Section 3. Measurement Errors

Transcription

1 eto 3 Measuremet Errors Egeerg Measuremets 3 Types of Errors Itrs errors develops durg the data aqusto proess. Extrs errors foud durg data trasfer ad storage ad are due to the orrupto of the sgal y ose. ystemat errors e strumet drft or as due to evrometal hages or strumet stalty. Radom errors e perturatos & upredtale effets o a measuremet sgal that our o oth sdes of the true value ut a e redued y averagg. Egeerg Measuremets 3

2 oures of systemat errors dsturaes the measured system aused y the measuremet proess toleraes of ompoets oetg leads wear agg oures of Errors evrometal fluees. Egeerg Measuremets 3 3 oures of radom errors Thermal ose - Johso ose due to moto of moleules resstve elemets Eletro ose - hot ose s due to of urret flutuatos a elemet Cotat ose oures of Errors Eletromaget Iterferee Vrato ad shok Egeerg Measuremets 3 4

3 Let x represet the th measuremet ( =...) the average value s defed as; where Radom Devato x xmea x x s the meda value (a reasg umer) xmeda x ( )/ f s odd x ( x x / / ) meda f s eve Egeerg Measuremets 3 5 The devato from the mea value s; ad the varae s; Radom Devato d x x V d where s the umer of data pots. The stadard devato s; V Egeerg Measuremets 3 6 3

4 Radom Devato The stadard error of the mea (e) s a estmate of the varalty expeted f the umer of measuremets s reased Numer of Data Pots.% e tadard Error (%).%.%.%.% Egeerg Measuremets 3 7 Hstogram A seres of measuremets ( = 5) have ee aqured. The umer of smlar measuremets have ee plaed s ad dsplayed. The mea ad stadard devato are omputed <x> = 45.6 =.9 Numer of Measuremets Measured Value Egeerg Measuremets 3 8 4

5 Frequey Dstruto Normalzg the umer of measuremets eah y the total umer the frequey of ourree s determed. A hstogram s trasformed to a frequey dstruto..5 Frequey of Ourree Measured Value Egeerg Measuremets 3 9 Proalty Dstruto Futo (pdf) As the umer of measuremets approah fty the urve eomes smooth urve ad s referred to as a pdf..5. Proalty Measured Value Egeerg Measuremets 3 5

6 Proalty Dstruto Futo (pdf) The proalty (P) that a measuremet s etwee x ad x+dx s foud y x where p(x) s the proalty dstruto futo (pdf). The proalty that all the aqured data falls wth the data set s % or. The mea value a e determed y P p ( x ) dx x p ( x ) dx x x mea x p ( x ) dx Egeerg Measuremets 3 Gaussa Dstruto Futo e most physal proesses are regular or otuous tme & spae a pdf a e desred y a ormal or Gaussa futo. p( x) e ( x x) 68.3 % of data pots s wth of the mea 95.4 % of data pots s wth of the mea 99.7 % of data pots s wth 3 of the mea Egeerg Measuremets 3 6

7 Error Estmate Estmated rage from measuremets; x x e 68.6% ofdeelevel x x e 95.4% ofdeelevel x x 3e 99.73% ofdeelevel tadard error from the mea: e N Comed effets of m urelated errors e e e... e m Egeerg Measuremets 3 3 Regresso Regresso s the proess of fdg a smple mathematal relatoshp y= f(x) etwee two varales x ad y ased o a seres of measured quattes x ad y ( =... ) Fttg the data wth oe of the followg geeral futoal forms; or m y f ( x) a x m j j f( x) a s( x ) j j j j j Egeerg Measuremets 3 4 7

8 The dfferee: d y f x ) A least-squares: ( d [ y f( x )] Regresso d or least-mea-squares dfferee) [ y f ( x )] Least-squares regresso (or fttg) m{ ( a... a... a m )} a a a m Egeerg Measuremets 3 5 Regresso Dsplaemet [mm] tal guess - -3 theory expermet -4 5 Posto [m] Mmum results the optmum oeffets Dsplaemet [mm] est-fttg urve theory expermet 5 Posto [m] Egeerg Measuremets 3 6 8

9 Uertaty Aalyss Aouts for that porto of the error that aot or s ot orreted for y alrato. - Determed expermetally -Calulated Egeerg Measuremets 3 7 Uertaty Aalyss Multple readgs wth a fxed kow put value. Uertaty = 3 sd tadard devato (sd) # samples Readgs Mea readg Egeerg Measuremets 3 8 9

10 Egeerg Measuremets 3 9 Uertaty Aalyss r u r u r u r u...) ( 3 f r Let The the overall uertaty s where s the reorded uertaty eah of the measured s u x Egeerg Measuremets 3 Wg Area Error Aalyss ad Fd the relatoshp for the uertaty the wg surfae area ().

11 Wg Area Error Aalyss Fd the relatoshp for the uertaty the wg surfae area (). Let = m ad = m. Let the uertaty = mm ad =.5 mm = Egeerg Measuremets 3