To find the mean, median, mode(s), and range of the data set \( 16, 22, 14, 12, 20, 19, 14, 11 \), we will perform the following calculations:
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Mean: \[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \] \[ \text{Sum} = 16 + 22 + 14 + 12 + 20 + 19 + 14 + 11 = 128 \] \[ \text{Number of values} = 8 \] \[ \text{Mean} = \frac{128}{8} = 16 \]
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Median: To find the median, we need to sort the data in ascending order: \[ 11, 12, 14, 14, 16, 19, 20, 22 \] Since there are 8 numbers (an even count), the median will be the average of the 4th and 5th numbers: \[ \text{Median} = \frac{14 + 16}{2} = \frac{30}{2} = 15 \]
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Mode(s): The mode is the number that appears most frequently in the data set. In the sorted data \( 11, 12, 14, 14, 16, 19, 20, 22 \), the number 14 appears twice, while all others appear only once. Therefore, the mode is: \[ \text{Mode} = 14 \]
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Range: The range is the difference between the maximum and minimum values in the data set. \[ \text{Max} = 22, \quad \text{Min} = 11 \] \[ \text{Range} = \text{Max} - \text{Min} = 22 - 11 = 11 \]
Summary of Results:
- Mean: \( 16 \)
- Median: \( 15 \)
- Mode: \( 14 \)
- Range: \( 11 \)