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Advanced Higher STATISTICS Linear Regression To see if there is a relationship between two variables, we draw a scatter-graph. It is then possible to draw a ‘best-fit’ line through the points on the graph. To draw the straight line, we try to have as many points as possible on either side of the line. This is pretty difficult so we calculate a line of regression instead: this is called linear regression.

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Advanced Higher STATISTICS The formula for drawing the linear regression The equation for a straight line is Y = a + bX ‘a’ is known as the ‘intercept’. o It is given by the formula: a = ӯ - bẊ ‘b’ is the ‘regression coefficient’ It is given by the formula: b = ∑(Ẋ-X)(y-ӯ) ∑(X-Ẋ) 2

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Advanced Higher STATISTICS First: draw yourself a scatter-graph to show the relationship between two variables SiteDischarge M3/sec Suspended load g/m3 10.4510.8 20.429.7 30.5111.2 40.5511.3 50.6812.5 60.7512.8 70.8913.0 80.7612.7 90.9613.0 101.2617.4 Increasing river discharge Increasing suspended load Scatter graph showing the relationship between discharge and suspended on the River Farg

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Advanced Higher STATISTICS Second: complete Pearsons Product to get the values you need for ‘b’ b = ∑(Ẋ-X)(y-ӯ) ∑(X-Ẋ) 2 SiteDischarge M3/sec Suspended load g/m3 10.4510.8 20.429.7 30.5111.2 40.5511.3 50.6812.5 60.7512.8 70.8913.0 80.7612.7 90.9613.0 101.2617.4

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Advanced Higher STATISTICS Second: complete Pearsons Product to get the values you need for ‘b’ b = ∑(Ẋ-X)(y-ӯ) ∑(X-Ẋ) 2 b = 4.65 0.62 b = 7.5

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Advanced Higher STATISTICS Third: calculate ‘a’ using the formula a = 12.44 – (7.5 x 0.723) a = 7.02 a = ӯ - bẊ

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Advanced Higher STATISTICS Fourth: figure out some ‘x’ & ‘y’ values To draw the straight line you have to calculate the values of ‘y’ for values of ‘x’ that you choose. For example, if x = 0, then y = a + (b x 0) For example, if x = 20, then y = a + (b x 20) To draw the straight line you have to calculate the values of ‘y’ for values of ‘x’ that you choose. if x = 0, then y = 7.02 + (7.5 x 0) = 7.02 if x = 0.6, then y = 7.02 + (7.5 x 0.6) = 11.52 if x = 1.2, then y = 7.02 + (7.5 x 1.2) = 16.02 b = 7.5 a = 7.02 if x = 0, y = 7.02 if x = 0.6, y = 11.52 if x = 1.2, y = 16.02

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Advanced Higher STATISTICS Fifth: plot your straight line! if x = 0, y = 7.02 if x = 0.6, y = 11.52 if x = 1.2, y = 16.02

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