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Published byIlene Alberta Joseph Modified over 6 years ago

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The Best-Fit Line Linear Regression

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PGCC CHM 103 Sinex How do you determine the best-fit line through data points? x-variable y-variable Fortunately technology, such as the graphing calculator and Excel, can do a better job than your eye and a ruler!

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PGCC CHM 103 Sinex The Equation of a Straight Line y = mx + b where m is the slope or y/ x and b is the y-intercept In some physical settings, b = 0 so the equation simplifies to: y = mx

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PGCC CHM 103 Sinex x-variable y-variable deviation = residual = y data point – y equation Linear regression minimizes the sum of the squared deviations y = mx + b

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PGCC CHM 103 Sinex Linear Regression Minimizes the sum of the square of the deviations for all the points and the best-fit line Judge the goodness of fit with r 2 r 2 x100 tells you the percent of the variation of the y-variable that is explained by the variation of the x- variable (a perfect fit has r 2 = 1)

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PGCC CHM 103 Sinex x-variable y-variable Goodness of Fit: Using r 2 r 2 is high r 2 is low How about the value of r 2 ?

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PGCC CHM 103 Sinex Strong direct relationship 99.1% of the y-variation is due to the variation of the x-variable

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PGCC CHM 103 Sinex Noisy indirect relationship Only 82% of the y-variation is due to the variation of the x-variable - what is the other 18% caused by?

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PGCC CHM 103 Sinex When there is no trend! No relationship!

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PGCC CHM 103 Sinex In Excel When the chart is active, go to chart, and select Add Trendline, choose the type and on option select display equation and display r 2 For calibration curves, select the set intercept = 0 option Does this make physical sense?

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PGCC CHM 103 Sinex Using the set intercept = 0 option lowers the r 2 value by a small amount and changes the slope slightly Does the set intercept = 0 option make a difference?

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PGCC CHM 103 Sinex The equation becomes A = mc or A = 0.89c 99.1% of the variation of the absorbance is due to the variation of the concentration.

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