CHEM3440Lec2F06 - CHEM*3440 Chemical Instrumentation Topic...

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Unformatted text preview: CHEM*3440 Chemical Instrumentation Topic 2 Statistics for Analytical Methods Statistics and Analytical Chemistry Random noise is part of any analytical measurement. Precise and Accurate answers demand a statistical analysis of the data. 1. How much analyte is present? 2. Is one test protocol better than another? 3. Are the results from this lab the same as from another? 4. Are the current results consistent with previous results? What You Should Know Calculate Mean Standard Deviation Work well with a spreadsheet program Find the slope and intercept of a linear least squares ¡t to a collection of data points with a spreadsheet program What We Want To Learn Answer “How much analyte is present?” Calibration Curve Standard Addition Internal Standard Calculate correct con¡dence limits. Understand when to use each experiment. Recognize when experimental data is ¢awed due to instrument failure. Linear Least Squares Fits What is meant by “linear”? What is meant by “least squares”? How do we determine the goodness-of-Ft? What is the variance-covariance matrix? How do we derive error limits or conFdence intervals for interpolated data? What are the dangers of extrapolation? Linear Several meanings 1. Polynomial where the highest power of the independent variable is 1. 2. In a multivariable function, each term depends on only one of the variables. 3. In a multiparameter function, each term depends on only one of the parameters In our case, it is the last description that is applicable. Linear con’t 1 Typical equation is y = mx + b This is linear in polynomial power, variable, and parameters. Can also perform Linear Least Squares analysis on y = ax 3 + bx 2 + cx + d It is still linear in variables and parameters. Cannot perform Linear Least Squares on y = ae-bx since its is not linear in parameters. Least Squares Given a set of data points, Fnd equation for linear model (always linear in parameters, often linear in polynomial power) that minimizes the error between the points and the line. 5 10 15 20 3.75 7.50 11.25 15.00 Least Squares con’t 1 Should minimize error in both x and y. Demands iterative solution. Instead, we assume error in x is negligible and ascribe the error to the dependent variable y. ˚ Error in x and y. Error in y. Matrix Formulation of Least Squares Measured data come in a set of ordered pairs (x i , y i ). Fit the data to a selected model. Can ¡t to any linear function, but most common to ¡t to a straight line: y = mx + b. Assume error is all in y (the dependent variable). For each x i there is a measured value, y i , and a ¡t value y i,¡t = mx i + b. The difference between the measured y i and the ¡tted y i,¡t is the error for the point, e i ....
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