02 STAT descriptive stats 4-APPENDIX

02 STAT descriptive stats 4-APPENDIX - DESCRIPTIVE...

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1 DESCRIPTIVE STATISTICS Dr. V.R. Bencivenga Economic Statistics Economics 329 DESCRIPTIVE STATISTICS PART 2 APPENDIX TWO PROOFS (OPTIONAL) 1. COVARIANCE INCREASES WHEN POINTS ARE “MORE TIGHTLY CLUSTERED” AROUND A LINE
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2 DESCRIPTIVE STATISTICS PROOF : The covariance increases when observations cluster more tightly around a line. Strategy: (1) Move observations. (2) Keep 2 Y s and Y unchanged (also keep 2 X s and X unchanged). (3) Show n 1 i i i ) Y Y ( ) X X ( increases, so s XY increases. Y Y X X We must keep s Y /s X unchanged, otherwise the slope of the “line” changes. We must keep X and Y unchanged, otherwise we’ll have to re -calculate all of the deviations!
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3 DESCRIPTIVE STATISTICS Move two observations together, vertically : moves down moves up These observations are below the mean, Y . Y Y X X
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4 DESCRIPTIVE STATISTICS Move two additional observations together, vertically: moves down moves up These observations are above Y . Y Y X X
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5 DESCRIPTIVE STATISTICS Y Y X X Focus on the observations below Y . To begin, is “far” below Y (a “big” negative deviation). Move toward Y . What is the impact on 2 Y s ? Moving closer to Y reduces 2 Y s a “lot” because 2 Y s squares the deviations.
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6 DESCRIPTIVE STATISTICS Given how far was moved, how far must move, for 2 Y s to be unchanged?
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