For a data set of real numbers

MATHEMATICS · Multi-part · AP · AP Statistics · Exploring One-Variable Data · Exploring One-Variable Data

Problem

For a data set of nn real numbers y1,y2,…,yny_1, y_2, \dots, y_n, two common criteria for selecting a single representative value cc are: Criterion A: minimize S(c)=∑i=1n(yi−c)2\displaystyle S(c)=\sum_{i=1}^{n}(y_i-c)^2 (sum of squared deviations). Criterion B: minimize T(c)=∑i=1n∣yi−c∣\displaystyle T(c)=\sum_{i=1}^{n}|y_i-c| (sum of absolute deviations). (a) Describe qualitatively how the location of the minimizing value of cc differs between Criterion A and Criterion B. Identify which common statistics arise from each criterion. (b) Provide a mathematical argument to explain why the minimizing cc for Criterion A is unique, whereas Criterion B may admit an entire interval of minimizers when the data set contains an even number of observations. (c) Discuss one practical advantage and one practical disadvantage of using Criterion A instead of Criterion B when summarizing real‐world data.

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