Let's analyze the data for Player A and Player B to determine the appropriate measures of variability and the consistency of each player.
### Step 1: Given Data
We have the following scores for each player:
- Player A: [tex]\(2, 1, 3, 8, 2, 1, 4, 3, 1\)[/tex]
- Player B: [tex]\(2, 3, 1, 3, 2, 2, 1, 3, 6\)[/tex]
### Step 2: Calculate the Range for Each Player
The range is the difference between the maximum and minimum values in the dataset.
- For Player A:
- Maximum value = 8
- Minimum value = 1
- Range [tex]\( = 8 - 1 = 7 \)[/tex]
- For Player B:
- Maximum value = 6
- Minimum value = 1
- Range [tex]\( = 6 - 1 = 5 \)[/tex]
### Interpretation of Range:
- Player A has a range of 7.
- Player B has a range of 5.
- The smaller the range, the more consistent the player is regarding range.
- Thus, Player B is more consistent based on the range.
### Step 3: Calculate the Interquartile Range (IQR)
To calculate the IQR, we first find the first quartile (Q1) and third quartile (Q3) for each player and then compute the difference [tex]\(IQR = Q3 - Q1\)[/tex].
- Player A:
- Sorted Data: [tex]\(1, 1, 1, 2, 2, 3, 3, 4, 8\)[/tex]
- [tex]\(Q1\)[/tex]: 25th percentile = 1.5
- [tex]\(Q3\)[/tex]: 75th percentile = 3.5
- [tex]\(IQR = 3.5 - 1.5 = 2.0\)[/tex]
- Player B:
- Sorted Data: [tex]\(1, 1, 2, 2, 2, 3, 3, 3, 6\)[/tex]
- [tex]\(Q1\)[/tex]: 25th percentile = 2.0
- [tex]\(Q3\)[/tex]: 75th percentile = 3.0
- [tex]\(IQR = 3.0 - 2.0 = 1.0\)[/tex]
### Interpretation of IQR:
- Player A has an IQR of 2.0.
- Player B has an IQR of 1.0.
- The smaller the IQR, the more consistent the player is regarding the IQR benchmark.
- Thus, Player B is more consistent based on the IQR.
### Conclusion:
To summarize:
- Player B has a smaller range (5) compared to Player A's range (7).
- Player B also has a smaller IQR (1.0) compared to Player A's IQR (2.0).
Thus, based on both the range and the IQR, Player B is the most consistent.
