Experimental Errors and Uncertainty: An Introduction Prepared for students in AE by J. M. Seitzman adapted from material by J.

Transcription

1 Eeriental Errors and Uncertaint: An Introdction Preared for stdents in AE 305 b J. M. Seitzan adated fro aterial b J. Craig Otline Errors and tes of error Statistic/robabilit: confidence levels Uncertaint analsis Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-

2 Eeriental Error Error: all easreents have soe ncertaint error = = eas - eact Objectives. Miniize error so that - within soe ncertaint (statistical confidence) or eas - eact eas. Estiate error (ncertaint) to deterine reliabilit, eaningflness of data eact eas Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised- Errors and tes of error

3 Accrac and Precision Accrac: also called ssteatic or bias error denotes soething reeatabl wrong with the easreent or eerient Precision: also called rando error or noise denotes errors that change randol each tie o tr to reeat eerient Good Accrac Good Precision Good Precision Poor Accrac (can calibrate) Good Accrac Poor Precision (can average) Poor Accrac Poor Precision Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-3 Errors and tes of error

4 Accrac/Ssteatic Errors Sorces Measring sste errors difference between odel of easring sste and realit cold be corrected, e.g., with better odel of easreent Measred sste errors inflence of ncontrolled or nacconted for variables in the eerient the easred data a be correct, bt a lead to an incorrect odel of the object/rocess being stdied Blnders han errors - isnderstandings Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-4 Errors and tes of error

5 Soe Ssteatic Measreent Errors =() Actal backlash Model (=const) Actal Model Actal Model onzero offset - Backgrond Model hsteresis Backlash & Hsteresis Ssteatic errors can be eliinated/reoved if the are known tie Drift (e.g., offset changing in ssteatic wa with tie) Model Actal onlinearit Qantization Error (digitized data iacts resoltion) Actal Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-5 Errors and tes of error

6 Soe Rando Measreent Errors (=const) Actal Model 0 tie Backgrond oise (offset changing randol with tie) Detector oise (rando change in sensitivit of device) After data acqired, nearl iossible to searate rando error (noise) sorces Eaine rando error with statistical ethods Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-6 Errors and tes of error

7 Other Related Ters Sensitivit Change in a easreent device s ott for a nit change in the easred (int) qantit, e.g., volts/torr for the Baratron Resoltion Sallest increent of change in a sste or roert that a easreent device can reliabl catre Dnaic Range Mai ott of a easreent device divided b its resoltion (or ini easreable signal) Sensitivit = 5000 si/70 = 8.5 si/degree 0 70 Resoltion = 50 si Dn. Range= 5000/50 = 00 Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-7

8 Uncertaint Probabilit and Statistics We do not know the eact error if we did, we wold correct it and be error free We st estiate the error or ncertaint in or easreent this reqires alication of soe basic robabilit and statistics Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-8

9 Freqenc Probabilit Distribtions When we ake easreents (i.e., take sales) of a sste a nber of ties, we will get a distribtion of reslts Ssteatic ncertaint # ties reading is in given Y range Rando ncertaint theoretical distribtion Prob. Distrib. Fnction f We ight even ake jst one easreent (sale) of a sste that has a distribtion of ossible states 0 eact Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-9 avg Statistics and robabilit - f d

10 Statistics and Probabilit Since we can not ake an infinite nber of easreents to deterine the tre robabilit distribtion we se statistics to ake estiates based on assed distribtion fnction Soe sefl analtic distribtion fnctions noral (Gassian) stdent s t log noral eonential Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-0

11 oral/gassian Probabilit Distribtion f() f ( ) ( - ) e - = ean = variance = standard deviation Coonl sed when easreents/easreent sste: ade fro an indeendent sstes, each with an kind of distribtion # sales taken is ver large (e.g., sale eans) ore Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised- Statistics and robabilit

12 oral Distribtions Probabilit Range What fraction of vales (cobined robabilit) lie within given range fro ean for a noral distribtion? f() One Siga: Two Siga: Three Siga: Prob( - ) Prob( - ) - Prob( ) f ( ) d f ( ) d Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised- Statistics and robabilit

13 Sale Statistics What if and are nknown (as is often the case)? se estiates fro easreents, and s ; s Sale ean Sale variance s i i i Mean Sqare i ( i - ) i - - Sqare of ean - se (-) for s becase we have indeendent i bt if also know ean then onl need to know (-) i to cote last reaining i onl (-) degrees of freedo for this calclation Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-3 Statistics and robabilit

14 Uncertaint Estiates Qestion: If one takes (large) readings and cotes, how confident can o be that the average is reall close to the tre ean ()? Confidence intervals are wa to describe this Prob = c% that lies in shaded area defined b Gassian s variance of a c distrib. - a c =? Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-4 Statistics and robabilit

15 For noral distribtion Confidence Levels Error Level ae Error Level Prob. that Error is Saller Prob. that Error is Larger Probable Error % : One Siga 68% ~ :3 90% error.65 90% :0 Two Siga.96 95% :0 Three Siga % :370 Mai Error :000 For Siga % :6000 Si Siga % :.0e9 back Si Siga is sed for an electronic anfactring rocesses Eale : c 95% a c.96, so. 96 With 95% confidence, will fall within. 96 Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-5 Statistics and robabilit

16 Eale Yo have a differential ressre transdcer o are sing to easre the q of a wind tnnel. The anfactrer sas that the transdcer is linear to within 0.05% of fll-scale (0 Torr). Yo easre the q 500 ties and find the average is.5 Torr and the rs is 0.05 Torr. What are the ssteatic and rando ncertainties of or easreent? Ssteatic sst =0.05% of 0 Torr = Torr Rando s / 0.5 = 0.05 Torr/ = 0.00 Torr sing a 95% confidence level, rand = Torr = 0.00 Torr Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-8 Single easreent eale

17 Cobining Bias and Precision Uncertainties We noted earlier that errors in each easred variable ( i ) will inclde both bias (ssteatic) and recision (rando) coonents These can sall be treated as indeendent and therefore the ncertainties for each can be cobined into a total: / - total - recision -bias i i note: if the are not indeendent, cobining in other was a be necessar In ters of confidence intervals a / - total c -bias i i i i Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-9 Uncertaint analsis

18 Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-0 Analsis of Cobined Uncertainties Man ties eeriental reslts are the reslt of several indeendent easreents cobined sing a theoretical forla (e.g., for ass flowrate throgh a ie). How do ncertainties in each variable contribte to whole? If =(,, ) is a linear fnction, a statistical theore states that: / For ncertainties, i, that are sall coared to i we can se a Talor Series eansion in i : /... ),..., ( ),..., ( so that is now a linear fnction of the ncertainties. Aling this in the first eqation ields: 4 D RT A Uncertaint analsis

19 Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised- Eale: Uncertaint Calclation Consider easreent of ass flowrate throgh a rond ie / Δ D t L T D t L T 4 4 v v D t L RT D RT A D D t L RT 4 L,t The fractional ncertaint in can then be shown to be:* *e.g., Uncertaint analsis,t,... f,

20 Eale: Uncertaint Calclation (cont d) incldes factor of for D ter Asse the following ncertainties ( -recision = s / / ) Variable Accrac Precision otes ( /) ( /) 0.4% 0.% Pressre transdcer with 8-bit digitizer T % 0.4% Te. transdcer with % fll-scale linearit error sed at half-scale t 0.0% % Accrate clock, bt starting/stoing ncertaint of 0.0 sec for 0.5 sec easreent L 0.% Onl easred once with rler having ai 0.5 reading error over 0.5 ie length D % Onl easred once with with rler having ai 0.5 reading error over 0.05 diaeter Sed ( ) /.5%.0% t eas. doinates recision error D eas. doinates accrac error 95% Confidence.5% 4.0% With 95% confidence level recision error doinant Total ncertaint in a overall easreent of Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised- / 4.7% accrac ssteatic recision rando Uncertaint analsis is then:

21 (Pa) Plotting Uncertaint - Error Bars Ar He T (K) Can have error bars in vertical and/or horizontal coordinates Coright ,008 b J. Seitzan and J. Craig. All rights reserved. errosr&ncertaintrevised-3 Uncertaint analsis