To determine the best value that represents all the measures of center (mean, median, and mode) for the given data set (48, 12, 11, 45, 48, 48, 43, 32), we first need to calculate each measure.
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Mean: The mean is calculated by summing all the values and dividing by the number of values.
\[ \text{Mean} = \frac{48 + 12 + 11 + 45 + 48 + 48 + 43 + 32}{8} = \frac{ 48 + 12 + 11 + 45 + 48 + 48 + 43 + 32}{8} = \frac{ 243}{8} = 30.375 \]
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Median: The median is the middle value when the numbers are arranged in order. First, arrange the data set:
\[ 11, 12, 32, 43, 45, 48, 48, 48 \]
Since there are 8 values (an even number), the median is the average of the 4th and 5th values:
\[ \text{Median} = \frac{43 + 45}{2} = \frac{88}{2} = 44 \]
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Mode: The mode is the value that appears most frequently in the data set. In this case, 48 appears three times, which is more frequent than any other number.
\[ \text{Mode} = 48 \]
Now, summarizing the measures of center we've calculated:
- Mean = 30.375
- Median = 44
- Mode = 48
The value that best represents all the measures of center is the Mode (48), as it is the most frequent value in the data set and is significantly higher than the mean and median values.
Hence, the best value that represents all the measures of center is 48.