Analytical Chem istry - DePauw University

Chapter 5 Standardizing Analytical Methods173Figure 5.9 Illustration showing three data points and twopossible straight-lines that might explain the data. The goalof a linear regression is to find the mathematical model, inthis case a straight-line, that best explains the data.where b 0 and b 1 are our estimates for the y-intercept and the slope, and ŷ isour prediction for the experimental value of y for any value of x. Because weassume that all uncertainty is the result of indeterminate errors affecting y,the difference between y and ŷ for each data point is the residual error,r, in the our mathematical model for a particular value of x.r = ( y − yˆ )i i iFigure 5.10 shows the residual errors for the three data points. The smallerthe total residual error, R, which we define asR = ∑( y − yˆ ) 2 i i5.16ithe better the fit between the straight-line and the data. In a linear regressionanalysis, we seek values of b 0 and b 1 that give the smallest total residualerror.ŷ 3r = ( y − yˆ )2 2 2ŷ ŷ 2 1y 2y 3r = ( y − yˆ )1 1 1y 1ŷ = b 0+ bx 1r = ( y − yˆ )3 3 3If you are reading this aloud, you pronounceŷ as y-hat.The reason for squaring the individual residualerrors is to prevent positive residualerror from canceling out negative residualerrors. You have seen this before in theequations for the sample and populationstandard deviations. You also can seefrom this equation why a linear regressionis sometimes called the method of leastsquares.Figure 5.10 Illustration showing the evaluation of a linear regression in which we assume that all uncertaintyis the result of indeterminate errors affecting y. The points in blue, y i , are the original data and thepoints in red, ŷ i, are the predicted values from the regression equation, ŷ = b 0+ bx 1.The smaller thetotal residual error (equation 5.16), the better the fit of the straight-line to the data.