The Mean: A Measure of Central Tendency

C

For the mean, the sum of the deviations from the average is always zero because the mean is the average.

The correct answer to this question is C. Mean.

The measure of central tendency refers to the value that represents the center or typical value of a dataset. There are several measures of central tendency, including the mean, median, mode, and geometric mean.

To understand why the sum of the deviations from the average will always be zero for the mean, let's first define some terms:

• Deviation: The deviation of each value from the average is calculated by subtracting the average from each individual value.
• Sum of deviations: This refers to adding up all the deviations of each value from the average.

Now, let's consider why the sum of the deviations is zero for the mean:

1. Mean: The mean is calculated by summing up all the values in a dataset and dividing the sum by the number of values. It represents the arithmetic average of the dataset.

When we calculate the deviation of each value from the mean, some values will be greater than the mean, while others will be smaller. However, since the mean is the sum of all values divided by the total count, the positive deviations will cancel out the negative deviations. Mathematically, the sum of positive deviations will be equal to the sum of negative deviations, resulting in a sum of deviations equal to zero.

For example, consider the following dataset: 1, 2, 3, 4, 5

The mean of this dataset is (1+2+3+4+5)/5 = 3.

The deviations from the mean for each value are as follows: 1 - 3 = -2 2 - 3 = -1 3 - 3 = 0 4 - 3 = 1 5 - 3 = 2

If we sum up all these deviations, we get: -2 + (-1) + 0 + 1 + 2 = 0.

This cancellation of positive and negative deviations is a property unique to the mean and does not hold true for other measures of central tendency. The sum of deviations for the median, mode, and geometric mean will not be zero in general.

To summarize, the sum of the deviations of each value from the mean always equals zero, making the mean the correct answer to this question.