When points lie nearly on a line, it is useful to determine an equation for a line that lies on or comes close to the points.
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Usually, there is no single line that passes through all the data points, so you try to find the line that best fits the data. This is called the best-fitting line . best-fitting line . There are several ways to find the best-fitting line for a given set of data points. In this lesson, you will use a graphical approach. – 8 8 6 4 2 – 2 – 4 – 6 0 2 4 6 – 2 – 4 – 6 – 8 F ITTING A L INE TO D ATA
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Write an equation of your line. The winning Olympic discus throws from 1908 to 1996 are plotted in the graph. Approximate the best-fitting line for these throws. Approximating a Best-Fitting Line D ISCUS T HROWS Years since 1900 Distance (ft) 0 8 16 24 32 40 48 56 64 72 80 88 96 104 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250
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S OLUTION Find two points that lie on the best-fitting line , such as ( 8, 138 ) and ( 96, 230 ) . Find the slope of the line through these points. Approximating a Best-Fitting Line Years since 1900 Distance (ft) 0 8 16 24 32 40 48 56 64 72 80 88 96 104 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 ( 96, 230 ) . (96, 230) ( 8, 138 ) ( 8, 138 )
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y = m x + b 129.6 = b Write slope intercept form. Substitute 1.05 for m , 8 for x , 138 for y . Simplify. Solve for b . 138 = ( 1.05 ) ( 8 ) + b y = m x + b 138 = 8.4 + b 92 88 = 1.05 Years since 1900 Distance (ft) 0 8 16 24 32 40 48 56 64 72 80 88 96 104 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 (96, 230) ( 8, 138 ) 230 – 138 96 – 8 = An equation of the best-fitting line is y = 1.05 x + 129.6. y 2 – y 1 x 2 – x 1 m = In most years, the winner of the discus throw was able to throw the discus farther than the previous winner. Approximating a Best-Fitting Line 230 – 138 96 – 8 = 92 88 = 1.05
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D ETERMINING THE C ORRELATION OF X AND Y In this scatter plot, x and y have a positive correlation , which means that the points can be approximated by a line with a positive slope .
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D ETERMINING THE C ORRELATION OF X AND Y In this scatter plot, x and y have a negative correlation , which means that the points can be approximated by a line with a negative slope .
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D ETERMINING THE C ORRELATION OF X AND Y In this scatter plot, x and y have relatively no correlation , which means that the points cannot be approximated by a line.
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D ETERMINING THE C ORRELATION OF X AND Y Positive Correlation No Correlation Negative Correlation T YPES OF C ORRELATION
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