Question
Which measures of center change when a value is deleted from these datasets?
Dataset 1: (0,1,0,1,3,2,0,1,0,1,0,2)
Dataset 2: (0,1,1,3,2,0,1,0,1,0,2)
Dataset 1: (0,1,0,1,3,2,0,1,0,1,0,2)
Dataset 2: (0,1,1,3,2,0,1,0,1,0,2)
Answers
GPT-4o mini
To analyze which measures of center change when a value is deleted from the datasets, we can consider the mean (average), median, and mode. Let's calculate these measures for both datasets.
### Dataset 1: (0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 0, 2)
1. **Mean**:
- Sum: \( 0 + 1 + 0 + 1 + 3 + 2 + 0 + 1 + 0 + 1 + 0 + 2 = 11 \)
- Count: 12
- Mean: \( \frac{11}{12} \approx 0.9167 \)
2. **Median**:
- Ordered: \( 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3 \)
- Middle values (6th and 7th): \( 1 \)
- Median: \( 1 \)
3. **Mode**:
- Most frequent value: \( 0 \) and \( 1 \)
- Mode: \( 0 \) and \( 1 \) (bimodal)
### Dataset 2: (0, 1, 1, 3, 2, 0, 1, 0, 1, 0, 2)
1. **Mean**:
- Sum: \( 0 + 1 + 1 + 3 + 2 + 0 + 1 + 0 + 1 + 0 + 2 = 11 \)
- Count: 11
- Mean: \( \frac{11}{11} = 1 \)
2. **Median**:
- Ordered: \( 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3 \)
- Middle value: \( 1 \) (6th value)
- Median: \( 1 \)
3. **Mode**:
- Most frequent value: \( 1 \)
- Mode: \( 1 \)
### Changes When One Value is Deleted
Let's analyze the effect of deleting one value from each dataset.
**Deleting a 0 from Dataset 1**:
- New Dataset: (1, 1, 3, 2, 0, 1, 0, 1, 0, 2)
- New Mean: \( \frac{11 - 0}{11} = \frac{11}{11} = 1 \)
- New Median: Ordered: (0, 0, 0, 1, 1, 1, 1, 1, 2, 3) - Median is \( 1 \)
- New Mode: Still \( 1 \) (but no longer \( 0 \))
**Deleting a 1 from Dataset 2**:
- New Dataset: (0, 1, 3, 2, 0, 1, 0, 1, 0, 2)
- New Mean: \( \frac{11 - 1}{10} = \frac{10}{10} = 1 \)
- New Median: Ordered: (0, 0, 0, 0, 1, 1, 2, 3) - New Median: \( 1 \)
- New Mode: \( 0 \) (but no longer \( 1 \))
### Summary of Change:
- **Dataset 1**:
- Mean, median, and mode can change.
- **Dataset 2**:
- Mean and median remain the same (1), but the mode changes.
In conclusion, all measures of center can change with the deletion of a value, particularly when considering which specific values are removed and their frequency in the datasets.
### Dataset 1: (0, 1, 0, 1, 3, 2, 0, 1, 0, 1, 0, 2)
1. **Mean**:
- Sum: \( 0 + 1 + 0 + 1 + 3 + 2 + 0 + 1 + 0 + 1 + 0 + 2 = 11 \)
- Count: 12
- Mean: \( \frac{11}{12} \approx 0.9167 \)
2. **Median**:
- Ordered: \( 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3 \)
- Middle values (6th and 7th): \( 1 \)
- Median: \( 1 \)
3. **Mode**:
- Most frequent value: \( 0 \) and \( 1 \)
- Mode: \( 0 \) and \( 1 \) (bimodal)
### Dataset 2: (0, 1, 1, 3, 2, 0, 1, 0, 1, 0, 2)
1. **Mean**:
- Sum: \( 0 + 1 + 1 + 3 + 2 + 0 + 1 + 0 + 1 + 0 + 2 = 11 \)
- Count: 11
- Mean: \( \frac{11}{11} = 1 \)
2. **Median**:
- Ordered: \( 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 3 \)
- Middle value: \( 1 \) (6th value)
- Median: \( 1 \)
3. **Mode**:
- Most frequent value: \( 1 \)
- Mode: \( 1 \)
### Changes When One Value is Deleted
Let's analyze the effect of deleting one value from each dataset.
**Deleting a 0 from Dataset 1**:
- New Dataset: (1, 1, 3, 2, 0, 1, 0, 1, 0, 2)
- New Mean: \( \frac{11 - 0}{11} = \frac{11}{11} = 1 \)
- New Median: Ordered: (0, 0, 0, 1, 1, 1, 1, 1, 2, 3) - Median is \( 1 \)
- New Mode: Still \( 1 \) (but no longer \( 0 \))
**Deleting a 1 from Dataset 2**:
- New Dataset: (0, 1, 3, 2, 0, 1, 0, 1, 0, 2)
- New Mean: \( \frac{11 - 1}{10} = \frac{10}{10} = 1 \)
- New Median: Ordered: (0, 0, 0, 0, 1, 1, 2, 3) - New Median: \( 1 \)
- New Mode: \( 0 \) (but no longer \( 1 \))
### Summary of Change:
- **Dataset 1**:
- Mean, median, and mode can change.
- **Dataset 2**:
- Mean and median remain the same (1), but the mode changes.
In conclusion, all measures of center can change with the deletion of a value, particularly when considering which specific values are removed and their frequency in the datasets.