Fit a straight line y = mx + c to x, y pairs by least squares and get r² — the fit behind Arrhenius plots, calibration lines and rate-law data.
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From the article Linear Regression y = mx + c — The Least-Squares Idea.
Five standard solutions are measured in a colorimeter. Concentration x is in arbitrary standard units and absorbance y has no unit.
Σx = 1 + 2 + 3 + 4 + 5 = 15, and n = 5.
Slope:
nΣxy = 5 × 11.03 = 55.15
(Σx)(Σy) = 15 × 3.02 = 45.30
numerator = 55.15 − 45.30 = 9.85
nΣx² = 5 × 55 = 275, (Σx)² = 15² = 225 → denominator = 275 − 225 = 50
m = 9.85 / 50 = 0.197
Intercept:
m(Σx) = 0.197 × 15 = 2.955
Σy − m(Σx) = 3.02 − 2.955 = 0.065
c = 0.065 / 5 = 0.013
Best-fit line: y = 0.197x + 0.013
Worked in full in Linear Regression y = mx + c — The Least-Squares Idea.
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