Statistics Practice Test 1

Concept:A measure of central tendency has the subtraction property if decreasing every observation by $k$ decreases the measure by exactly $k$.Explanation:Let the original observations be $x_1, x_2, \ldots, x_n$.After decreasing each observation by $k$, the new observations are $x_1-k, x_2-k, \ldots, x_n-k$.Arithmetic mean: $\bar{x}_{\text{new}} = \frac{\sum (x_i-k)}{n} = \bar{x} - k$.Median: the order remains the same, so the middle value also decreases by $k$.Mid-range: $\frac{\max(x_i-k)+\min(x_i-k)}{2} = \frac{\max x_i+\min x_i}{2}-k$.Geometric mean: $GM_{\text{new}} = \sqrt[n]{(x_1-k)(x_2-k)\cdots(x_n-k)}$.This is not generally equal to $GM_{\text{old}} - k$.Therefore, the geometric mean does not satisfy the given property.Answer:Geometric mean