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Unformatted text preview: e values indicate a negative relationship. Covariance The covariance is computed as follows: sxy xy ( xi x ) ( yi y ) n 1 ( xi m x ) ( yi m y ) N for samples for populations Correlation Coefficient Correlation is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it does not mean that one variable is the cause of the other. Correlation Coefficient The correlation coefficient is computed as follows: rxy sxy sx s y for samples xy xy x y for populations Correlation Coefficient The coefficient can take on values between -1 and +1. Values near -1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. The closer the correlation is to zero, the weaker the relationship. Covariance and Correlation Coefficient Example: A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score. Average Driving Average Distance (yds.) 18-Hole Score 277.6 69 259.5 71 269.1 70 267.0 70 255.6 71 272.9 69 Covariance and Correlation Coefficient x y 277.6 259.5 269.1 267.0 255.6 272.9 69 71 70 70 71 69 Average 267.0 70.0 Std. Dev. 8.2192 .8944 ( xi x ) ( yi y ) ( xi x )( yi y ) 10.65 -7.45 2.15 0.05 -11.35 5.95 -1.0 1.0 0 0 1.0 -1.0 -10.65 -7.45 0 0 -11.35 -5.95 Total -35.40 Covariance and Correlation Coefficient Sample Covariance sxy (x x )( y y ) 35.40 i i n1 61 7.08 Sample Correlation Coefficient sxy 7.08 rxy -.9631 sx sy (8.2192)(.8944) The Weighted Mean and Working with Grouped Data • • • • Weighted Mean Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. In the computation of a grade point average (GPA), the weights are the number of credit hours earned for each grade. When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. Weighted Mean w x x w ii i where: xi = value of observation i wi = weight for observation i Grouped Data The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. Mean for Grouped Data Sample Data fM x i i n Population Data fM m i i N where: fi = frequency of class i Mi = midpoint of class i Sample Mean for Grouped Data Example:Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the form of a frequency distribution. Rent (\$) 420-439 440-459 460-479 480-499 500-519 520-539 540-559 560-579 580-599 600-619 Frequency 8 17 12 8 7 4 2 4 2 6 Sample Mean for Grouped Data Rent (\$) 420-439 440-459 460-479 480-499 500-519 520-539 540-559 560-579 580-599 600-619 Total fi 8 17 12 8 7 4 2 4 2 6 70 Mi 429.5 449.5 469.5 489.5 509.5 529.5 549.5 569.5 589.5 609.5 f iMi 3436.0 7641.5 5634.0 3916.0 3566.5 2118.0 1099.0 2278.0 1179.0 3657.0 34525.0 34,525 x 493.21 70 This approximation differs by \$2.41 from the actual sample mean of \$490.80. Variance for Grouped Data For Sample Data 2 f i ( Mi x ) s2 n 1 For Population Data 2 f i ( Mi m ) 2 N Sample Variance for Grouped Data Rent (\$) 420-439 440-459 460-479 480-499 500-519 520-539 540-559 560-579 580-599 600-619 Total fi 8 17 12 8 7 4 2 4 2 6 70 Mi 429.5 449.5 469.5 489.5 509.5 529.5 549.5 569.5 589.5 609.5 Mi - x -63.7 -43.7 -23.7 -3.7 16.3 36.3 56.3 76.3 96.3 116.3 (M i - x )2 f i (M i - x )2 4058.96 32471.71 1910.56 32479.59 562.16 6745.97 13.76 110.11 265.36 1857.55 1316.96 5267.86 3168.56 6337.13 5820.16 23280.66 9271.76 18543.53 13523.36 81140.18 208234.29 continued Sample Variance for Grouped Data Sample Variance s2 = 208,234.29/(70 – 1) = 3,017.89 Sample Standard Deviation s 3,017.89 54.94 This approximation differs by only \$.20 from the actual standard deviation of \$54.74....
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