Evaluato of ucertaty measuremets Laboratory of Physcs I Faculty of Physcs Warsaw Uversty of Techology Warszawa, 05
Itroducto The am of the measuremet s to determe the measured value. Thus, the measuremet begs wth specfyg the quatty to be measured, the method used for measuremet (e.g. comparatve, dfferetal, etc.) ad the measuremet procedure (set of steps descrbed detal ad appled whle measurg wth the selected measurg method). I geeral, the result of a measuremet s oly a appromato or estmate of the value of the specfc quatty subject to measuremet, that s, the measurad. Thus, the result of measuremet s complete oly whe accompaed by a quattatve statemet of ts ucertaty. Iteratoal Stadard Orgazato (ISO) prepared Gude to the Epresso of Ucertaty Measuremet, whch s deftve documet descrbg orms ad procedures the measuremets ucertaty evaluato. Based o the teratoal ISO stadard, Polsh orm Wyrażae epewośc pomaru. Przewodk was accepted the 999.
Sources of ucertaty a measuremet complete defto of the measurad; mperfect realzato of the defto of the measurad; orepresetatve samplg the sample measured may ot represet the defed measurad; adequate kowledge of the effects of evrometal codtos o the measuremet or mperfect measuremet of evrometal codtos; persoal bas readg aalogue strumets; fte strumet resoluto or dscrmato threshold; eact values of measuremet stadards ad referece materals; eact values of costats ad other parameters obtaed from eteral sources ad used the data-reducto algorthm; appromatos ad assumptos corporated the measuremet method ad procedure; varatos repeated observatos of the measurad uder apparetly detcal codtos.
Types of measuremets Drect measuremet measured quatty ca be drectly compared wth the eteral stadard, or the measuremet s made usg a sgle strumet gvg result straghtaway seres of measuremets gross error Idrect measuremet measurg oe or more physcal quattes to determe quatty depedet o them
Basc deftos () Measuremet ucertaty - parameter assocated wth the result of measuremet characterzg dsperso of the values attrbuted to the measured quatty Stadard ucertaty u( the ucertaty of measuremet epressed as a stadard devato. Ucertaty ca be reported three dfferet ways: u, u( or u(accelerato), where quatty ca be epressed also words ( the eample s accelerato). Please ote, that u s a umber, ot a fucto.
Idrect measuremets Laboratorum Fzyk, Wydzał Fzyk, Poltechka Warszawska
Basc deftos () Type A evaluato of ucertaty the evaluato of ucertaty by the statstcal aalyss of seres of observatos. Result of a seres of measuremets: mea value Assumptos: Dstrbuto fucto s symmetrcal probablty for results smaller as well as bgger tha mea value are the same The bgger devato from the mea value the lower probablty Result: for bgger umber of measuremets observed dstrbuto of data pots s smlar to Gauss fucto Eample of a Type A evaluato of ucertaty: the stadard devato of a seres of depedet observatos ca be calculated, or least squares method ca be appled to ft the data wth a curve ad determe ts parameters ad ther stadard ucertates.
Gauss dstrbuto Dstrbuto for cotous varable : ( ep μ epected value σ stadard devato ( (d 3 3 ( d ( d ( d 0,683 0, 997 0, 954 Gauss dstrbuto for fte umber of pots: epected value s equal to mea value, stadard devato s equal to stadard devato of a mea value Type A stadard ucertaty for a seres of measuremets s equal to stadard devato of a mea value u( s ( ) (
Basc deftos (3) Type B evaluato of ucertaty the evaluato of ucertaty by meas other tha the statstcal aalyss of seres of observatos, thus usg method other tha type A. Type B evaluato of stadard ucertaty s usually based o scetfc judgmet based o eperece ad geeral kowledge, ad s a skll that ca be leared wth practce. Assumpto: uform dstrbuto probablty s costat the whole terval determed by measuremet ad calbrato ucertaty calbrato ucertaty (due to measuremet devce vestgator ucertaty (due to vestgator s epermetal sklls e ) u( 3 ( 3 Combato of ucertates u( s ( 3 ( ) 3 e
Uform dstrbuto Probablty desty the terval a to b s costat ad dfferet from zero ad equal to zero outsde ths terval Desty probablty fucto for uform dstrbuto: ( ( 3 3 3 ( 0 outsde ths rage Epected value: Varace: a b b a Type B stadard ucertaty s equal to stadard devato a = - b = u( ( 3 3
Type B stadard ucertaty () mechacal devces Rulers, mcrometers, calpers Thermometer, baromether Stopper calbrato ucertaty : half of the scale terval Aaloge devces u( 3
Type B stadard ucertaty () aalogue devces Measuremet rage mamal value to be measured for the set rage. Class of the strumet descrbes the precso of the measuremet devce covertg measured sgal to value preseted o a scale. Class descrbes ucertaty the percetage of the measuremet rage. Calbrato ucertaty: class rage 00 u( 3 Ivestgator ucertaty: e umber rage of scale tervals e u( 3
Type B stadard ucertaty (3) dgtal devces Measuremet ucertaty for dgtal devces: measured value z measuremet rage c cz c, c devce costats e.g. c = 0,%, c = 0,0% u( 3 Avalabe fuctos Measuremet rage
Ucertaty evaluato drect measuremets summary Perform measuremet (sgle or seres) Type A ucertaty Measuremet result mea value Stadard ucertaty stadard devato of the mea value u( s ( ) ( Type B ucertaty Calbrato ucertaty Ivestgator ucrtaty e u( 3 ( 3 Combato of ucertates u( s ( 3 ( ) 3 e
Idrect measuremets Laboratorum Fzyk, Wydzał Fzyk, Poltechka Warszawska
Basc deftos (4) Combed stadard ucertaty u c ( stadard ucertaty of the value calculated based o measuremets of other quattes ucertaty propagato rule Measuremets of correlated quattes Measuremets of ucorrelated quattes I the Physcs Laboratory all measuremets are ucorellated
Ucertaty evaluato drect measuremet summary Measure k quattes drectly (sgle or seres) z f (,,..., k ) Calculate mea value ad stadard ucertaty u( ) for every quatty usg Type A or Type B evaluato method,,..., k u( ), u( ),..., u( k ) Calculate fal value of studed quatty z z f (,,..., k ) Calculate combed ucertaty u c (z) (ucertaty propagato law) u c ( z) j k f ( j j ) u ( j ) Eample for two quattes u c ( z) f (, y) u ( f (, y) y u ( y)
Basc deftos (5) Epaded ucertaty U( or U c ( the measure of ucertaty that defes terval about the measuremet, that may be epected to ecompass a large fracto of the dstrbuto Stadard ucertaty u( defes terval about the measured value, where the true value est wth probablty: 68% for Type A ucertaty 58% for Type B ucertaty Epaded ucertaty: Allows to compare results from dfferet laboratores Allows to compare results wth referece database or theoretcal value Useful for commercal purposes Requred for dustry, health ad securty regulatos
Basc deftos (6) Coverage factor k umber used to multply stadard ucertaty to calculate epaded ucertaty Typcally k vares from to 3. I the most cases the Physcs Laboratory k = should be used. Epaded ucertaty U( defes terval about the measured value, where the true value est wth probablty for k = : 95% for Type A ucertaty 00% for Type B ucertaty (00% also for k=,73!) U( k u(
Reportg measuremet results Laboratorum Fzyk, Wydzał Fzyk, Poltechka Warszawska
Reportg measuremet results () Ucertaty s preseted wth accuracy (ouded) to two sgfcat dgts The measuremet result (the most probable value) s preseted wth a accuracy specfed by the ucertaty, whch meas that the last dgt of the measuremet result ad the measuremets ucertaty must be at the same decmal place. Roudg of ucertates ad measuremet results follows the mathematcal rules of roudg Stadard ucertaty t =,364 s, u(t) = 0,03 s t =,364(3) s, recommeded otato t =,364(0,03) s Epaded ucertaty t =,364 s, U(t) = 0,046 s (k = ) = t = (,364±0,046) s. recommeded otato ot requred
Reportg measuremet results () eamples Measuremet Proper Reportg a = 3,735 m/s; u(a) = 0,4678 m/s a = 3,74 m/s; u(a) = 0,5 m/s a = 3,74(0,5) m/s a = 3,74(5) m/s b = 3785 m; u(b) = 330 m C = 0,0000045 F; u c (C) = 0,00000056 F T = 373,43 K; u(t) =,3456 K b = 3800 m; u(b) = 300 m b = 3800(300) m b = 3,8(,3) 0 3 m b = 3,8(,3) km C=0,00000 F; u c (C)=0,00000056 F C =,00(0,56) 0-6 F C =,00(56) 0-6 F C =,00(56) μf T = 373,4 K; u(t) =,3 K T = 373,4(,3) K U(T) = 4,7 K T = (373,4 ± 4,7) K
Hypothess verfcato Laboratorum Fzyk, Wydzał Fzyk, Poltechka Warszawska
Lear fucto hypothess Graphcal Least squares method Statstcal tests
Graphcal test The most smple Plot theoretcal model fucto. I should cross ucertates bars for more the /3 of epermetal data pots If ot hypothess should be rejected
Least squares method () Goal: to verfy the theoretcal model depedece betwee measured quattes s vald Assumpto: every model ca be coverted to lear type fucto y = a + b Method: least squares to fd the le for whch the sum of squared devatos of epermetal pots from ths le s the smallest to fd le whch s the closest to all epermetal pots Results: a, b ad ucertaty u(a) ad ucertaty u(b) (Type A stadard ucertaty)
Least squares method () ~ a y b y a ~ ~ ~ ~ d y a y ~ ~ ~ s s d s a b a y = a + b
Least squares method(3)
Test () Test fucto Defto Statstcal weght Lear type fucto Sgfcace value probablty of hypothess rejecto Value the rage of to 0 Determed by the vestgator (typcally 0,05) Depeds o umber of freedom (umber of measuremet pots mus umber of calculated parameters) w ( y w ( y y( )) B( ) A) w [ u( y )] Crtcal value χ crtcal (lsted the table for every sgfcace value ad umber of freedom) Test fucto χ χ crtcal there s o argumets to reject hypothess χ > χ kcrtcal hypothess should be rejected