Which measures of center change when a value is deleted from these datasets?

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.