To find the measures of center for the dataset: 20, 30, 32, 16, 31, 32, 13, 20, 28, 32, 15, 18, 20, 21, 32, we will calculate the mean, median, and mode.
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Mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} = \frac{20 + 30 + 32 + 16 + 31 + 32 + 13 + 20 + 28 + 32 + 15 + 18 + 20 + 21 + 32}{15} \] \[ \text{Sum} = 20 + 30 + 32 + 16 + 31 + 32 + 13 + 20 + 28 + 32 + 15 + 18 + 20 + 21 + 32 = 414 \] \[ \text{Mean} = \frac{414}{15} = 27.6 \]
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Median: First, we sort the numbers: \[ 13, 15, 16, 18, 20, 20, 20, 21, 28, 30, 31, 32, 32, 32, 32 \] The median is the middle number. Since there are 15 numbers (odd), the median is the 8th number: \[ \text{Median} = 21 \]
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Mode: The mode is the number that appears most frequently. In this dataset, 32 appears 5 times, which is more than any other number: \[ \text{Mode} = 32 \]
So, the measures of center are:
- Mean: 27.6
- Median: 21
- Mode: 32
The final response is: mean: 27.6; median: 21; mode: 32