ANALYTICAL CHEMISTRY LECTURE
ANALYTICAL CHEMISTRY LECTURE CHEM 321
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Chapter 3 Experimental Errors Chapter 4 Statistics Sttpg tm a Typ ta Quantitative Almatyt t if gtData of unknown quality are useless gtAll laboratory measurements contain experimental error gtlt is necessary to determine the magnitude of the accuracy and reliability in your measurements gtThen you can make a iudgment about their usefulness Terms amp Definitions Replicates iwe er more determinatlens on the same sample Example 31 One student measures Fe III concentrations six times The results are listed below 194 195 19 6 198 201 203 ppm parts per million 6 replicates 6 measurements The quotmiddlequot or quotcentralquot value for a group of results in 1 i1 gt Mean average erarithmetie mean gt Median the middle value of replicate data gt If an add number of replicates the middle value ef replicate data gt If an even number of replicates the middle have values are averaged to obtain the median Calculation Mean and Median Example 32 measurements of Fe III cancentrations 194 195 19 6 198 201 203 ppm parts per million What are the mean and median of these measurements Mean 194195 196198201203 6 1978 ppm 193 ppm 6 replicates An even number cf replicates ll 196198 2 Median 197 ppm Calculation Mean and Median Example 33 measurements of Fe III concentrations 194 195 19 6 198 201 ppm parts per millipn What are the mean and median of these measurements 19p419 5 196198201 5 Mean 1968 ppm 197 ppm 5 replicates An odd number 0f replicates 1 Median 196 ppm Any Questions Terms amp Definitions gt Precision describes the regroducibilig of measurements How close are results which have been obtained in exactly the same way The reproducibility is derived from the deviation from the mean Deviation from the mean ii Ii El to Standard deviation 5 Variance F inef cient of variation Terms amp Definitions gt Accuracy the closeness of the measurement t0 the true or accepted value This quotclosenessquot called as the error a oluge or relative error of a result to its true value gtabsclute error gtrelative error Terms amp De nitions gt Outlier Occasional ermr that abviausly differs signi cantly fram the resi cf the results Precision amp Accuracy High rZIII39E I fu Inw IE I39IQ39IZquotBIE39JF 115311 UL39L39IJE39diizfl high pracisinn Mean amp True Value 3 1978 x 2000 0 O O O O O 196 200 204 ppm ironIH Absolute and Relative Errors gt Absolute Error E the difference between the experimental value and the true value Has a sign and experimental units Eamp g Experimental value true acceptable value gtRelative Error E the absolute error corrected for the size of the measurement or expressed as the framedx or partsper thousand ppt of the true value g1 has a sign but no units 33quot x100 1 parts per hundred pph EIr x100 parts per thousand ppt Er x1000 Calculation Absolute and Relative Errors Example 34 measurements of Fe concentrations 194 195 19 6 198 201 203 ppm Assumed we already knew the true value pf Fe II I concentration 31200 ppm What are absolute and relative errors of each measurement E 194 200 06 ppm E 195 200 05 ppm E 196 200 04 ppm E 198 200 02 ppm E 201 200 01 ppm E 203 200 03 ppm Er 0620x100 3 Er 0520 x100 25 3 Er 0420 x100 2 Er 0220 x100 1 Er0120x100 05 Er 0320x100 15 2 Example 35 A mathod of analysis yields weights for gold that are low by 03 mg Calculate the percent relative error caused by this uncertainty if the weight of gold in the sample is a 800 mg b 500 mg c 100 mg d 25 mg w lt100 t Er 03 rug500 mg x100 006 006 pph 06 ppt Any Questions Mes of Errors gt Systematic or determinate errors affect accuracy gt Random or indeterminate errors affect precision gt Gross errors or blunders Lead to outlier s and require statistical techniques to be rejected Systematic or Determinate Errors 1 Instrument BITOI39S failure to calibrate degradatien 01 parts in the instrument power uctuations ate 2 Method errors errors due to ma ideal physical or chemieal behavior completeness and speed of reamicn interfering side reactions sampling pmblems 3 Personal errors accur where measurements require judgment result frem prejudice celor acuity prcblems Systematic or determinate errors P Potential Instrument Errors gtVariation in temperature gtContamination of the equipment gtPower uctuations gtComponent failure All of these can be corrected by calibration or proper instrumentation maintenance Systematic or determinate errors gt Method Errors gtSlow or incomplete reactions gtUnstable species gtNonspeci c reagents gtSide reactions These can be corrected with proper method development Systematic or determinate errors gtPersonal Errors gtMisreading of data gtImpr0per calibration gtPoor technique sample preparation gtPersonal bias gtImproper calculation of results These are blunders that can be minimized or eliminated with proper training and experience The Effect of Systematic Error normally quotbiasedquot and often very quotreproduciblequot 1 Constant errors E is of the same magnitude regardless of the size of the measurement This error can be minimized when larger samples are used In other words the relative ermr decreases with increasing amount of analyte Er EsX x100 Constant eg Solubility loss in gravimetric analysis eg Reading a bu rat 2 Proportional errors E increases or decreases with increasing or decreasing sample size respectively In other words the relative error remains constant Proportional Typically a contaminant or interference in the sample Detection of Systematic Method Errors 1 Analysis cf standard samples 2 Independent Analysis Analysis using a quotReference Methodquot sr quotReference Lab 3 Blank determinations 4 Variation in sample size detects constant error only Gross Error gtGress errors cause an experimental value to be discarded gt Lead to outlier s and require statistical techniques to be rejected gtExamples of gross error are an obviously quotoverrun end pointquot titration instrument breakdown loss of a crucial sample and discovery that a quotpurequot reagent was actually contaminated gtWe do NOT use data obtained when cross error has occurred during collection Random Errors gt caused by uncontrollable variables which normally cannot be de ned gtThe accumulated effect causes replicate measurements to uctuate randomly around the mean gt Random errors give rise to a normal or gaussian curve 1 Results can be evaluated using statistics gt Usually statistical analysis assumes a normal distribution Term amp De nition The Nature of Random Errors also called quotindeterminatequot and follow a predictable pattern gt Error is the deviation from the quottrue value l gt Random error results in values that are higher or lower than the quottrue valuequot The Statistical Treatment of Random Error A The Population and the Sample Data gt The population data is an in nite number of observations all the possible results in the universe gt The sample data is a nite number of observations that are hopefully representative of the population A Normal or Gaussian Curve Normal distribution mam i 15 633 ufthe data 20 955 13 997 L 7 Frequency 3930 20 10 0 10 20 36 The Statistical Treatment of Random Error Properties of a Gaussian Curve has a population mean u and a population standard deviation Population meant In the absence of systematic error u is the true value for the measurement The sample meanj approaches u when the number of observations approach in nity 2 Population standard deviation Standard Deviation v The Population of Standard Deviation a Samgle Standard Deviation as a measure of precision gt Reliability of the sample standard deviation 5 increases with the number of replicates N gt For N greater than 20 s 2 e gt Measuring 20 replicates is usually net practical Standard Deviation Standard Deviation s Other measures of grecision e Standard deviation 3r Variance 32 Sr Relative standard deviation 3r Coef cient of variation 3 Spread or range Other measures of precision Variance s The advantage of working with variance is that variances from independent sources of variation may be summed to obtain a total variance for a measurement N 2 xi i 2 variance 52 i N 1 Other measures of precision Relative standard deviation RSD RSD 1000 ppt RSD 100 pph Coefficient of variation CV RSD x100 X Spread or Range W iThe differencze batween the largest and the smallest values in the set of data 3 Another term that is accasionally used t0 described the precision of a set of replicate data Calculation SI VarianceI RSDI CVl Range Example 36 measurements of Fe III concentrations I 195 19 6 194 198 201 203 ppm parts per million What are the standard deviation variance RSD coefficient of variation CV and range w of the data set m b Standard Deviation s Nl 195 196194198201 203 6 Mean 1978 ppm N replicates 6 Nl 61 5 number of degrees of freedom g 195 19 78 mas 1973 19 4 193ar 193 way 261 1938 ma 1973 quotV 61 0396 040 PPm Standard Error of a Mean or Standard Deviation of a Mean Sm rThe standard deviation of each mean is known as the standard error of the mean or Standard Deviation of a Mean o o 5 Standard Dev1atlon of a mean Sm Ht shows that the standard error of the mean is inversely proportional to the square root of the number of data replicates N Calculation SI SmI VarianceI RSDI CVI Range Example 37 measurements of Fe III concentrations I 195 19 6 194 198 201 203 ppm parts per million What are the standard deviation SmI variance RSD coefficient of variation CV and range w of the data set 195 1978 196 l978 194 1973 l98 l978 201 1978Y 2031978 61 F 0396 040 ppm Standard Deviation of a mean sm w016 ppm J6 Calculation of Sm the standard error of the mean aia on me Cabumiou of 1 leL mpu r a For 50 trails S 1 Sm 3 4 5 00056 3 m 00008 x m m 7 W 2 w W 3 4 i 993 53 80X10 L 3 33 2 b For 125 trails 1 22 a 0 0058 0001 Mam mmmc 9 982 mL WWW L 12X103 mL 3mm umnuu o moss mL m mu m m mdcr ubmmxd umm mm WWW nlm Pooled standard deviation Combine standard deviation from different experiments to obtain a reliable estimate of the precision of a method M NZ N 2 l 2 2Xi Xl ZXj X2 ZXk X3 3971 j1 k1 s 397 pooled if N1NZN3Nt Example 38 Tm m l39CLll39 in samples nf wen sh taken 15mm Ellhcsagm kt Bay was demar min ed by 1 mc md based up n the Elh rpti Df radiation by g mus cle mcnml mercury Calculzn u a pwuulcd SIiII l t Hi the atundard den39iatiml fm hie THEEth Etrzmmjl upun Lhc rst hr mlumn Hf dam Number cutquot hI aru hpm H 3 Sum Inf qumrcs nf Dm39i 39liama 39um Mam Sill39i lplu Specimcn Muagurm El39ig Cnntcznti ppm gt 1 3 LEI 153 Mini 2 1 196 193 H133 ML I 393 E 7i 335 4 i EDIE 2 I 1M3 199 17915 5 151 Hii wiltJ E 3 235 244 2 1H 2 43 144 T 4 l1l5EI391211U391 Number m madmlremenm 33 i 13739 H J I 5 33r li 2018 15m 243 i 1 3f 130253 El 0 15 Jmm U116 ll 10 1 l4 l mm Sum mlquot gimme T 121915 The valum in tha last WEI cirlum s fur Sp il 1 ware cnmpumd as Enl luwsr 3 H1quot Bl If E LSD 012 3316 153 111393 101336 0033 11643111 SITE Sum Elf squares MESS 5 132 1 T3 1 3 5 The other data in cramming 4 and 5 wan htained similarly Then 131233 110115 0132 91m 1 313114 Mm 31mm 25 39i ll ppm Hg