Statistics and Chemical Measurements: Quantifying Uncertainty. Normal or Gaussian Distribution The Bell Curve

Transcription

1 Statitic ad Chemical Meauremet: Quatifyig Ucertaity The bottom lie: Do we trut our reult? Should we (or ayoe ele)? Why? What i Quality Aurace? What i Quality Cotrol? Normal or Gauia Ditributio The Bell Curve IF oly radom error are preet, data will follow a Gauia Ditributio Thi ditributio i decribed by: y e ( x) Two importat parameter for a Gauia ditributio : populatio mea or average The mea defie : populatio tadard deviatio The tadard deviatio defie Frequecy of Obervatio Stadard Deviatio from the Mea

2 Normal or Gauia Ditributio The Bell Curve Experimetal determiatio of ad i urealitic, becaue they are baed o a ifiite data et. SO, a more realitic goal i to calculate a arithmetic mea: xi i x, where i the umber of ample. It i alo more realitic to calculate a ample tadard deviatio: xi x i Why -? Remember e? Kow how to calculate o your calculator! 3 Relatig Stadard Deviatio, Gauia Ditributio ad Probability For ANY Gauia curve ( ormal ditributio, radom error): 68.3% of meauremet are withi td. dev. ( or ) 95.5% of meauremet are withi td. dev. 99.7% of meauremet are withi 3 td. dev. We ca predict the odd of fidig a value withi a pecific rage. It all boil dow to area uder the curve!. Pick a rage o the x-axi of the curve. Itegrate the area uder thi rage (Table 4-) 3. Thi area i the probability of obervig a value omewhere i thi rage. 4

3 Relatig Stadard Deviatio, Gauia Ditributio ad Probability For example: 50% of the value hould be > the mea, ad % hould be betwee the mea ad +3. Sice 34.3% of the obervatio fall betwee the mea ad +, ad 47.73% fall betwee the mea ad +, what fractio fall betwee + ad +? 5 So jut how good are your data? How do you kow (tatitically)? Whe we determie a average (with ome aociated error), how ure are we that the "true value" i cloe to thi average? What factor ifluece thi cofidece? The mot commo tatitical tool for determiig that the "true" value i cloe to our calculated mea i the cofidece iterval. x t The cofidece iterval preet a rage about the mea withi which there i a fixed probability of fidig. 6 3

4 Cofidece Iterval t x Value for t are tabulated baed o everal cofidece level ad variou umber of degree of freedom. Degree of Freedom Cofidece Level (%) NOTE: eve though the umber of meauremet () i ued i the CI calculatio, t i determied baed o the degree of freedom (-). How ca we work to miimize the rage calculated at a give cofidece iterval? How would you cut the CI i half experimetally? 7 Are two et of data really differet? How do we tell? Geerally bae our determiatio of the 95% cofidece iterval. If there i greater tha 95% probability that the data are the ame, we ay they do ot differ. Le tha 95% probability idicate tatitically differet reult. Ivolve calculatig a "t" (t calculated ) ad comparig the reult to tabulated value for t (t table or t critical ). Null Hypothei : Three differet coideratio:. Comparig a meaured reult with a "Kow" or "True" value.. Comparig two differet method. 3. Comparig differece of multiple ample ad two or more method. 8 4

5 Comparig a meaured reult with a "Kow" or "True" value. Key quetio: Doe the true value fall withi our cofidece limit? Ueful for comparig a reult to a tadard (i.e. SRM) Rearrage cofidece limit calculatio x t t calculated kow value x If t calculated > t table at 95% cofidece, the reult are tatitically differet. 9 Comparig Two Differet Method x If the reult of method A (, ) are differet from the reult of method B ( x, ), i thi differece igificat? Mut coider both the mea ad tadard deviatio Still compare t calculated ad t table, but ue ew calculatio t calculated x x pooled pooled x x x x i et A j et B + - = umber of degree of freedom If t calculated > t table at 95% cofidece, the reult are tatitically differet. Thi aume i the ame for both data et. If ot, the calculatio chage. How do we kow? F-Tet 0 5

6 F-Tet for Comparig Stadard Deviatio F calc = ( ) F alway ( ) Compare F calc with F table, if F calc >F table, differece i igificat! Comparig Differece of Multiple Sample ad Two or More Method. Oly idividual ample have bee ru, o replicate. The bai for our deciio become the average differece betwee the two method. t calculated d d d d d i i If t calculated > t table at 95% cofidece, the reult are tatitically differet. 6

7 Tet for Data Validity: Tetig for outlier Ueful whe oe piece of data appear to be outide a reaoable rage. Tet for tatitical probability that the outlier i a member of the ame populatio of the coitet data Thee are tatitical tet, but are till ubjective ad hould be ued carefully to avoid elimiatig ueful data!!! I. Q-Tet Q calculated gap rage gap i the differece b/w outlier ad earet value rage i total pread of the data. Compare Q calculated with Q table (typically ue 90% cofidece) If Q calculated i greater tha Q table, there i a tatitical probability that the outlier i a ivalid data poit ad may be dicarded. If Q calculated i le tha Q table, the data poit hould be retaied. 3 Tet for Data Validity: Tetig for Outlier II. Grubb Tet G calculated upect value x Compare G calculated with G table If G calculated i greater tha G table, there i a tatitical probability that the outlier i a ivalid data poit ad hould be dicarded. If G calculated i le tha G table, the data poit hould be retaied. Care mut be take to avoid dimiig ueful data! Commo See hould be the guide! 4 7

8 Spreadheet Tip ad Hit Excel i great, but o amout of calculatio ca alvage bad data! Whe eterig calculatio, ue parethee at will! SQRT(3+A5/) i differet tha SQRT((5+A5)/)!! Be ure order of operatio will be followed correctly. Expoet. Multiplicatio ad Diviio (left to right) 3. Additio ad Subtractio Documet your preadheet by icludig cell formula for critical calculatio Ue abolute referece whe helpful The dollar ig lock a row or colum i.e. $B$5 will refer to cell B5 i ay calculatio, but B$5 will allow the colum to vary while the row tay locked at 5 Lear commo built-i fuctio Thig like SUM, STDEV, AVERAGE Check out the IertFuctio meu i Excel Help or right-clickig ca come i hady, too! 5 8