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Unformatted text preview: 56.19 ‐3.19 10.18 65 80 89.66 ‐9.66 93.31 44 78 67.69 10.31 106.29 64 93 88.61 4.39 19.27 60 88 84.43 3.57 12.74 40 58 63.51 ‐5.51 30.36 ‐‐ ‐‐ ‐‐ 0 272.1 SSE = sum of squared errors (or residuals) 272.1 187 Measuring Strength and Direction of a Linear Relationship with Correlation The correlation coefficient r is a measure of strength of the linear relationship between y and x. Properties about the Correlation Coefficient r 1. r ranges from ... –1 to +1 (and it is unitless) 2. Sign of r indicates ... direction of the association 3. Magnitude of r indicates ... strength (r = -0.8 and r = +0.8 indicate equally strong linear associations) A “strong” r is discipline specific r = 0.8 might be an important (or strong) correlation in engineering r = 0.6 might be a strong correlation in psychology or medical research 4. r ONLY measures the strength of the LINEAR relationship. Some pictures: y r = +0.7 y r = -0.4 y x x The formula for the correlation: (but we will get it from computer output or from r2) Exam Scores Example: r = ___0.889____ Interpretation: A fairly strong positive linear association between exam 2 scores and final exam scores. 188 r0 x The square of the correlation r 2 The squared correlation coefficient r 2 always has a value between __0 and 1 __ and is sometimes presented as a percent. It can be shown that the square of the correlation is related to the sums of squares that arise in regression. The responses (the final exam scores) in data set are not all the same ‐ they do vary. We would measure the total variation in these responses as SSTO y y 2 (this was the last column total in our calculation table that we said we would use later). Total
in the y’s Variation not
accounted for Part of the reason why the final exam scores vary is because there is a linear relationship between final exam scores and exam 2 scores, and the study included students with different exam 2 scores. When we found the least squares regression line, there was still some small variation re...
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