results (e.g., test scores, heights).
- The second dataset (from 60 to 100) displays a distribution that starts at a higher minimum and peaks higher, which could indicate a generally higher performance or trait.

5. **Skewness and Symmetry**:
- Since both distributions are described as symmetric, we can infer that their variations are evenly distributed around their medians. However, the first dataset has a longer tail on the lower end (reaching down to 25), while the second dataset is limited to higher values with its minimum set at 60.

### Conclusion
In conclusion, the data shows that the first distribution is characterized by lower overall values, a greater range, and greater variability within its middle 50% compared to the second dataset. In contrast, the second dataset reflects higher values, a narrower interquartile range, and a more concentrated distribution around its median. This comparison could inform further analysis depending on the context of the data, such as performance metrics, socioeconomic data, or any other quantitative measures.