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m Corrected sum starting point in matrix formula |
Removed incorrect hat on y in best fit equation |
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Substituting the above expressions for <math>\widehat{\alpha}</math> and <math>\widehat{\beta}</math> into the original solution yields
: <math>\frac{
This shows that {{math|''r''<sub>''xy''</sub>}} is the slope of the regression line of the [[Standard score|standardized]] data points (and that this line passes through the origin). Since <math>-1 \leq r_{xy} \leq 1</math> then we get that if x is some measurement and y is a followup measurement from the same item, then we expect that y (on average) will be closer to the mean measurement than it was to the original value of x. This phenomenon is known as [[Regression_toward_the_mean#Definition_for_simple_linear_regression_of_data_points|regressions toward the mean]].
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