Answer :
To analyze the home run data for Babe Ruth and Hank Aaron, we want to summarize the data using key statistical measures. Here are the key steps and results for summarizing the data:
### Babe Ruth's Home Run Statistics:
1. Data: 0, 4, 3, 2, 11, 29, 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22, 6
2. Number of Seasons Played: 22
Let's look at the following statistical measures:
- Mean (Average): The mean number of home runs per year.
- [tex]\(\text{Mean} \approx 32.45\)[/tex]
- Median: The middle value when the home run counts are ordered.
- [tex]\(\text{Median} = 38.0\)[/tex]
- Standard Deviation: A measure of the amount of variation or dispersion of the home run counts.
- [tex]\(\text{Standard Deviation} \approx 19.75\)[/tex]
### Hank Aaron's Home Run Statistics:
1. Data: 13, 27, 26, 44, 30, 39, 40, 34, 45, 44, 24, 32, 44, 39, 29, 44, 38, 47, 34, 40, 20, 12, 10
2. Number of Seasons Played: 23
The statistical measures for Hank Aaron are:
- Mean (Average): The mean number of home runs per year.
- [tex]\(\text{Mean} \approx 32.83\)[/tex]
- Median: The middle value when the home run counts are ordered.
- [tex]\(\text{Median} = 34.0\)[/tex]
- Standard Deviation: A measure of the amount of variation or dispersion of the home run counts.
- [tex]\(\text{Standard Deviation} \approx 10.94\)[/tex]
### Comparative Analysis:
1. Mean Comparison: Both players have very similar average home runs per season, with Hank Aaron slightly higher at approximately 32.83 compared to Babe Ruth's 32.45.
2. Median Comparison: The median home runs also reflect close performance, with Hank Aaron at 34.0 and Babe Ruth at 38.0.
3. Standard Deviation Comparison: Babe Ruth's home runs show more variability (standard deviation of about 19.75) compared to Hank Aaron (standard deviation of about 10.94). This indicates Babe Ruth had more fluctuations in his yearly home run counts.
### Best Representation of Data:
A Box Plot would be an excellent choice to represent this data visually. It provides a clear summary of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values. It also displays outliers and the spread of the data, making it easy to compare the distributions and variabilities between Babe Ruth and Hank Aaron.
### Box Plot Explanation:
- Babe Ruth's Box Plot: It would show a larger spread, reflecting the higher standard deviation, with some extreme outliers representing his peak home run seasons.
- Hank Aaron's Box Plot: It would show a more concentrated distribution with tighter spread due to the lower standard deviation.
In summary:
1. Calculate mean, median, and standard deviation to understand central tendency and variability.
2. Use a box plot to visually compare the two distributions.
3. Note the close means and medians but differing variabilities and interpret these in the context of player performance.
This method provides a comprehensive and insightful comparison of the home run capabilities of Babe Ruth and Hank Aaron.
### Babe Ruth's Home Run Statistics:
1. Data: 0, 4, 3, 2, 11, 29, 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22, 6
2. Number of Seasons Played: 22
Let's look at the following statistical measures:
- Mean (Average): The mean number of home runs per year.
- [tex]\(\text{Mean} \approx 32.45\)[/tex]
- Median: The middle value when the home run counts are ordered.
- [tex]\(\text{Median} = 38.0\)[/tex]
- Standard Deviation: A measure of the amount of variation or dispersion of the home run counts.
- [tex]\(\text{Standard Deviation} \approx 19.75\)[/tex]
### Hank Aaron's Home Run Statistics:
1. Data: 13, 27, 26, 44, 30, 39, 40, 34, 45, 44, 24, 32, 44, 39, 29, 44, 38, 47, 34, 40, 20, 12, 10
2. Number of Seasons Played: 23
The statistical measures for Hank Aaron are:
- Mean (Average): The mean number of home runs per year.
- [tex]\(\text{Mean} \approx 32.83\)[/tex]
- Median: The middle value when the home run counts are ordered.
- [tex]\(\text{Median} = 34.0\)[/tex]
- Standard Deviation: A measure of the amount of variation or dispersion of the home run counts.
- [tex]\(\text{Standard Deviation} \approx 10.94\)[/tex]
### Comparative Analysis:
1. Mean Comparison: Both players have very similar average home runs per season, with Hank Aaron slightly higher at approximately 32.83 compared to Babe Ruth's 32.45.
2. Median Comparison: The median home runs also reflect close performance, with Hank Aaron at 34.0 and Babe Ruth at 38.0.
3. Standard Deviation Comparison: Babe Ruth's home runs show more variability (standard deviation of about 19.75) compared to Hank Aaron (standard deviation of about 10.94). This indicates Babe Ruth had more fluctuations in his yearly home run counts.
### Best Representation of Data:
A Box Plot would be an excellent choice to represent this data visually. It provides a clear summary of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values. It also displays outliers and the spread of the data, making it easy to compare the distributions and variabilities between Babe Ruth and Hank Aaron.
### Box Plot Explanation:
- Babe Ruth's Box Plot: It would show a larger spread, reflecting the higher standard deviation, with some extreme outliers representing his peak home run seasons.
- Hank Aaron's Box Plot: It would show a more concentrated distribution with tighter spread due to the lower standard deviation.
In summary:
1. Calculate mean, median, and standard deviation to understand central tendency and variability.
2. Use a box plot to visually compare the two distributions.
3. Note the close means and medians but differing variabilities and interpret these in the context of player performance.
This method provides a comprehensive and insightful comparison of the home run capabilities of Babe Ruth and Hank Aaron.
