EVALUATION OF UNCERTAINITY IN MEASUREMENTS

Transcription

1 Adrzej Kubaczyk Laboratory of Physcs I Faculty of Physcs Warsaw Uversty of Techology EVALUATION OF UNCERTAINITY IN MEASUREMENTS (Studet s gude to Physcs Laboratores) Chapter - Itroducto Iteratoal Stadard Orgazato (ISO) prepared Gude to the Expresso 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. Ths gudebook s addressed to Physcs Laboratores studets ad ts ma goal s to provde formato ecessary to uderstad teratoal regulatos cocerg measuremets of physcal quattes as well as evaluato of measuremet ucertates the everyday laboratory expermets. Chapter - Base deftos - Measuremet set of actvtes determg the measured value. - Measuremet ucertaty - parameter assocated wth the result of measuremet characterzg dsperso of the values attrbuted to the measured quatty (measurad). - Stadard ucertaty u(x) the ucertaty of measuremet expressed as a stadard devato. Ucertaty ca be reported three dfferet ways: u, u(x) or u(accelerato), where quatty x ca be expressed also words ( the example x s accelerato). Please ote, that u s a umber, ot a fucto. - Type A evaluato of ucertaty the evaluato of ucertaty by the statstcal aalyss of seres of observatos. - 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. - Combed stadard ucertaty u c (x) represets the estmated stadard devato of the result ad s obtaed by combg the dvdual stadard ucertates (both type A ad type B), usg the usual method combg stadard devatos. - Expaded ucertaty U(x) or U c (x) the measure of ucertaty that defes terval about the measuremet that may be expected to ecompass a large fracto of the dstrbuto of values that Warsaw, October 07

2 Evaluato of ucertaty measuremets could reasoably be attrbuted to the measurad. The expaded ucertaty s especally applcable for some commercal, dustral ad regulatory applcatos. - Coverage factor k umerc multpler of the stadard ucertaty used for expaded ucertaty determato. Typcally, k s the rage to 3. For the most laboratory cases t s recommeded to use k =, whch defes a terval havg a level of cofdece of approxmately 95%. Chapter 3 - Ucertaty measuremets 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 approxmato 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. It must be uderled ad always remembered, that report of measuremet result cossts of determed value of the measured quatty ad measuremet ucertaty. There are may possble sources of ucertaty a measuremet, cludg:. complete defto of the measurad;. mperfect realzato of the defto of the measurad; 3. o-represetatve samplg the sample measured may ot represet the defed measurad; 4. adequate kowledge of the effects of evrometal codtos o the measuremet or mperfect measuremet of evrometal codtos; 5. persoal bas readg aalogue strumets; 6. fte strumet resoluto or dscrmato threshold; 7. exact values of measuremet stadards ad referece materals; 8. exact values of costats ad other parameters obtaed from exteral sources ad used the data-reducto algorthm; 9. approxmatos ad assumptos corporated the measuremet method ad procedure; 0. varatos repeated observatos of the measurad uder apparetly detcal codtos. These sources are ot ecessarly depedet, ad some of sources (sources from to 9) may cotrbute to source 0. Two categores of ucertaty ca be dstgushed based o ther method of evaluato, "Type A" ad "Type B" []. Type A evaluato of stadard ucertaty may be based o every statstcal data aalyss methods. For example, the stadard devato of a seres of depedet observatos ca be calculated, or at least squares method ca be appled to ft the data wth a curve ad determe ts parameters ad ther stadard ucertates. Type B evaluato of stadard ucertaty s usually based o scetfc judgmet takg to accout all avalable formato cludg: - prevous measuremet data; - experece wth or geeral kowledge of the behavor ad propertes of relevat materals ad strumets; - maufacturer's specfcatos; - data provded calbrato ad other certfcates; - ucertates assged to referece data take from hadbooks. The proper use of the pool of avalable formato for a Type B evaluato of stadard ucertaty calls for sght based o experece ad geeral kowledge, ad s a skll that ca be leared wth practce. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory

3 Evaluato of ucertaty measuremets Chapter 4 - Evaluato of ucertates If the measured quatty ca be drectly compared wth the exteral stadard, or f the measuremet s made usg a sgle strumet gvg result straghtaway, s called drect measuremet. Ths type of measuremet clude: legth measuremet wth a ruler, dameter measuremet of the rod usg a mcrometer, tme measuremet wth a tmer, measuremet of electrc curret wth a ampermeter or a voltage measuremet wth a voltmeter. I may cases, t s ecessary to measure oe or more physcal quattes to determe quatty depedet o them. Ths type of measuremet s called drect measuremet. These clude for example: the measuremet (determato) of resstace by measurg curret ad voltage, determato of the cylder volume by measurg ts dameter ad heght, the measuremet of gravtatoal accelerato based o the legth ad the perod of oscllato pedulum. Methods used for calculatg the measuremet ucertaty deped o whether the measuremets were made drectly or drectly. 4. Drect measuremets Cosder put quatty X determed a drect way, whch value s estmated from depedet measuremets x, x,,..., x. If oe of the measured values dffer sgfcatly from the other (gross error), t should be dsregarded ad must ot be take to accout further calculatos. I the most cases gross errors are caused by the vestgator (e.g. readg V stead of. V) or by a mometary dsrupto of the measuremet codtos. The decso to recogze gross error depeds o the vestgator ad s usually take at the stage of terpretato of the results. Type A evaluato of stadard ucertaty The set of depedet measuremets x, x,,..., x ca cosdered as a -elemet radom sample of the fte set of measuremets. If the probablty dstrbuto for x s descrbed by Gaussa fucto (see Appedx B), the followg way of data aalyss ca be appled. The arthmetc mea value should be cosdered as a result of the measuremet: x x x. () It should be oted, that the bgger umber of measuremets results a better mea value. Stadard devato of the mea value of the measuremet of quatty X s equal: where s x s called the stadard devato of the mea value. Type B evaluato of stadard ucertaty u( x) sx ( x x), () ( ) Qute ofte the laboratory work oly oe measuremet s performed (or a sgle measuremet of each measured quattes) or measured values show o spreadg. Ths ca happe especally, whe the measurg devce has low accuracy. For example, whe measurg the thckess of the plate wth mcrometer screw, set of dfferet results wll be obtaed, but f the mllmeter ruler wll be used to measure the same plate we wll get always the same result. The accuracy of used meter devce determes the calbrato ucertaty Δx (also called the ucertaty level). Ths s the umber specfed by the maufacturer of the measurg devce or estmated o the bass of the scale terval used the devce. The probablty of ay result wth the rage defed by measured value ad calbrato accuracy s the same. Such a probablty dstrbuto s called a uform dstrbuto, ad the stadard devato ths case s defed by formula x / 3 (Appedx B). The Type B stadard ucertaty s equal to ths value: Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 3

4 Evaluato of ucertaty measuremets x ( x) u( x) (3) 3 3 The other source of Type B measuremet ucertaty may be vestgator ucertaty Δx e ad ts value s estmated o the bass of vestgators expermetal sklls ad type of performed measuremet. The stadard ucertaty ths case s also calculated usg the formula (3), where Δx should be replaced wth Δx e. I the case, where two sources of Type B ucertaty are observed both stadard devatos should be added: Combato of ucertates ( x) ( x ) u( x) e. (3a) 3 3 If there are two types of ucertates the same expermet (Type A - dsperso of results ad Type B - the ucertaty of calbrato ad vestgator), they should be added usg followg formula for the stadard total ucertaty: ( x) ( xe) u( x) sx. (4) 3 3 It should be oted, that f oe of the calculated ucertates s more tha oe order of magtude smaller t could be eglected. Equato (4) should be appled oly the case, whe all of calculated stadard devatos are the same order of magtude. Determato of calbrato ucertates ad vestgator ucertates for basc strumets used the laboratory - Mechacal devces (rulers, mcrometers, calpers) - Δx s equal to half of the terval. Calbrato ucertaty for mercury barometers, thermometers, stoppers etc. s defed the same way. - Aalogue devces - calbrato ucertaty Δx ca be determed usg the class of the strumet ad chose measurg rage: class rage x. (5) 00 Oly the vestgator observg the poter durg the measuremet process ca estmate the vestgator ucertaty. - Dgtal devces (electroc) - measuremet ucertaty s defed the techcal data of the devce specfed the maual. It depeds mostly o the measured value of x ad the smaller extet o the used rage: c x c z, (6) x where c ad c are devce costats specfed user maual - for the most of dgtal voltmeters used the laboratory c = 0.05% ad c = 0.0%, however may cases oe ca apply c = 0. Please ote that order to determe Type B stadard ucertaty, calculated above Δx values should be dvded by 3 ad possbly use the law of propagato of ucertaty. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 4

5 Evaluato of ucertaty measuremets 4. Idrect measuremets Idrect measuremet of the quatty Z (called output) s performed by measurg k values measured drectly (called put), marked as x, x,..., x k. Desred value z depeds o x k : z f x, x,..., x ) or z f x ). ( k For every drectly measured quatty, ts mea value x,..., ( k, x xk ad stadard ucertates u(x ), u(x ),..., u(x k ) should be determed. Stadard ucertates ca be calculated by both, Type A ad Type B methods. Obvously, the case of method B, there s o average value but oly the measuremet result. The measured value of Z s calculated usg the formula: z f x, x,..., x ). ( k The ucertaty of Z s called the combed ucertaty u c ad s calculated usg the followg formula (the law of propagato of ucertaty): k f ( x j ) uc( z) u ( x j ). (7) j x j Whe the dervatve fuctos are calculated, x j values should be replaced wth the mea values I the case of two drectly measured quattes (x ad y) combed ucertaty s equal: u c x j. f ( x, y) f ( x, y) ( z) u ( x) u ( y). (8) x y I the case of drect measuremets oe ca cosder measuremets of correlated ad ucorrelated quattes (correlated ad ucorrelated measuremets). I the ucorrelated measuremets every quatty s measured drectly but a dfferet, depedet expermet (e.g. measured ad calculated at dfferet tme). As a example of ucorrelated measuremets we could cosder expermets for determato of the gravtatoal accelerato usg a pedulum. There are two depedet measuremets of the legth of the pedulum ad the oscllato perod ad based o ths the accelerato s calculated usg followg formula: g. 4 l T I the correlated measuremets measured quattes are somehow depedet o each other. I ths case t s mportat to perform all measuremets of put quattes uder exactly the same expermetal codtos, wthout troducg ay chages the measuremet system. However, vrtually all expermets the Physcs Laboratory oly ucorrelated measuremets are performed ad therefore combed ucertaty should be calculated usg formula (7) or (8). 4.3 Exteded ucertaty The stadard ucertaty u(x) defes the terval from x u(x) to x u(x), wth whch the value of X ca be asserted to le wth the probablty of 68% for the Type A ucertaty, ad wth the probablty of 58% for Type B ucertaty (these values are the result of Gaussa ad uform probablty dstrbutos, respectvely). The stadard ucertaty s a measure of the accuracy of measuremets ad allows for the comparso of dfferet methods of measuremet. To make possble comparso of measuremet results obtaed dfferet laboratores ad uder dfferet codtos, the cocept of the exteded ucertaty U was troduced. Exteded ucertaty s commoly used to allow comparso of measured results wth the referece data, but also for commercal purposes ad to establsh stadards of dustral health, safety, etc. Exteded ucertaty U(x) s equal to stadard ucertaty value u(x) multpled by the coverage factor k, so that the - Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 5

6 Evaluato of ucertaty measuremets terval x ±U(x) the large fracto of the dstrbuto of measured values could reasoably be attrbuted to X. U( x) k u( x), (9) I the most cases the laboratory practce the coverage factor k = should be used, sce for ths value the probablty of fdg the true value of X the terval x ±U(x) s equal to 95% to Type A ucertaty ad 00% for Type B ucertaty (probablty equal to 00% for Type B ucertaty s acheved already for k =.73!). For the laboratory referece devces usually exteded ucertaty wth the coverage factor to k = 3 s preseted as a measure of ucertaty. I ths case the stadard ucertaty, correspodg to a probablty of 68%, s equal to /3 of the specfed value of ucertaty. 4.4 Reportg ucertaty It should be oted, that the report of measuremet result must cossts of measured value ad measuremet ucertaty ad both of quattes should be expressed usg SI uts (see Appedx A). To report results of measuremets properly oe should start from the correct recordg of ucertaty. Ucertaty s preseted wth accuracy (rouded) 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 follows the mathematcal rules of roudg: dgts 0-4 are rouded dow ad the 5-9 dgts are rouded up. Stadard ucertaty ca be reported a umber of ways: () t =.364 s, u(t) = 0.03 s () t =.364(3) s, ths s the most commo ad recommeded way to report results for scetfc publcatos ad referece data (3) t =.364(0.03) s I otato () the brackets two sgfcat dgts of stadard ucertaty are preseted whereas the otato (3) stadard ucertaty s preseted full form but wth the same accuracy of sgfcat dgts. Exteded ucertaty s oly oted wth the symbol ±. For the above metoed example U(t) = k u(t), t =.364 s, U(t) = s (k = ), = t = (.364±0.046) s. Chapter 5 - Measuremets of fuctoal type relatos I the typcal laboratory cases of drect measuremets are multple, almost smultaeous measuremets of two quattes x ad y depedet o each other. For dfferet values of x dfferet values of y wll be obtaed ad pars of umbers (x, y ) wll be the result of the measuremet. If there s a kow fucto relatg measured quattes x ad y (e.g. y = ax ) t s possble to determe fucto parameters ( ths example, the parameter a by fttg collected data wth the kow fuctoal relatoshp. The fttg procedure provdes also stadard ucertaty of determed parameters (Type A ucertaty). However ths ucertaty does t deped o the ucertaty of the measured values x, y ad to take t to accout complex ucertaty u c (z) has to be calculated for oe of the measured pars x, y (Type B ucertaty). Two types of ucertates should be combed usg ucertaty propagato rule. It s also possble to verfy theoretcal depedece betwee two measured quattes by fttg collected data wth the kow fucto. For example oe ca check the applcablty of Ohm's law by measuremet of curret-voltage characterstc for ukow resstace. For ths purposes dfferet methods of fttg ca be appled wth the most kow ad wdely used least squares method, whch wll be descrbed later ths gudebook. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 6

7 Evaluato of ucertaty measuremets 5. Lear type fuctos To verfy theoretcal model, depedece betwee measured quattes collected data pots have to be ftted to the kow fucto. I some cases ths relato ca be qute complcated cludg also mplct type fucto. Usually, the physcal model also provdes a rage of values, for whch model equato s vald. The task of the vestgator s to perform as may measuremets as possble the rage of applcablty of the model to obta the best ft. Moder computer software allows fttg ay fuctoal relatoshp to the set of measuremets. However, the most of relatoshps physcs ca be reduced to a lear form (learzed). Such a learzato s based o the trasformato of fucto y = f(x) to aother fucto Y = F(X), whch wll take the form of a frst degree polyomal Y = BX + A. Some examples of trasformato to lear depedece: y d s x X s x, Y y, B d y y y x y 0e X x, Y l, B 0 x y 0 y y0 e X l, Y x, B y Metoed above trasformatos are uque for fucto y = f(x). Two sets of expermetal data (x, x, x 3,... x ) ad (y, y, y 3,... y ), should be trasformed to (X, X, X 3,... X ) ad (Y, Y, Y 3,... Y ) ad plotted Y=F(X). I the ext step fttg procedure usg lear type fucto has to be appled. 5. Least squares method The most commoly used method of lear regresso aalyss s the least squares method descrbed detals Appedx C. The goal of ths method s to determe parameters of modellg fucto to le as close as possble to all expermetal pots. For ths purpose sum of squared resduals s calculated, where resdual s the dfferece betwee a observed value y ad the value provded by the model (Bx + A). Parameters A ad B are modfed teratvely to mmze sum of squared resduals. [ y ( Bx A)] Studets ca prepare ther ow procedures to apply least squares method, followg the formato the Appedx C, but t s much more coveet to use some software wth lear ft or lear regresso optos. As a result slope of the le (parameter B) ad ts tercept (parameter A) as well as ther stadard ucertates u(b) ad u(a) ca be obtaed. It should be oted, that there are dfferet types of least squares methods. I the smplest case ucertates of measured values x ad y are ukow, but assumed to be equal. I ths case stadard ucertates u(b) ad u(a) does t deped o the ucertates of measured values. The least squares method s oe of the statstcal methods ad therefore t gves a Type A ucertaty. Appedx D shows a example of usg least squares method wth McroCal Org 8 software. m. 5.3 Verfcato of lear depedece hypothess Lear regresso models, cludg least squares method, ca be used to model expermetal data. However, ot always lear fucto offers proper ft to measured values. Most of the fttg software packages calculates also value of correlato factor, whch s a umber from the rage [-, ] descrbg correlato betwee varables. Ufortuately the most laboratory cases ths parameter s close to or - ad does t provde eough formato o the devato from learty. Therefore other test should be performed to verfy lear depedece for measured values. Oe of the possbltes s the graphcal aalyss. At frst data pots should be plotted, cludg ucertaty le segmets of measured quattes. If the theoretcal model le crosses ucertaty le segmets for less tha /3 of expermetal data pots, hypothess of lear depedece should be rejected, eve f the correlato factor s hgh. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 7

8 Evaluato of ucertaty measuremets The most commoly method used for hypothess verfcato s χ test. The varable χ s defed: w ( y y( x )), (0) where y(x ) value of the tested hypothess fucto at pot x, w s a statstcal weght value for pot umber, whch s calculated accordg to followg formula: w [ u( y )]. () Let us cosder verfcato of lear depedece usg χ test. I ths case value of ( y y( x )) s equal to squared dfferece betwee measured value y ad value of tested fucto at pot x = x, ( y B( x ) A). For every measuremet pot ths value dvded by the squared stadard ucertaty should t be bgger tha. I fact value bgger tha meas, that ftted le does t cross ucertaty le segmet for ths pot. If the ftted le crosses all ucertaty le segmets, test fucto χ should ot be bgger tha umber of measured pots. Calculated value of χ fucto provdes formato o correlato betwee expermetal data ad ftted le. The secod very mportat parameter of χ test s statstcal sgfcace value α. Sgfcace level descrbes level of cofdece about lear depedece of measured data. The value of sgfcace vares from 0 to ad s a put parameter set by vestgator. For the hgh value of α also may good pots (lyg close to model le) wll be rejected ad for the very small values also pots ot fttg to model le wll be cosdered. I the Laboratory expermets typcal value of α = 0.05 should be appled. It s also strogly suggested to aalyse seres of data ot smaller tha = 6 pots. I the typcal case χ value determed for expermetal data pots s compared wth crtcal value χ crtcal for gve sgfcace value ad gve umber of degree of freedom. Degree of freedom s equal to umber of data pots reduced by umber of parameters used fttg procedure (for least squares procedure there are two parameters B ad A). Crtcal values χ crtcal for dfferet sgfcace value ad degree of freedom values are preseted Appedx E.. χ χ crtcal - hypothess o lear depedece betwee measured values ca be accepted. χ > χ crtcal - hypothess o lear depedece betwee measured values should be rejected If the value of χ fucto for expermetal data s sgfcatly hgher tha crtcal value, oe should cosder f the measuremet ucertaty s t too bg or measuremets should t be repeated wth hgher accuracy measuremet devces. If the lear depedece hypothess s rejected, the other model should be tested. A example of χ test s preseted the Example 3 the ext Chapter. SUMMARY: least squares method appled for correlated quattes allows determg value of parameter correlatg these quattes. However, at frst ftted le should be plotted ad vestgator should check f t crosses ucertaty le segmets. The secod stroger codto s a χ test, whch allows acceptg or rejectg hypothess regardg set sgfcace level. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 8

9 Evaluato of ucertaty measuremets Examples Reportg measuremet results Results of expermet ad calculatos a = m/s; u(a) = m/s b = 3785 m; u(b) = 330 m C = F; u c (C) = F T = K; u(t) =.3456 K R = ; u c (R) = x =.345 A; u(x) = A y =. A; u(y) = A Reportg results a = 3.74 m/s; u(a) = 0.5 m/s a = 3.74(5) m/s a = 3.74(0.5) m/s b = 3800 m; u(b) = 300 m b = 3800(300) m b = 3.8(.3) 0 3 m b = 3.8(.3) km C= F; u c (C)= F C =.00(56) 0-6 F C =.00(0.56) 0-6 F C =.00(56) μf T = K; u(t) =.3 K T = 373.4(3) K U(T) = 4.7 K (k=) T = (373.4 ± 4.7) K R = 7886 ; u c (R) = 67 R = 7886(67) R = 7.886(0.067) k U c (R) = 30 (k=) R = (7890 ± 30) R = (7.89 ± 0.3) k x =.345 A; u(x) = A x =.345() A x =.345(0.000) A y =.000 A; u(y) = A y =.000() A y =.000(0.000) A NOTES: () Notato uderled s RECOMMENDED ad should be used () Notato bolded refers oly to exteded ucertaty (3) It s allowed also to report results by text e.g. The speed of soud ar s equal to 3.74 m/s wth stadard ucertaty 0.5 m/s Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 9

10 Evaluato of ucertaty measuremets Example The dmeso of squared cross-secto rod was measured wth callper wth accuracy of 0. mm. Obtaed results are:.5;.3;.6;.5;.3;.5;.7;.3;.7;.4;.3. Determe sze of the rod. Report result properly. Measured quatty (the legth of rod sde - d) was measured drectly by a seres of measuremets. The result of expermet s a mea value accordg to formula (): d d d mm Sce the callper s terval s equal to 0. mm calbrato ucertaty Δd s twce smaller 0.05 mm. Type B ucertaty ca be calculated: 0.05 u( d) d mm 3 3 Sce dstrbuto of results s observed Type A stadard ucertaty should be calculated usg formula (): u( d) s d ( ) () ( d d ) ( d.46364) mm Both ucertates Type A ad Type B are the same order of magtude, therefore ucertaty propagato rule should be used to determe overall stadard ucertaty: ( d) u( d) s mm. d 3 Stadard ucertaty ca be preseted dfferet ways: u = mm u(d) = mm u(sde of the rod) = mm Proper reportg of the result: d =.464 mm. u(d) = mm d =.464(55) mm d =.464(0.055) mm If the exteded ucertaty s requred (e.g. to be compared wth lterature referece data), the result should be reported: U(d) = k u(d). d =.46 mm. U(d) = 0. mm (k = ). = d = (.46±0.) mm. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 0

11 Evaluato of ucertaty measuremets Example [4] To determe gravtatoal accelerato, tme of flght of a body dropped from heght h was measured. The heght h was measured 3 tmes wth a le ad every tme the same value of 70 mm was obtaed. Tme of flght was measured 5 tmes usg a stopper wth a accuracy 0.00 s ad recorded values are: t = 0.509, t = 0.5, t 3 = 0.50, t 4 = 0.504, t 5 = 0.50 s. The ucertaty related wth the vestgator ad start ad stop momets was estmated to be 0.0 s. Calculate gravtatoal accelerato ad ts ucertaty. h Gravtatoal accelerato g ca be calculated usg formula g. At frst the average t heght h ad average tme t s calculated usg formula (): h 70mm. 7m, t s. Now, based o h ad t values the g value ca be calculated:.7 g m/s. To determe ucertaty of drectly measured g stadard ucertates of tme ad heght should be calculated at frst. Stadard ucertaty u(t): Type A ucertaty: Formula () wll be used: 5 3 u ( t) st ( t t ) ( t 0.507).0350 s =.035 ms. ( ) 5 4 Type B ucertaty: The ucertaty related wth the vestgator ad hs decso whe to start ad stop measurg tme of flght was estmated to be Δt e = 0,0 s = 0 ms (accuracy of the stopper ca be eglected sce s oe order of magtude smaller). Therefore Type B stadard ucertaty s equal: te 0 u ( t) ms. 3 3 It should be oted, that both types of ucertates are the same order of magtude ad therefore ucertaty propagato rule should be used to determe overall stadard ucertaty: u( t) The result of tme measuremet: t = 0.507(6) s, =5. 6. ms. Stadard ucertaty u(h): I ths case there s o dstrbuto of results ad stadard ucertaty of heght measuremet should be determed as the Type B ucertaty. The smallest terval the le s equal to mm but takg to accout other factors (o-vertcal algmet of le, vestgators error) ucertaty of ths measuremet should be estmated as Δh = mm. Therefore stadard ucertaty Type B s equal: u( h) h.5 mm. 3 3 Measuremet of heght ca be reported: h = 70.0() mm or h =.700() m, =. Combed ucertaty of gravtatoal accelerato u c (g): Sce gravtatoal accelerato s measured based o two drect ad ucorrelated measuremets ucertaty propagato rule should be used (7): u c g g 4h ( g) u ( h) u ( t) u ( h) u ( t) 3 h t t t Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory

12 Evaluato of ucertaty measuremets m/s If we compare two compoets of combed ucertaty uder square root, oe ca clearly fd that the frst oe related wth the heght ucertaty s eglgbly small compared to tme ucertaty compoet. Notato reportg gravtatoal accelerato measuremet results: g = 9.87 m/s, u c (g) = 0.4 m/s g = 9.87(4) m/s g = 9.87(0.4) m/s Exteded ucertaty U c (g): Accordg to formula (9): U c (g) = u c (g) = 0.37 m/s = m/s. The fal result of gravtatoal accelerato: g = (9.87±0.47) m/s. Ths value ca be compared wth referece data accordg to whch gravtatoal accelerato Warsaw s equal to 9,80665 m/s. Ths value lay wth rage defed by exteded ucertaty, ad therefore t ca be cosdered as a correct measuremet. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory

13 Evaluato of ucertaty measuremets Example 3 To verfy lear type depedece betwee electrc voltage appled to the resstor ad measured curret through ths resstor two seres of I ad U measuremets were performed for the same resstor of a omal resstace equal to 0 ad a class 5% (exteded ucertaty s equal to 0.5 ). I two seres two dfferet ammeters were used. I the frst expermet aalogue ammeter of class.5 measuremet rage.5 A, ad wth 50 tervals was used. I the secod expermet dgtal ammeter was used ad ths case formula (6) ca be used to determe devce ucertaty wth c = 0.% ad c = 0.% for the measuremet rage 0 A. The voltage was measured wth aalogue voltmeter of a rage 5 V ad class, both cases. Measured values are preseted table. Is t possble to cofrm exstece of lear depedece betwee I ad U? Does the measured resstace correspod to omal value? Measuremet results: SERIES I SERIES II U (V) I (A) u (y ) u (x ) U (V) I (A) u (y ) u (x ) Valdty of Ohm's law was assumed U = R I ad I(U) plots were prepared for both data seres. The lear fttg procedure usg Mcrocal Org software was appled for expermetal data (see sert the fgures above) ad obtaed parameters of the ft are lsted below: Seres I Seres II B = B = A = A = χ = 5.39 χ = 6.3 I both cases, fttg les cross all of the ucertaty boxes ad the correlato factor for both seres s smlar ad close to (see Appedx D). I addto, slope values (parameter B) for both seres are very smlar, thus lear type of depedece seems to be cofrmed. However, there s a sgfcat dfferece the χ parameter values. For the studed case umber of degrees of freedom s equal to 0 ( measuremet pots mus ftted parameters) ad for the sgfcace value 0.05 crtcal Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 3

14 Evaluato of ucertaty measuremets value of χ s equal to 8.3. Therefore, based o the frst expermetal seres lear depedece hypothess caot be rejected 5.39 < 8.3. However, based o the secod expermetal seres, wth a hgher accuracy devces, lear depedece hypothess have to be rejected. It s worth to ote, that small vale of χ for the frst data seres s suggestg, that the ucertaty of measuremets are too hgh to use t verfcato of learty hypothess. Ideed, oly secod seres of measuremets was good eough to reject metoed hypothess. Ths somehow surprsg result ca be also cofrmed other calculatos. If we cosder, that for a appled voltage equal to 0 o curret should be measured, the fttg lear fucto has to cross addtoally (0,0) pot ad cosequetly tercept parameter A should be zero. I ths case, umber of degrees of freedom s actually ( measuremet pots mus for oe ftted parameter), ad for sgfcace value of 0.05 crtcal value for χ s equal to 9.7. The ew fttg parameters are lsted below: Seres I Seres II B = B = χ = 6.59 χ = 9.7 The χ value for the frst data seres almost dd t chage for the ew ft, but for the secod seres there s almost 50% chage of χ value cofrmg, that hypothess o lear depedece betwee U ad I should be rejected. Obtaed results suggest, that the measured resstace s chagg wth the appled voltage. Ideed, addtoal expermet showed that the temperature of the resstor was creasg due to the Joule-Lez heat ad therefore ts resstace was ca. 0% hgher at the ed of expermet. Expermetal data pots were ftted wth a ew model fucto cludg temperature chages I = U/R k U (k s a parameter) (Fg.) ad a resstace was determed to be 9.7 wth a stadard ucertaty of 0.. Obtaed resstace result ca be reported: R = (9.7 ± 0.4). Ths agree wth the techcal specfcato for ths resstor: R = (0.0 ± 0.5). Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 4

15 Evaluato of ucertaty measuremets Fg.. Polyomal fucto fttg of data from Example 3. I would lke to thak Prof. Irma Śledzńska ad Potr Paeck PhD for ther help, labor ad commtmet durg creatg ths gude. Also I would lke to thak Prof. Adrzej Zęba for hs detaled ad crtcal revso of the text. Refereces [] Wyrażae epewośc pomaru. Przewodk, Główy Urząd Mar, Warszawa 999. [] Gudele for Evaluatg ad Expressg the Ucertaty of NIST Measuremets Results, NIST Techcal Note 97 [] H. SZYDŁOWSKI, Nepewośc w pomarach, Wydawctwo Naukowe UAM, Pozań 00. [3] A. ZIĘBA, Pracowa fzycza, Wydawctwo AGH, Kraków 00. [4] M. LEWANDOWSKA, Aalza epewośc pomarowych, the teret. [5] Wkpeda Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 5

16 Evaluato of ucertaty measuremets APPENDIX A SI Uts A. Base uts METRE (m) - The dstace travelled by lght vacuum / s (983) KILOGRAM (kg) - The mass of the teratoal prototype klogram placed Sevres (90). SECOND (s) - The durato of perods of the radato correspodg to the trasto betwee the two hyperfe levels of the groud state of the cesum 33 atom 33 Cs (964). KELVIN (K) - /73.6 of the thermodyamc temperature of the trple pot of water (967/68). MOL (mol) - The amout of substace of a system whch cotas as may elemetary ettes as there are atoms 0,0 klogram of carbo, C (97). AMPER (A) - The costat curret whch, f mataed two straght parallel coductors of fte legth, of eglgble crcular cross-secto, ad placed m apart vacuum, would produce betwee these coductors a force equal to 0 7 ewtos per meter of legth (948). CANDELA (cd) - The lumous testy, a gve drecto, of a source that emts moochromatc radato of frequecy hertz ad that has a radat testy that drecto of /683 watt per sterada (979). A. Derved uts RADIAN (rd) - Oe rada s the agle subteded at the ceter of a crcle by a arc that s equal legth to the radus of the crcle. STERADIAN (sr) - The sold agle subteded at the ceter of a ut sphere by a ut area o ts surface. A.3 Prefxes Prefx ame Symbol Factor Exa E 0 8 = Peta P 0 5 = Tera T 0 = Gga G 0 9 = Mega M 0 6 = Klo k 0 3 = 000 Hecto h 0 = 00 Deca da 0 = Dec d 0 - = 0. Cet c 0 - = 0.0 Mll m 0-3 = 0.00 Mcro 0-6 = Nao 0-9 = Pco p 0 - = Femto f 0-5 = Atto a 0-8 = Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 6

17 Evaluato of ucertaty measuremets APPENDIX B B. Gaussa dstrbuto (ormal dstrbuto) If we assume that durg our measuremets, obtag a result bgger or smaller compared to the mea value has the same probablty, ad the results far away from the mea value are less probable the oes close to the mea value, the dstrbuto of results for bg umber of measuremets may by estmated by the Gaussa curve: ( x) ( x ) exp. Fucto (x) s called the Gaussa or ormal dstrbuto. It depeds o two parameters μ ad σ (mea ad varace). The tegral of ths fucto fulfll the codto: (x)dx. Ths codto meas, that fdg a result of the measuremet the terval from x to x+dx s equal (x)dx, ad the probablty of fdg ay result the terval from - to must be equal. Parameters ad ca be easly terpreted aalytcally ad statstcally. For value x = fucto (x) has ts maxmum. Parameter defes two pots ad +, where the Gaussa graph has flecto pots. Hece, the value ca be regarded as the measure of the dstrbuto wdth. From the statstcal pot of vew, s the expectato value E(X), ad the parameter s the square root of the varace D (X), called the stadard devato. The tegrals of the fucto (x) preseted below defe probablty of fdg the specfc umber of the measuremets (68.3%. 95.4% ad 99.7%) the tervals, whch wdth s the stadard devato multple: ( x ) dx 0.683, ( x ) dx , ( ) dx x. Gaussa dstrbuto s cotuous probablty dstrbuto, whch ca properly approxmate the expermetal dsperso of the measuremets resultg from the causes descrbed Chapter 3. Ths dstrbuto ca be adapted to fte umber of measuremets to evaluate Type A measuremet ucertaty. I such case, the expected value of the dstrbuto s mea value (), ad the stadard devato s the stadard devato of the mea value (). [5 ] Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 7

18 Evaluato of ucertaty measuremets B. Uform dstrbuto (x) x [5 ] The uform dstrbuto (also kow as rectagular) s a symmetrc probablty dstrbuto, whch probablty desty fucto s costat (ot zero) the terval from a to b, ad outsde ths terval s equal to zero. The probablty desty fucto of the uform dstrbuto s: ( x ) 3 for 3 3 ( x) 0 for other x, x, where μ ad σ (mea ad varace) are as follows: a b ( b a), ad. If we assume that for the Type B measuremet ucertaty the calbrato ucertaty determes the terval wth Δx wdth about the μ value, the the formula (3) results drectly from the varace defto (substtute a = - Δx b = Δx). Note for the qustve persos: For the uform dstrbuto, the coverage factors whch guaratee probablty for fdg specfc umber of measuremets (95.4% ad 99.7%) the tervals beg the multple of the stadard devato are dfferet tha for the ormal dstrbuto. 95% gets for k =.65, 99% for k =.7, ad for k =.73 the probablty s equal 00%. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 8

19 Evaluato of ucertaty measuremets Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 9 APPENDIX C The method of least squares The method of least squares s the most popular aalytcal method for fttg the lear fucto to the expermetal pots. Its ame s coected wth the crtera of the fttg qualty the parameters of the lear fucto are chose such maer to mmze the sum of the squares resduals, a resdual beg the dfferece betwee a observed value y ad ftted value (Bx + A) provded by the model fucto. m. )] ( [ A Bx y S To fd out the parameters B ad A the codto for the two varables fucto mmum s used: 0 B S ad 0 A S. Calculatg both partal dervatves creates the lear system of equatos the two ukow varables B ad A. Further calculatos are preseted the form eablg maual calculatos. For every measured pot, the values of auxlary fuctos X ~, Y ~ ad d ~ must be calculated: X X X ~, Y Y Y ~, Y BX Y d ~ ~. Next the value of the slope B ad the tercept A (ordate of the pot whch the lear fucto crosses axs OY) ca be obtaed: X B Y A X X Y B ~ ~ ~. Formulas: ~ ) ( ) ( ~ ~ ) ( X X B u A u X d B u defe the stadard ucertates of B ad A. Nowadays, these calculatos ca be doe usg ay software wth lear regresso or lear ft. O fg. the results of the lear ft made by McroCal Org software are preseted.

20 Evaluato of ucertaty measuremets APPENDIX D Lear regresso the McroCal Org 8.0 software O the fg. the least square method the McroCal Org 8 software (usg fucto lear ft) s preseted. Basc results of the calculato are dsplayed the table, whch appears automatcally the wdow Graph. Detaled formato about the fttg ad all statstcal parameter ca be foud the wdow Book, tab Data. I the table we ca fd the followg formato: - Equato fucto ftted to the expermetal data. I our example we have lear fucto y = a + b*x. - Weght the method of calculatg the statstcal weght of the specfc measured pot. Istrumetal meas, that weght w s calculatg as a recprocal of the squared ucertaty y (value take from the ucertaty colum for quatty Y). - Resdual Sum of Squares value of the fucto χ (to dsplay ths value the table wth results, the opto Resdual Sum of Square Quattes to Compute>Ft statstcs the wdow Ft Lear must be checked. - Adj. R-Square umber that dcates how well data ft chose statstcal model. dcates that the regresso le perfectly fts the data, whle 0 dcates that the le does ot ft the data at all. - Value ad Stadard Error ucertates for a ad b. - Itercept (a) ad Slope (b). Fg. Lear ft the McroCal Org software Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory 0

21 Number of degrees of freedom Evaluato of ucertaty measuremets APPENDIX E Crtcal values for dfferet sgfcace levels α ad the umber of degrees of freedom Sgfcace level Shaded gray colum cotas the crtcal values χ most frequetly used the studets laboratory. Traslato made by Wojcech Wróbel, PhD the collaborato wth Adrzej Kubaczyk, MSc. Faculty of Physcs, Warsaw Uversty of Techology Physcs Laboratory