Measures of relative standing l median relies on

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Measures of relative standing l Median relies on ranking of data to measure location l Can generalize this notion to percentiles l P th percentile is the value for which P percent of observations are less than that value l Median is the 50th percentile l 25th & 75th percentiles called lower & upper quartiles l Difference between upper & lower quartiles called the inter-quartile range - another measure of spread
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16 Measures of relative standing… l Simple example l Denote 25th, 50th & 75th percentiles by Q1, Q2 & Q3 l Suppose n = 8, & ordered data are x1, x2, x3, x4, x5, x6, x7, x8 l Need to divide data into quarters l Thus Q1= (x2+x3)/2, Q2= (x4+x5)/2, Q3= (x6+x7)/2 l IQR = Q3 – Q1 l See Keller Ex 4.11 for another example
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17 Measures of association l Do large values of x tend to be associated with large values of y ? l Graphical answer from scatter plots l Covariance is a numerical measure l Positive (negative) covariance  positive (negative) linear association l Zero covariance  no linear association ( 29 ( 29 ( 29 ( 29 1 covariance Sample covariance Population 1 1 n- y y x x s N y x n i i i xy N i y i x i xy = = - - = - - = μ μ σ
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18 Measures of association… l Covariance is not scale free l Is covariance of 500 big? l Covariance between height & weight depends on units used l Correlation coefficient is standardized measure of association that is unit- free l 1 (-1)  perfect positive (negative) linear relationship 1 1 and 1 1 n correlatio Sample n correlatio Population y x - - = = r s s s r y x xy xy ρ σ σ σ ρ
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19 Correlations in each of these scatter plots? (Keller Fig. 3.13)
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20 Least squares: The problem l Have ( y i, x i) pairs for i = 1, … , n l Portrayed graphically in scatter plot l Interested in linear relationship between y & x l How do you determine the intercept & slope in this relationship? l Choose values that give the best fit l What do you mean best fit? l One approach is minimize residual sum of squares l Method called least squares l Basis of regression analysis (see Keller Ch 16)
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21 Least squares: The diagram x b b y 1 0 ˆ + = y x w w w w b 0 e 1
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22 Least squares: The optimization problem squares of sum residual the minimize that estimates slope and intercept be will solution Thus ) ˆ minimize to chosen are & where ˆ Assume 2 1 1 0 1 0 i i n i i i y y ( b b x b b y - + = =
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23 Least squares: The solution x b y b s s b x xy 1 0 2 1 - = = l Note: l Point of the means will lie on line of best fit l b 1 will have same sign as covariance (correlation) between y & x l Zero covariance (correlation)  b 1 = 0  ?
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