Calibration Curves. Lecture #5 - Overview. Construction of Calibration Curves. Construction of Calibration Curves. Construction of Calibration Curves

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1 Lecture #5 - Overvew Statstcs - Part 3 Statstcal Tools n Quanttatve Analyss The Method of Least Squares Calbraton Curves Usng a Spreadsheet for Least Squares Analytcal Response Measure of Unknown Calbraton Curves Concentraton of Standard = Known Amount/Concentraton of Standard Amount/Concentraton of Unknown Constructon of Calbraton Curves Standard Solutons = Solutons contanng known concentratons of analyte(s) Blank Solutons = Solutons contanng all the reagents and solvents used n the analyss, but no delberately added analyte Constructon of Calbraton Curves Step 1: Prepare known samples of analyte coverng a range of concentratons expected for unknowns. Measure the response of the analytcal procedure for these standards. e.g. Seral luton 1x 1/5x 1/5x 1/15x 1/65x Blank Measure response wth analytcal procedure Constructon of Calbraton Curves Step 1: Prepare known samples of analyte coverng a range of concentratons expected for unknowns. Measure the response of the analytcal procedure for these standards. Step : Subtract the (average) response of the blank samples from each measured standard to obtan the corrected value. Constructon of Calbraton Curves Step 1: Prepare known samples of analyte coverng a range of concentratons expected for unknowns. Measure the response of the analytcal procedure for these standards. Step : Subtract the average response of the blank samples from each measured standard to obtan the corrected value. Corrected = Measured - Blank Step 3: Make a graph of corrected versus concentraton of standard, and use the method of least squares procedure to fnd the best straght lne through the lnear porton of the data. Step 4: To determne the concentraton of an unknown, analyze the unknown sample along wth a blank, subtract the blank to obtan the corrected value and use the corrected value to determne the concentraton based on your calbraton curve. 1

2 Calbraton Curves Analytcal Response Measure of Unknown Amount/Concentraton of Unknown Concentraton of Standard = Known Amount/Concentraton of Standard to draw the best straght lne through expermental data ponts that have some scatter and do not le perfectly on a straght lne Method of Least Squares Vertcal evaton = d = y - y = y - (mx + b) y = mx + b d = (y - y) = (y - mx - b) y y-ntercept (b) Δx Δy Slope (m) = Δy Δx We wsh to mnmze to mnmze the magntude of the devatons (regardless of sgn) so we square the terms. Ths s where Method of least Squares takes ts name. x Σ(x y ) Σx Slope: m = Σy n etermnants Σ(x ) Σ(x y ) Intercept: b = Σx Σy A C B A - BC = Σ(x ) Σx Σx n

3 m = nσ(x y ) - Σx Σy nσ (x ) - (Σx ) b = Σ(x )Σy - (Σx y )Σx nσ (x ) - (Σx ) best ft lne. Amount Proten (mg) Absorbance Corrected* * Absorbance - Average Blank (= ) best ft lne Σ data ponts m = nσ(x y ) - Σx Σy nσ (x ) - (Σx ) = (6)(.07) - (75)(1.5) (6)(1375) - (75) m = best ft lne. b = Σ(x )Σy - (Σx y )Σx Σ nσ (x ) - (Σx ) = (1375)(1.5) - (.07)(75) (6)(1375) - (75) b =

4 best ft lne. m = b = y = ( )x + ( ) to draw the best straght lne through expermental data ponts that have some scatter and do not le perfectly on a straght lne y = mx + b σ y (x,y ) Vertcal evaton (d ) = y - y y d = y - y = y - (mx + b) (d ) = (y - mx - b) x σ y s y = Σ(d 1 - d) (degrees of freedom) s y = Σ(d 1 ) (degrees of freedom) s y = Σ(d 1 ) n- 4

5 d (=y - mx - b) d Σ s y = Σ(d 1 ) n- = ( )/(6-) = = s m = s y n s b = s y Σ(x ) Σ(x ) Σx d = Σx n = Σ = (1375 x 6) - (75 x 75) s = 65 y = d Σ s y = , =65 s m = s y n = ( ) (6) (65) = s m = d Σ s y = , =65 s b = s y Σ(x ) = ( ) (1375) (65) = s b =

6 Lnearty m = ± = ± b = ± = ± Lnear Range vs. ynamc Range ynamc Range Lnear Range etermnng Lnearty Square of Correlaton Coeffcent R = [Σ(x - x)(y - y)] Σ(x - x) Σ(y - y) R Hgh (>0.95) R close to 1 (e.g. 0.99, 0.98, 0.95) R Low (<<0.95) 6

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