Unformatted text preview: Statistics 21, Summation and Correlation The correlation coefficient r can be written either 1 n (xi - x) (yi - y ) n i=1 SDx SDy The proof is as follows: 1 n (xi - x) (yi - y ) n i=1 SDx SDy = = = = 1 n SDx SDy 1 SDx SDy
1 n n i=1 n i=1 or 1 n n i=1 xi y i - x y SDx SDy = 1 n SDx SDy
n n n xi yi - xi y - xyi + xy i=1 n n xi y i - y i=1 xi - x i=1 yi +
i=1 n i=1 xy + 1 n xy n i=1 xi y i n -y i=1 n i=1 xi n + - x yi n xi yi - yx - xy + xy SDx SDy xi yi - xy 1 n n i=1 SDx SDy The first formula is like that in the text and is probably easier to understand intuitively. The second formula is often easier to compute. In the previous handout we solved for the x2 term in the SD to use when comi bining two lists of numbers into a long list. Similarly, the xi yi term in r can be used to solve problems which involve combining two sets of data. Homework due Wednesday February 8 From the text, do Ch 8: A1, B6, D1, Review 1, Ch 9: A10, B3, Review 3,8. Also do the following: 1. A square of fabric, 20 inches on each side, is cut into 10 rectangles. The rectangles have average length 4 inches, with an SD of 2 inches. The widths have an average of 9 with an SD of 4. Find the correlation of length and width for these rectangles. Note that length width (xi yi ) is area and think about the total area of all 10 rectangles. 2. Another square of fabric, 12 inches on each side, is cut into 6 rectangles. The average length is 4 inches with an SD of 2 inches; the average width is 5 inches with an SD of 3 inches. Find the correlation between length and width for all 16 rectangles in this problem and problem 1. 1 ...
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