Error propagation in analytical calibration methods
[Operating instructions] [Cell definitions and equations] [Student assignment handout]
[Operating instructions] [Cell definitions and equations] [Student assignment handout]
A set of spreadsheets that perform a Monte-Carlo simulation (e.g. random-number driven) of the precision of analysis based on widely-used calibration methods including single standard, bracket, and standard addition methods. Simulation includes additive and multiplicative interference (systematic errors) and random errors in signal and in volumetric measurements. Students observe how errors combine, attempt to optimize precision and accuracy of the measurement.
You can download several versions of this spreadsheet model, including:
To operate the Monte-Carlo simulation, set the values of the independent variables, and then click on the "20 repeat runs" button. This simulates the 20 spearate standard addition experiments with random errors caused by Es and Ev. The results are shown in the table on the right of the screen.
Assumptions: 1. The only sources of error are random errors in volume and signal measurement. 2. Errors are a fixed percentage of the quantity measured (fixed relative error rather than fixed absolute error).
Inputs: mo Analytical curve slope without interference z Interference factor (zero -> no interference) Io Interferent concentration in original sample Ev Random volumetric error (% RSD ) Es Signal measurement error (% RSD) Cx True analyte concentration in sample C1s Concentration of standard solution 1 C2s Concentration of standard solution 2 blank (Uncorrected) blank signal Outputs: Analytical curve slope in actual sample m = mo-z*Io Signal given by standard 1 S1s = log((1+0.0001)/(10^(-(C1s*m))+0.0001))*(1+0.01*2.5*Ev*(rand()-rand()))*(1+0.01*2.5*Es*(rand()-rand())) Signal given by standard 2 S2s = log((1+0.0001)/(10^(-(C2s*m))+0.0001))*(1+0.01*2.5*Ev*(rand()-rand()))*(1+0.01*2.5*Es*(rand()-rand())) Signal given by sample Sx = log((1+0.0001)/(10^(-(blank+Cx*mo))+0.0001))*(1+0.01*2.5*Ev*(rand()-rand()))*(1+0.01*2.5*Es*(rand()-rand())) Measured analyte concentration in sample Cm = C1s+(C2s-C1s)*(Sx-S1s)/(S2s-S1s) Relative percent accuracy accuracy = (Cm-Cx)/Cx Relative % effect of interference on signal recovery = m/mo Array calculations: Average: mean = avg(R51C9..R70C9) Standard deviation: s = std(R51C9..R70C9) Relative standard deviation: %RSD = s/mean Accuracy = (mean-Cx)/Cx Total error = %RSD+abs(Accuracy) Button script: repaint off define count column numbers select range R51C9..R74C9 remove data unselect repaint range R51C9..R74C9 repaint on for count = 1 to 20 recalc put Cm into "R"&50+count&"C9" end for put "=avg(R51C9..R70C9)" into R71C9 put "=std(R51C9..R70C9)" into R72C9 put "=std(R51C9..R70C9)/avg(R51C9..R70C9)" into R73C9 put "=(R71C9-Cx)/Cx" into R74C9
Student assignment (for Standard Addition Method): Monte-Carlo Simulation of the Single Standard Addition Method "The standard addition procedure is a powerful technique that is often used improperly due to a failure to understand the assumptions involved." Ingle and Crouch, Chapter 6, page 179. To help you appreciate the capabilities and limitations of the standard addition procedure, I have prepared a numerical simulation of the method on FileServer2:Chem 623:Chapter 6:Standard Addition. 1. The model is based on the text, page 178-179 and Equation 6-16. The same terminology is used, with the following modifications: Ss is used for the signal measured after standard addition instead of Sx+s. Cx means the true analyte concentration (the unknown in the simulated experiment); the experimental quantity calculated by equation Equation 6-16, which is supposed to be a measure of Cx, is called result. The volumes Vx and Vs mean the actual volumes; nomVx and nomVs are the nominal volumes, that is, the labeled volumes of the pipettes and flasks. 2. The simulation includes the effect of a multiplicative interference (Io = interferent concentration) and additive interference, i.e. blank error (blank = uncorrected blank signal), and random errors in volume and signal measurement. Errors are assumed to be a fixed percentage of the quantity measured (fixed relative error rather than fixed absolute error). The analytical curve is assumed to be linear. 3. The following are the independent variable that you can change: mo Analytical curve slope without interference z Interference factor (zero -> no interference) Io Interferent concentration in original sample Ev Random volumetric error (% RSD ) Es Signal measurement error (% RSD) Cx Analyte concentration in original sample solution Cs Analyte concentration of standard solution blank (Uncorrected) blank signal nomVx Nominal volume of sample solution before addition nomVs Nominal volume of standard added to sample To change any of these, click on the number (not on the symbol) in the spreadsheet, type a new value, and press the enter key. The other quantities in the spreadsheet are dependent variables that are calculated from these independent variables. The most important of these is result, which is the experimental estimate of Cx calculated by equation Equation 6-16. In this simulation we will compare result to the correct value Cx to see how well Equation 6-16 works. 4. Choose any value of Cx and nomVx you like, then set Cs = ten-fold or so larger than Cx. Start with the ideal case of no interference (Io = 0; blank = 0) and no random errors (Ev = 0 and Es = 0). Verify that result = Cx for arbitrary Cs, nomVx, and nomVs. 5. Introduce a multiplicative interference by making Io > 0 and z > 0, keeping blank = 0. (The recovery expresses by what percent the analytical signal is changed by the interference). Does result = Cx? Try arbitrary values of Io, z, Cx, Cs, nomVx, and nomVs and notice the effect on result. 6. Introduce an additive interference by making blank > 0. Compare result and Cx. What do you conclude about the ability of the standard addition method to compensate for additive and multiplicative interferences? 7. Introduce random errors into the volumetric measurement (Ev) and the signal measurement (Es). To start with make both 1% RSD (Ev = Es =1). Set Io > 0 and z > 0, keeping blank = 0 to simulate a multiplicative interference only. Click on the 20 repeat runs button to simulate 20 separate standard addition measurements. (Quick repeat does the same thing, only faster). The table on the right shows the result of each measurement, and at the bottom computes the mean, standard deviation (s), percent relative standard deviation, and the accuracy (% difference between the mean and Cx). Why is it that if you perform several successive 20-run simulations under fixed conditions, the standard deviation is the exactly the same each time? How could the simulation be designed to make the standard deviation more reproducible? 8. Vary Cs and nomVs and observe the effect on the percent relative standard deviation of the 20 repeats. Is there an optimum value of Cs and nomVs that minimizes this error? On the basis of your observations, formulate a rule that allows you to predict the optimum value of Cs and nomVs. 9. Why is it that, even under the best condition, the % RSD of result is greater than Es or Ev?