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Explain the centering property of the mean, referencing the provided table to demonstrate.
Mathematics
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This table illustrates the Centering Property of the Mean.
- The Property: The text above the table explains that if you calculate the difference (or deviation) between each individual value in a dataset and the mean of that dataset, and then sum up all those differences, the total will always be zero. This unique property holds true only for the mean, not for any other value you might choose.
- How the Table Demonstrates It:
- The table assumes that the mean of a dataset is 12.
- Each entry in the table (e.g., ) calculates the deviation of a specific data point (like 0, 2, 3, etc.) from this assumed mean of 12.
- The first column lists data points less than 12, resulting in negative deviations, which sum to -71.
- The second column lists data points around 12, also resulting in negative or zero deviations, summing to -17.
- The third column lists data points greater than or equal to 12, resulting in positive or zero deviations, summing to 88.
- The Result: When you add the sums of all these deviations together: (-71) + (-17) + (88) = -88 + 88 = 0. This confirms the property that the sum of deviations from the mean is always zero.
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