D. Harvey - Modern Analytical Chemistry (794078), страница 31
Текст из файла (страница 31)
60)degrees of freedom (p. 80)detection limit (p. 95)determinate error (p. 58)Dixon’s Q-test (p. 93)error (p. 64)F-test (p. 87)heterogeneous (p. 58)histogram (p. 77)homogeneous (p. 72)indeterminate error (p. 62)limit of identification (p. 95)limit of quantitation (p. 96)mean (p. 54)measurement error (p. 58)median (p. 55)method error (p. 58)normal distribution (p. 73)null hypothesis (p. 83)one-tailed significance test (p.
84)outlier (p. 93)paired data (p. 88)paired t-test (p. 92)personal error (p. 60)population (p. 71)probability distribution (p. 71)proportional determinate error (p. 61)range (p. 56)repeatability (p. 62)reproducibility (p. 62)sample (p. 71)sampling error (p. 58)significance test (p. 83)standard deviation (p. 56)standard reference material (p. 61)tolerance (p. 58)t-test (p. 85)two-tailed significance test (p. 84)type 1 error (p.
84)type 2 error (p. 84)uncertainty (p. 64)unpaired data (p. 88)variance (p. 57)4I SUMMARYThe data we collect are characterized by their central tendency(where the values are clustered), and their spread (the variation ofindividual values around the central value). Central tendency is reported by stating the mean or median.
The range, standard deviation, or variance may be used to report the data’s spread. Data alsoare characterized by their errors, which include determinate errorsaffecting the data’s accuracy, and indeterminate errors affectingthe data’s precision. A propagation of uncertainty allows us to estimate the affect of these determinate and indeterminate errors onresults determined from our data.The distribution of the results of an analysis around a centralvalue is often described by a probability distribution, two examples1400-CH04 9/8/99 3:55 PM Page 97Chapter 4 Evaluating Analytical Dataof which are the binomial distribution and the normal distribution.Knowing the type of distribution allows us to determine the probability of obtaining results within a specified range.
For a normaldistribution this range is best expressed as a confidence interval.A statistical analysis allows us to determine whether ourresults are significantly different from known values, or fromvalues obtained by other analysts, by other methods of analysis, or for other samples. A t-test is used to compare mean values, and an F-test to compare precisions.
Comparisons betweentwo sets of data require an initial evaluation of whether the datais paired or unpaired. For unpaired data it is also necessary todecide if the standard deviations can be pooled. A decisionabout whether to retain an outlying value can be made usingDixon’s Q-test.Finally, we have seen that the detection limit is a statisticalstatement about the smallest amount of analyte that can be detected with confidence.
A detection limit is not exact because itsvalue depends on how willing we are to falsely report the analyte’spresence or absence in a sample. When reporting a detectionlimit, you should clearly indicate how you arrived at its value.Suggested EXPERIMENTSThe following experiments may be used to introduce the statistical analysis of data in the analytical chemistrylaboratory.
Each experiment is annotated with a brief description of the data collected and the type of statisticalanalysis used in evaluating the data.Cunningham, C. C.; Brown, G. R.; St Pierre, L. E.“Evaluation of Experimental Data,” J. Chem. Educ. 1981, 58,509–511.ExperimentsExperiments4J97In this experiment students determine the density of glassmarbles and the radius of the bore of a glass capillary tube.Density is determined by measuring a marble’s mass andvolume, the latter by measuring a marble’s diameter andassuming a spherical shape. Results are compared with thoseexpected for a normal distribution. The radius of a glasscapillary tube is determined using Poiseuille’s equation bymeasuring the volume flow rate of water as a function of thehydrostatic head. In both experiments the experimentallyobtained standard deviation is compared with that estimatedby a propagation of uncertainty.Gordus, A.
A. “Statistical Evaluation of Class Data for TwoBuret Readings,” J. Chem. Educ. 1987, 64, 376–377.The volumes of water in two burets are read, and thedifference between the volumes are calculated. Studentsanalyze the data by drawing histograms for each of the threevolumes, comparing results with those predicted for a normaldistribution.Harvey, D. T.
“Statistical Evaluation of Acid/BaseIndicators,” J. Chem. Educ. 1991, 68, 329–331.In this experiment students standardize a solution of HCl bytitration using several different indicators to signal thetitration’s end point. A statistical analysis of the data using ttests and F-tests allows students to compare results obtainedusing the same indicator, with results obtained using differentindicators. The results of this experiment can be used laterwhen discussing the selection of appropriate indicators.O’Reilley, J. E. “The Length of a Pestle,” J.
Chem. Educ. 1986,63, 894–896.In this experiment students measure the length of a pestleusing a wooden meter stick, a stainless-steel ruler, and avernier caliper. The data collected in this experiment providean opportunity to discuss significant figures and sources oferror. Statistical analysis includes the Q-test, t-test, and F-test.Paselk, R.
A. “An Experiment for Introducing Statistics toStudents of Analytical and Clinical Chemistry,” J. Chem.Educ. 1985, 62, 536.Students use a commercial diluter to prepare five sets ofdilutions of a stock dye solution (each set contains tenreplicates) using two different diluters.
Results are comparedusing t-tests and F-tests.Richardson, T. H. “Reproducible Bad Data for Instruction inStatistical Methods,” J. Chem. Educ. 1991, 68, 310–311.This experiment uses the change in the mass of a U.S. pennyto create data sets with outliers. Students are given a sampleof ten pennies, nine of which are from one population. TheQ-test is used to verify that the outlier can be rejected. Classdata from each of the two populations of pennies are pooledand compared with results predicted for a normaldistribution.Sheeran, D. “Copper Content in Synthetic CopperCarbonate: A Statistical Comparison of Experimental andExpected Results,” J. Chem.
Educ. 1998, 75, 453–456.In this experiment students synthesize basic copper(II)carbonate and determine the %w/w Cu by reducing thecopper to Cu. A statistical analysis of the results shows thatthe synthesis does not produce CuCO3, the compound thatmany predict to be the product (although it does not exist).Results are shown to be consistent with a hemihydrate ofmalachite, Cu2(OH)2(CO3) • 1/2H2O, or azurite,Cu3(OH)2(CO3)2.—Continued1400-CH04 9/8/99 3:55 PM Page 9898ExperimentsModern Analytical ChemistryContinued from page 97Spencer, R.
D. “The Dependence of Strength in Plastics uponPolymer Chain Length and Chain Orientation,” J. Chem.Educ. 1984, 61, 555–563.The stretching properties of polymers are investigated byexamining the effect of polymer orientation, polymer chainlength, stretching rate, and temperature. Homogeneity ofpolymer films and consistency between lots of polymer filmsalso are investigated. Statistical analysis of data includes Qtests and t-tests.Thomasson, K.; Lofthus-Merschman, S.; Humbert, M.; et al.“Applying Statistics in the Undergraduate ChemistryLaboratory: Experiments with Food Dyes,” J.
Chem. Educ.1998, 75, 231–233.The absorbance of solutions of food dyes is used to explorethe treatment of outliers and the application of the t-test forcomparing means.Vitha, M. F.; Carr, P. W. “A Laboratory Exercise in StatisticalAnalysis of Data,” J. Chem. Educ. 1997, 74, 998–1000.Students determine the average weight of vitamin E pills usingseveral different methods (one at a time, in sets of ten pills,and in sets of 100 pills).
The data collected by the class arepooled together, plotted as histograms, and compared withresults predicted by a normal distribution. The histograms andstandard deviations for the pooled data also show the effect ofsample size on the standard error of the mean.4K PROBLEMS1. The following masses were recorded for 12 different U.S.quarters (all given in grams):5.6835.6205.5515.5495.5365.5525.5485.5395.5545.5525.6845.632Report the mean, median, range, standard deviation, andvariance for these data.2. Shown in the following rows are results for the determinationof acetaminophen (in milligrams) in ten separate tablets ofExcedrin Extra Strength Pain Reliever.18224.3261.7240.4229.4246.3255.5239.4235.5253.1249.7(a) Report the mean, median, range, standard deviation, and–variance for these data. (b) Assuming that X and s2 are good2approximations for µ and σ , and that the population isnormally distributed, what percentage of tablets are expectedto contain more than the standard amount of 250 mgacetaminophen per tablet?3.
Salem and Galan have developed a new method fordetermining the amount of morphine hydrochloride intablets.19 Results, in milligrams, for several tablets containingdifferent nominal dosages follow100-mgtablets99.1794.3195.9294.5593.8360-mgtablets30-mgtablets10-mgtablets54.2155.6257.4057.5152.5928.5126.2525.9228.6224.9319.068.839.08(a) For each dosage, calculate the mean and standarddeviation for the milligrams of morphine hydrochloride per–tablet.
(b) Assuming that X and s2 are good approximations2for µ and σ , and that the population is normally distributed,what percentage of tablets at each dosage level are expected tocontain more than the nominal amount of morphinehydrochloride per tablet?4. Daskalakis and co-workers recently evaluated severalprocedures for digesting the tissues of oysters and musselsprior to analyzing the samples for silver.20 One of themethods used to evaluate the procedure is a spikerecovery in which a known amount of silver is addedto the tissue sample and the percent of the added silverfound on analysis is reported. Ideally, spike recoveriesshould fall within the range 100 ± 15%. The resultsfor one method are106%108%92%99%104%101%93%93%Assuming that the spike recoveries are normally distributed,what is the probability that any single spike recovery will bewithin the accepted range?5.
The formula weight (FW) of a gas can be determined usingthe following form of the ideal gas lawFW =gRTPVwhere g is the mass in grams, R is the gas constant, T is thetemperature in kelvins, P is the pressure in atmospheres, andV is the volume in liters. In a typical analysis the followingdata are obtained (with estimated uncertainties inparentheses)1400-CH04 9/8/99 3:55 PM Page 9999Chapter 4 Evaluating Analytical DatagRTPV=====0.118 (±0.002)0.082056 (±0.000001)298.2 (±0.1)0.724 (±0.005)0.250 (±0.005)(a) What is the compound’s formula weight and its estimateduncertainty? (b) To which variable(s) should you direct yourattention if you wish to improve the uncertainty in thecompound’s molecular weight?6. A standard solution of Mn2+ was prepared by dissolving0.250 g of Mn in 10 mL of concentrated HNO3 (measuredwith a graduated cylinder).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
