Dealing with quantitative data and problem solving life is a story problem! Attacking Quantitative Problems
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1 Daling with quantitati data and problm soling lif is a story problm! A larg portion of scinc inols quantitati data that has both alu and units. Units can sa your butt! Nd handl on mtric prfixs Dimnsional Analysis Dis calculation using conrsion factors so unndd units cancl and ncssary units ar lft bhind. How many cm ar in km? Appndix A-5 Practic, Practic, Practic!!! Attacking Quantitati Problms. Rad th qustion. Idntify th rquird quantity (and units). Dis a way to us gin information to gt rquird quantity (and units) st up ncssary calculations Chck to b sur you gt th corrct units!. Insrt alus into quations and sol 5. Look at th answr is it rasonabl (ordr of magnitud, unit)? 6. Rad th qustion again Exampl: If th diamtr of a singl carbon atom is 5 pm, how many atoms would b in a lin 00.0 m long (about th diamtr of a human hair)? How much would this lin wigh if a singl carbon atom wighs.99 x 0 - g?
2 What can a rsult tll us? What is th "truth"? W don't (and can't) dtrmin th "tru alu" of any componnt in a mixtur. All masurmnts ha som associatd rror (uncrtainty). t Rprsntation of rsults must illustrat this rror; othrwis alu is uslss! W can work to undrstand (and impro) th prcision of a masurmnt. Prcision = Comparing multipl mthods incrass confidnc in th accuracy of our rsults. Accuracy = Rprsntation of Quantitati Data Significant Figurs: Numbr of digits ncssary to prsnt a rsult with th appropriat accuracy. Last digit has uncrtainty, at last! Rsults ar prsntd to th appropriat numbr of significant figurs, and oftn accompanid by th rror in th final digit. Valu Error.g ppm Whn rounding, look at all digits byond th last plac ndd. Always round alus that ar xactly halfway to th narst n digit
3 Whn looking at a numbr: Guidlins for Sig. Figs.. all nonzro digits ar significant,. all zros that t appar btwn nonzro digits it of a numbr ar significant,. all zros at th nd of a numbr on th right hand sid of th dcimal point ar significant,. all othr zros ar not significant. Exampls: How many significant figurs in ach alu? Sig. Figs. and Calculations In doing calculations, th numbr sig figs is limitd by th last crtain pic of data. Addition/Subtraction Multiplication/Diision Exampl: How many sig. figs. should b in th answrs to th following: = 58.6 x. = NOTE: Wait to truncat rsults until th nd of th calculation! WHY??? Indicat xtra digits by subscripting: Exact numbrs don t impact sig. figs. 6
4 Typs of Error (Uncrtainty) Uncrtaintis fall into two main classs: Systmatic Error: Random Error Two gnral approachs to quantifying rror: Absolut Uncrtainty Rlati Uncrtainty, (% Rlati Uncrtainty) 7 Statistics and Quantitati Data Statistics allow an stimation of uncrtainty in our data. AS LONG AS: Foundation of statistics: Normal Distribution Two paramtrs dfin th distribution: : : quncy of Obsrations Fr Statistical analyss rlat xprimntal data to this idal distribution. BUT, thr ar som challngs! Obsrd Valu 8
5 Statistics in Practic: Rmmbr th ky assumption! Dscribing th prcision of our data. Arag, or Arithmtic Man: n is th numbr of sampls xi i x n Sampl Standard Diation: s i x i x n 9 Dscribing th uncrtainty in our data. How good is it? Whn w dtrmin an arag (with som associatd rror), how sur ar w that th "tru alu" is clos to this arag? What factors influnc this confidnc? Th most common statistical tool for dtrmining that th "tru" alu is clos to our calculatd man is th confidnc intral. Th confidnc intral prsnts a rang about th man, within which thr is a fixd probability of finding th tru alu, m. ts x n 0 5
6 Mor on Confidnc Limits Valus for t ar tabulatd basd on sral confidnc lls and arious numbrs of dgrs of frdom. Confidnc Ll (%) Dgrs of Frdom NOTE: n though th numbr of masurmnts (n) is usd in th CI calculation, t is dtrmind basd on th dgrs of frdom (n-). Lt s look at som data: What is th arag? What is th standard diation (s)? What is th 95% confidnc limit? Error Propagation (aka propagation of uncrtainty) How do rrors in indiidual alus affct an analysis? Effcts dpnd on how th data is usd. Ruls com from calculus! l Error Propagation in Addition and Subtraction: Orall rror is basd on th absolut uncrtaintis of indiidual alus. Exampl: ??? 6
7 7 Mor Error Propagation Error Propagation in Multiplication and Diision Concpt is th sam as addition and subtraction, xcpt rlati uncrtaintis ar usd rlati uncrtaintis ar usd % % % %
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