Question
Darius and Barb are playing a video game in which the higher score wins the game. Their scores are shown below.
Darius’s scores: 96, 54, 120, 87, 123
Barb’s scores: 92, 98, 96, 94, 110
Barb says that she is the winner. Darius says that it is a tie. Who is correct?
Barb is correct if both the mean and median scores are considered.
Barb is correct if only the median score is considered.
Darius is correct if both the mean and median scores are considered.
Darius is correct if only the median score is considered.
Darius’s scores: 96, 54, 120, 87, 123
Barb’s scores: 92, 98, 96, 94, 110
Barb says that she is the winner. Darius says that it is a tie. Who is correct?
Barb is correct if both the mean and median scores are considered.
Barb is correct if only the median score is considered.
Darius is correct if both the mean and median scores are considered.
Darius is correct if only the median score is considered.
Answers
GPT-4o mini
To determine who is correct, we need to calculate the mean and median of both Darius's and Barb's scores and compare them.
**Darius's scores: 96, 54, 120, 87, 123**
1. **Mean**:
\[
\text{Mean} = \frac{96 + 54 + 120 + 87 + 123}{5} = \frac{480}{5} = 96
\]
2. **Median**:
To find the median, we first sort the scores: 54, 87, 96, 120, 123.
Since there are 5 scores (an odd number), the median is the middle score:
\[
\text{Median} = 96
\]
---
**Barb's scores: 92, 98, 96, 94, 110**
1. **Mean**:
\[
\text{Mean} = \frac{92 + 98 + 96 + 94 + 110}{5} = \frac{490}{5} = 98
\]
2. **Median**:
To find the median, we sort the scores: 92, 94, 96, 98, 110.
The median is the middle score:
\[
\text{Median} = 96
\]
---
**Comparison:**
- **Means**:
- Darius: 96
- Barb: 98
Therefore, Barb has a higher mean score.
- **Medians**:
- Darius: 96
- Barb: 96
Therefore, both players have the same median score.
**Conclusion:**
- For the mean, Barb wins (98 > 96).
- For the median, it's a tie (96 = 96).
Since the problem states that "Barb is correct if both the mean and median scores are considered" and "Darius is correct if only the median score is considered," it follows that **Barb is ultimately the winner** because her mean score is higher.
**Darius's scores: 96, 54, 120, 87, 123**
1. **Mean**:
\[
\text{Mean} = \frac{96 + 54 + 120 + 87 + 123}{5} = \frac{480}{5} = 96
\]
2. **Median**:
To find the median, we first sort the scores: 54, 87, 96, 120, 123.
Since there are 5 scores (an odd number), the median is the middle score:
\[
\text{Median} = 96
\]
---
**Barb's scores: 92, 98, 96, 94, 110**
1. **Mean**:
\[
\text{Mean} = \frac{92 + 98 + 96 + 94 + 110}{5} = \frac{490}{5} = 98
\]
2. **Median**:
To find the median, we sort the scores: 92, 94, 96, 98, 110.
The median is the middle score:
\[
\text{Median} = 96
\]
---
**Comparison:**
- **Means**:
- Darius: 96
- Barb: 98
Therefore, Barb has a higher mean score.
- **Medians**:
- Darius: 96
- Barb: 96
Therefore, both players have the same median score.
**Conclusion:**
- For the mean, Barb wins (98 > 96).
- For the median, it's a tie (96 = 96).
Since the problem states that "Barb is correct if both the mean and median scores are considered" and "Darius is correct if only the median score is considered," it follows that **Barb is ultimately the winner** because her mean score is higher.