/7/06 Aalzg Two-Dmesoal Data The most commo aaltcal measuremets volve the determato of a ukow cocetrato based o the respose of a aaltcal procedure (usuall strumetal). Such a measuremet requres calbrato, or the preparato of a calbrato curve. Determato of the respose of the method to solutos of kow cocetrato (stadards). Oce the respose for the stadards s kow, the cocetrato of a ukow ca be determed IF the cocetrato/respose relatoshp s well defed. Ideall prefer a lear relatoshp does t have to be lear as log as ou kow what t s, ca ofte force olear relatoshps to appear lear b approprate expermet desg Aalzg Two-Dmesoal Data Respose Cocetrato Importat questos to ask:. How do we defe the best le?. How do errors our data affect ths le? 3. How cofdet ca we be of the ukow cocetrato that we calculate from our calbrato curve?
/7/06 Aalzg Two-Dmesoal Data Example: Prote determato usg spectrophotometr. IMPORTANT: Absorbace Prote mass Prote (g) 0.00 9.36 8.7 8.08 37.44 Absorbace 0.466 0.756 0.843.6.80 Our objectve s to draw the best ft le through the data, but how? Mmze devato (spread) of the data aroud the le Absorbace.40.30.0.0.00 0.90 0.80 0.70 0.60 0.50 0.40 0 0 0 30 40 Prote (mg) Mathematcall, ths s a least squares aalss Work to mmze the square of the devato (to remove effects of sg) from our calculated le. Qualtatvel ths s eas, quattatvel; thgs are a lttle more challegg. 3 Tpcall workg to defe a straght le, = mx +b Assume that values for x have lttle error, but more error s assocated wth values for. Sce our data have some scatter, each datum ma devate from the le the -drecto. Ths s also called a resdual (d ) d = le = (mx + b) We reall wat to mmze the square of the devatos (actuall the SUM of the squares): (d ) = ( -mx -b) (d ) = -mx - b + mx b + m x + b (d ) = - mx - b + mx b + m x + b How do we do ths? 4
/7/06 (d ) = - mx - b + mx b + m x + b Two parameters, so two partal dervatves to set equal to zero: = - x + x b + mx =-x + bx + mx = 0 =- + mx + b = - + mx + b = 0 Ths produces two equatos ad two ukows (m, ad b) we should be able to solve ths sstem! 5 (d ) = -mx - b + mx b + m x + b Wth a lttle had-wavg (ad the magc of calculus ad lear algebra), we are able to mmze the equato above ad solve for m ad b, whe we do, we get: (x ) m x D x b x (x ) D Each operato volves takg the determat of a matrx x D x x a c b d a d b c There s ol oe soluto to the sstem of equatos So ol oe least squares le! 6 3
/7/06 Lets appl ths to our example data: x D x m x 4380.48 Now lets calculate some pots based o our le: Prote (g) Absorbace x x x 0.00 0.466 0.000 0.000 9.36 0.756 7.076 87.609 8.7 0.843 5.780 350.438 8.08.6 34.46 788.486 37.44.80 47.90 40.75 93.60 4.57 05.06 68.88 (x) x x D 0.045 (x ) b D 0. x 4946 Prote (g) Absorbace x calc d d 0.00 0.466 0.495-0.09 0.00088 9.36 0.756 0.704 0.05 0.00663 8.7 0.843 0.94-0.07 0.005069 8.08.6.4 0.0 0.00404 37.44.80.334-0.054 0.00894 93.60 4.57 4.57 0.000 0.0 Absorbace.40.30.0.0.00 0.90 0.80 0.70 0.60 0.50 0.40 0 0 0 30 40 Prote (mg) 7 How relable are m, b, ad values we determe based o our calbrato curve? The majort of our cofdece depeds o the scatter of values about the le, or the stadard devato, s (also called s r, st. dev. about regresso). (d d) d s Lke usual, the umber of degrees of freedom s the deomator. Wh - degrees of freedom? Other std. devs. deped o s s m s D s b s x D x x s x x x x s sx m k D D D m k m where k s the umber of replcate measuremets of the ukow ad s the umber of calbrato pots. 8 4
/7/06 Cofdece Lmts for m, b, x calc Cofdece Lmts for m, b, x calc m ± ts m b ± ts b x calc ±ts x t s for - degrees of freedom 9 R ad Such Plottg Excel (or o m calculator) gves me R (or R) values. What the #$%@ do these mea? R (or r ): Coeffcet of Determato s the fracto of the scatter the data that ca be descrbed b the lear relatoshp. R compares the varato of the data from the least-squares le to that due to radom scatter: R le A R close to does t guaratee good precso m ad b.5.5 =(0.56±0.07)x + (0.7±0.) R = 0.954.5.5 =(0.39±0.05)x + (0.0±0.) R = 0.957 0.5 0.5 0 0 4 6-0.5 x 0 0 4 6-0.5 x 0 5