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Suppose we have some bivariate quantitative information (x1, y1), . . . , (xn, yn) for which the correlation coefficient indicates some direct association. It is organic to desire to write down clearly the equation of the ideal line via the data – the question is what is this line. The a lot of prevalent meaning provided to *finest *in this search for the line is *the line whose full square error is the smallest feasible. *We make this idea specific in two steps

DEFINITION 3.1.1. Given a bivariate quantitative dataset (x1, y1), . . . , (xn, yn) and also a candiday line ( haty = mx+b) passing via this dataset, a **residual **is the difference in y-works with of an actual information suggest (xi, yi) and the line’s y value at the exact same x-coordinate. That is, if the y-coordinate of the line as soon as x = xi is ( haty_i = mx_i + b), then the residual is the measure of error provided by ( error_i = y_i - haty_i).

Keep in mind we use the convention below and somewhere else of composing ( haty) for the y-coordinate on an approximating line, while the plain y variable is left for actual data worths, like yi.

Here is an example of what residuals look like

It appears pretty clear that tbelow is quite a solid linear association between these two vari- ables, as is born out by the correlation coefficient, r = .935 (computed with **LibreOffice Calc**’s CORREL). Using then STDEV.S and AVERAGE, we discover that the coefficients of the LSRL for this information, ( haty = mx + b) are

( m = r fracs_ys_x = .935 frac18.70123.207 = .754) and ( b = ary - arxm =71 − 58 · .754 = 26.976)

We have the right to likewise use **LibreOffice Calc**’s Insert Trfinish Line, through Sexactly how Equation, to obtain all this done immediately.

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Note that once **LibreOffice Calc **writes the equation of the LSRL, it supplies f (x) in area of ( haty), as we would.