Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal
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1 9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd Effects of Sstematc Errors Constant Errors Become more serous as sze of measurement get smaller Proportonal Errors Interferng contamnants I f the contamnant becomes larger, the sgnal becomes larger.
2 9/3/009 Detecton of Sstematc and Personal Errors Calbraton Care and Self-dscplne Instrument readngs otebook entres Calculatons Phscal dsabltes--color blndness Bas Dffcult to detect Analze standard samples Do an ndependent analss Determne a blank Var the sample sze Applng Statstcs to Data Evaluaton Gross error or segment of populaton? Defne the confdence nterval. Fnd the number of replcates necessar to ensure that the mean falls wthn a predetermned nterval. What s the probablt that an epermental mean and a true value or a two epermental means are dfferent. Calbrate
3 9/3/009 Gross Errors The Q -test: rejectng outlers Gross Errors The Q -test: rejectng outlers Q ep q q n d w quest. result - nearest neghbor range The Q -test: An Eample A calcte sample elds the followng data for the determnaton of calcum as CaO: 55.95, 56.00, 56.04, 56.08, and Should we reject 56.3? Q ep q q n
4 9/3/009 The Q -test: The Q -Table (5-) umber of Q crt Observatons 90% 95% 99% The Q -test: An Eample A calcte sample elds the followng data for the determnaton of calcum as CaO: 55.95, 56.00, 56.04, 56.08, and Should we reject 56.3? Q ep q q n What s the crteron? If Q ep > Q crt, reject. If Q ep < Q crt, accept. Q ep = 0.54; Q crt = 0.64, so accept. Here 4
5 9/3/009 Can we reject data? Blnd applcaton of statstcal tests s no better than dong nothng. Use good judgement based on eperence. If ou know that somethng went wrong wth a sample and the sample produces an outler, then rejecton ma be warranted. Be cautous about rejectng data for an reason. Recommendatons Keep good records and eamne the data carefull. If possble, estmate the precson of the method. Repeat the analss f tme and sample are avalable. Compare wth frst data. If not feasble, appl the Q -test. Recommendatons If Q -test ndcated retenton, consder reportng the medan. The medan allows ncluson of all of the data wthout undue nfluence from the outler. The medan of a set of 3 measurements from a normal dstrbuton gves a better estmate than the mean of the remanng values after an outler s rejected. 5
6 9/3/009 Confdence Lmts and Intervals Confdence Lmts are lmts around an epermentall determned mean wthn whch the true mean les wth a gve degree of probablt. The confdence nterval s the nterval around the mean defned b the confdence lmts. Confdence lmts f s s a good estmate of CL for (sngle measurement) z CL for (mean of measurements) z 50% Confdence Lmts 6
7 9/3/009 80% Confdence Lmts 90% Confdence Lmts 95% Confdence Lmts 7
8 9/3/009 99% Confdence Lmts Confdence lmts f s s not a good estmate of t Student' s t (analogous to z) ts CL for (mean of measurements) Values of Student's t Degrees of Probablt Level Freedom 90% 95% 99% 99.8% (z)
9 9/3/009 Fndng the Confdence Interval: An Eample Determnaton of the alcohol content n blood gves the followng data: % C H 5 OH: 0.084, 0.089, and (a) If the precson of the method s unknown, fnd the 95% confdence lmts of the mean. (b) Perform the same calculaton f the the standard devaton s = % C H 5 OH. (How could we determne s?) (a) unknown (use t ) s 95%CL 0.084% C H OH % C H OH 5 5 ts (4.30)(0.0050) % C H OH 5 Values of Student's t Degrees of Probablt Level Freedom 90% 95% 99% 99.8% (z)
10 9/3/009 (b) s = % (use z ) 95%CL 0.084% C H OH % C H OH 5 5 z (.96)(0.0050) % C H OH 5 Values of Student's t Degrees of Probablt Level Freedom 90% 95% 99% 99.8% (z) Comparng a mean to the true value: The ull Hpothess The null hpothess assumes that two measurments are the same. An numercal dfference s assumed to be due to random error. If the observed dfference s greater than or equal to the dfference that would occur 5% of the tme, the null hpothess s rejected, and the dfference s judged sgnfcant. 0
11 9/3/009 The Crtcal Value We rearrange the equaton for theconfdencenterval. ts ts Compare the dfference to the crtcal value The dfference s compared to the crtcal value ts / the desred probablt level. at If s greater than the crtcal value, the null hpothess s rejected. An Eample: The Determnaton of Sulfur n Kerosenes A known sample contanng 0.3% sulfur was analzed and the results for four samples were: 0., 0.8, 0.5, and 0.9 %S. Is there bas n the method? Let s do a spreadsheet.
12 9/3/009 The Spreadsheet (5%) True Val. Data t(95%, 3 df) Dfference ts/sqrt() Mean 0.6 Std. Dev If we wsh to be wrong no more than 5% of the tme, we must reject the null hpothess, and there s sstematc error. What about %? Here True Val. Data t(99%, 3 df) Dfference ts/sqrt() Mean 0.6 Std. Dev If we wsh to be wrong no more than % of the tme, we must accept the null hpothess, and there s no sstematc error. Comparng Two Epermental Means ts pooled d.f.
13 9/3/009 Least-Squares for Analzng Lnear Calbratons: = m +b Least-squares assumes that there s relatvel lttle error n the measurement. The mathematcs of the dervaton of the equatons mnmzes the sum of the squares of the devatons (the resduals ) of the ponts from the best lne n the drecton onl. From calculus, take the partal dervatves of the equaton for the sum of squares wth respect to m and b, set t equal to zero, and solve for the varables. 3
14 9/3/009 The Intermedate Equatons (See pp. 6-) S S S and The Results. Slope : m s r S S S or the standard error of m S 4. The standard devaton of. Intercept: b 3. The standard devaton about regresson, where the estmate: the slope: d.f. s m m s S r The Standard Devaton about Regresson Analogous to the standard devaton Measure of the scatter of ponts Precson smlar to ndvdual data s r S m S lne m b 4
15 9/3/009 More Results 5. The standard devaton of the ntercept: s b 6. The standard devaton of resultsfromthe calbraton curve : s b c s r sr m M M M M c m S where no. replcatesof the unknown 5
16 9/3/009 Assgnment 7-, 7-4, 7-6, 7-, 7-6, 7-9 SS p. 64 6
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