The sum of the deviations of each data value from this measure of central location will always be zero.

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The question asks about a measure of central location with the property that the sum of the deviations of each data value from it is always zero.
Option 1: mode. The mode is simply the most frequently occurring value in the data set. Deviations from the mode do not have to sum to zero, and there is no general guarantee that they will; this makes the mode an unlikely candidate for the described property.
Option 2: mean. The mean (arithmetic average) has a key defining property: the sum over all observations of (each value minus the mean) equals zero, i.e., Σ(xi - x̄) = 0. This arises from the equation n*x̄ = Σxi, which implies the centered deviations balance out to zero.
Option 3: standard deviation. The standard deviation is computed from the squared deviations from the mean, namely the square root of the average of (xi - x̄)². Because it uses squared deviations, the sum of the original (unsquared) deviations does not necessarily equal zero, so this option does not fit the described property.
Option 4: median. The median is the middle value (or average of the two middle values) when data are ordered. There is no general guarantee that the sum of deviations from the median will be zero, so this option does not satisfy the condition either.
In summary, among the listed measures, only the mean has the formal property that the deviations from it sum to zero, while the other measures do not guarantee this behavior.

IS 2010-001 Spring 2025 Exam 1

The sum of the deviations of each data value from this measure of central location will always be zero.

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