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8: Calibrating Data

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  • A calibration curve is one of the most important tools in analytical chemistry as it allows us to determine the concentration of an analyte in a sample by measuring the signal it generates when placed in an instrument, such as a spectrophotometer. To determine the analyte's concentration we must know the relationship between the signal we measure , \(S\), and the analyte's concentration, \(C_A\), which we can write as

    \[S = k_A C_A + S_{blank} \nonumber\]

    where \(k_A\) is the calibration curve's sensitivity and \(S_{blank}\) is the signal in the absence of analyte.

    How do we find the best estimate for this relationship between the signal and the concentration of analyte? When a calibration curve is a straight-line, we represent it using the following mathematical model

    \[y = \beta_0 + \beta_1 x \nonumber \]

    where y is the analyte’s measured signal, S, and x is the analyte’s known concentration, \(C_A\), in a series of standard solutions. The constants \(\beta_0\) and \(\beta_1\) are, respectively, the calibration curve’s expected y-intercept and its expected slope. Because of uncertainty in our measurements, the best we can do is to estimate values for \(\beta_0\) and \(\beta_1\), which we represent as b0 and b1. The goal of a linear regression analysis is to determine the best estimates for b0 and b1.

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