Analytical Chem istry - DePauw University

178 Analytical Chemistry 2.0Minimizing Un c e r t a i n t y in Calibration Cu r ve sTo minimize the uncertainty in a calibration curve’s slope and y-intercept,you should evenly space your standards over a wide range of analyte concentrations.A close examination of equation 5.20 and equation 5.21 willhelp you appreciate why this is true. The denominators of both equationsinclude the term ∑( x − x)2 i. The larger the value of this term—whichyou accomplish by increasing the range of x around its mean value—thesmaller the standard deviations in the slope and the y-intercept. Furthermore,to minimize the uncertainty in the y‐intercept, it also helps to decreasethe value of the term ∑ x iin equation 5.21, which you accomplishby including standards for lower concentrations of the analyte.Ob t a i n i n g t h e An a l y t e’s Co n c e n t r a t i o n Fr o m a Re g r e s s i o n Eq u a t i o nEquation 5.25 is written in terms of a calibrationexperiment. A more general formof the equation, written in terms of x andy, is given here.sxs 1 1r= + +b m n12( Y − y)∑2 2( b) ( x − x )A close examination of equation 5.25should convince you that the uncertaintyin C A is smallest when the sample’s averagesignal, Ssamp, is equal to the averagesignal for the standards, Sstd. When practical,you should plan your calibrationcurve so that S samp falls in the middle ofthe calibration curve.1iiOnce we have our regression equation, it is easy to determine the concentrationof analyte in a sample. When using a normal calibration curve, forexample, we measure the signal for our sample, S samp , and calculate theanalyte’s concentration, C A , using the regression equation.CAS=sampb−b015.24What is less obvious is how to report a confidence interval for C A thatexpresses the uncertainty in our analysis. To calculate a confidence intervalwe need to know the standard deviation in the analyte’s concentration, s CA,which is given by the following equationsCAsr1 1= + +b m n1( S −Ssamp std )(2( ) −1 ∑b C Cistdi2std)25.25where m is the number of replicate used to establish the sample’s averagesignal ( S samp), n is the number of calibration standards, S stdis the averagesignal for the calibration standards, and C stdiand C stdare the individual andmean concentrations for the calibration standards. 7 Knowing the value ofs CA, the confidence interval for the analyte’s concentration isµ C= C ± tsA A C Awhere m C A is the expected value of C A in the absence of determinate errors,and with the value of t based on the desired level of confidence and n–2degrees of freedom.7 (a) Miller, J. N. Analyst 1991, 116, 3–14; (b) Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics,Wiley-Interscience: New York, 1986, pp. 126-127; (c) Analytical Methods Committee“Uncertainties in concentrations estimated from calibration experiments,” AMC TechnicalBrief, March 2006 (http://www.rsc.org/images/Brief22_tcm18-51117.pdf)