Quantitative analysis requires : sound knowledge of chemistry : possibility of interferences WHY do we need to use STATISTICS in Anal. Chem.?
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1 Ch 4. Statstcs 4.1 Quattatve aalyss requres : soud kowledge of chemstry : possblty of terfereces WHY do we eed to use STATISTICS Aal. Chem.? ucertaty ests. wll we accept ucertaty always? f ot, from how wll we dsregard the data? by statstcal treatmet Radom Evets follows Gaussa Dstrbuto 4-1 Gaussa Dstrbuto 4. test of the lfe tmes of 4768 lght bulbs 1) mea value & stadard devato * mea : : or average
2 4-1 Gaussa Dstrbuto (Cot.) 4.3 * stadard dev. : s : measures how closely the data are clustered aroud the mea s ( ) 1-1 : degrees of freedom for a fte set of data: s (mea) (mu, (sgma, or popular mea) popular stadard devato) : var ace 4.4
3 4-1 Gaussa Dstrbuto (Cot.) 4.5 ) std.dev. & probablty Gaussa curve tells the broadess of Gaussa curve a gaussa curve area uder 1 = 68.3 % = 95.5 % 3 = 99.7 % 1 ( ) ep( y ) 4-1 Gaussa Dstrbuto (Cot.) 4.6 3) std.dev. of mea more measuremets more cofdet o average 1 (early the true value) ucertaty decreases by : = umber of meas. stadard devato of mea = : s = std.dev. * relatve stadard devato = (RSD) or to percetage = s = C.V. precso of mea = average devato of mea = d ( d ) s s 100
4 4- Cofdece Itervals 4.7 1) cofdece terval : a epresso statg that true mea,, s lkely to le wth a certa dstace our measuremets, s (stead of, ) True mea () s lkely to le wth a certa rage from Cofdece tervals t s 4.8 E. The cotet of carbohydrate a glycoprote (a prote wth sugars attached to t) s determed to be 1.6, 11.9, 13.0, 1.7, ad 1.5 g per 100 g of prote replcated aalyss. Fd the 50% ad 90% cofdece tervals for the carbohydrate cotet. mea = 1.5, std = 0.4
5 4-3 Comparso of meas wth Studet's t (from dfferet measuremets) 4.9 : tool for epressg cofdece terval for comparg results from other epermetal tech. Normally, 95% cofdece level : Two results do ot dffer from each other IF there s 95% chace that our cocluso s correct. Case 1. t test : measured result wth kow value 4.10 : whe we test a ew aalytcal method, we wat to see f t agrees to a kow value. e) N cotet; kow value : % (from std. Materal) measured value : 0.039, 0.03, , % The 95% cofdece terval? ths terval does't cover , thus, measured value are dfferet from kow val. Not wth the radom error boudary. (t mples there ests systematc errors)
6 1. <t-test> You are developg a procedure for determg traces of copper 4.11 bologcal materals usg a wet dgesto followed by measuremets by atomc absorpto spectrophotometry. I order to test the valdty of the method, you obta a NIST orchard leaves stadard referece materal ad aalyze ths materal. Fve replcas are sampled ad aalyzed, ad the mea of the results s foud to be ppm wth a stadard devato of 0.7ppm. The lsted value s 11.7ppm. Does your method gves a statstcally correct value at the 95% cofdece level? Case. t test: comparg replcate measuremets (test of two sets of measuremets) : test the two techques are statstcally the SAME or NOT 4.1 for two sets of data, 1, measuremets t 1 S pooled 1 1 S pooled ( 1) ( j ) s1 ( 1 1) s ( 1 1 1) If t cal > t table (wth 95%) ths dfferece s sgfcat (out of radom error rage) there ests systematc error
7 4.13 E) The average mass of troge from ar Table 4-3 s = g, wth a stadard devato of s 1 = , (for 1 =7 measuremets). The average mass from chemcal sources s =.9947 g, wth a stadard devato of s = (for =8 measuremets). <t-test> A ew gravmetrc method s developed for ro (II) whch the ro 4.14 s precptated crystalle form wth a orgaocarbo "cage" compoud. The accuracy of the method s checked by aalyzg the ro a ore sample ad comparg wth the results usg the stadard precptato wth ammoa ad weghg of FeO3. The results, reported as % Fe for each aalyss, were as follows. Test method Referece Method 0.10% 18.89% =19.65% =19.4% Is there a dfferece betwee the two methods?
8 Case 3; Comparg dvdual dffereces 4.15 Two dfferet methods o several dfferet samples (o duplcato) Cholesterol cotet (g/l) Plasma sample Method A Method B Dfferece (d ) d =+0.06 t cal d S d s d ( d d ) 1 Is my red blood cell cout hgh today? 4.16 Red cell couts o fve ormal days : 5.1, 5.3, 4.8, 5.4, ad cells/l =5.16 s=0.3 Today s value = cells/l t cal today' s cout S d What s the probablty of fdg t=4.8 for 4 degrees of freedom? See table 4.: at 4 degrees of freedom, 4.8 les betwee 98 & 99% There s less tha a % probablty of observg a cout of cells/l o ormal days. reasoable to coclude that today s cout s elevated.
9 4-4 Comparso of st.dev. wth the F test 4.17 F test ---- check two std.devs are sgfcatly dfferet each other. S F calc If F S calc > F table the sgfcat Grubbs test for a outler 4.18 durg measuremets of mass lost of zc, we eed to dscard some questoable data 10., 10.8, 11.6, 9.9, 9.4, 7.8, 10.0, 9., 11.3, 9.5, 10.6, 11.6 If G calc > G tab, the rejected.
10 4-7. Method of Least Squares Fdg the BEST STRAIGHT LINE ; correlato betwee data pots 1) Method of Least Squares y = m + b m: slope, b: y-tercept each data --- (, y ) vertcal devato = d = y -y = y -(m + b) 4-7. Method of Least Squares 4.0 we wat to MINIMIZE d (whether postve or eg.) -- drect summato of each d? o good method of mamum lkelhood : Assume a gaussa dstrbuto wth std.dev.. for the observatos about the actual value y( ) at = the probablty P P 1 ep 1 y y mamze the probablty? mmze the sum the epoetal
11 4-7. Method of Least Squares 4.1 d d = (y -y) = (y -m -b) mmzg (assume ) m b METHOD OF LEAST SQUARES m b ( y ) ( ) ( ) ( ) y (y) ( ) ( ) y 4-7. Method of Least Squares 4. ) How relable are least-squares parameters? estmate UNCERTAINTY slope & tercept std. dev. of y (d ) s y y degrees of freedom ( - ) m b ( ( y ) ( y ( ) ( ) ) )
12 4-8. Calbrato Curves 4.3 Std. Soluto : solutos wth kow cocetratos How to buld calbrato? 1. prepare a seres of std. Solutos (varyg coc.) measure absorbace.. subtract the absorbace of blak soluto 4-8. Calbrato Curves Plot the absorbaces vs. Cocetrato the do least squares.
13 4-8. Calbrato Curves 4.5 Ucertaty Propagato Calbrato curve m : slope Depeds o # of calbrato pots. Lowest error data from the ceter of calbrato Homework F, 13, 14, 16, 0, 33, Addtoal Problems Set
14 Addtoal Problems Set The followg replcate calcum determatos o a blood sample usg AAS ad a ew colormetrc method were reported. Is there a sgfcat dfferece the precso of the two methods? AAS (mg/dl) 10.9, 10.1, 10.6, 11., 9.7, 10.0 Colormetrc (mg/dl) 9., 10.5,9.7, 11.5,11.6, 9.3, 10.1, 11.. Studets measured the cocetrato of HCl a soluto by ttratos usg dfferet dcators to fd the ed pot. Is the dfferece betwee dcators 1 ad sgfcat at the 95% cofdece level? Aswer the same questo for dcator ad Idcator 1. Bromothymol blue. Methyl red 3. Bromocresol gree Mea HCl cocetrato (M) (+std.dev.) Number of Measuremets
15 A Stadard Referece Materal s certfed to cota 94.6 ppm of a orgac cotamat sol. Your aalyss gves values of 98.6, 98.4, 97., 94.6, ad 96. ppm. Do your results dffer from the epected results at the 95% cofdece level? If you made oe more measuremet ad foud 94.5, would your cocluso chage?
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