EXPERIMENTAL ERRORS. There are primarily two kinds of errors that affect experimental results:

Size: px

Start display at page:

Download "EXPERIMENTAL ERRORS. There are primarily two kinds of errors that affect experimental results:"

Abigayle Edwards
6 years ago
Views:

1 EXPERIMENTAL ERRORS The followg s a very smple troducto to the topcs of Systematc ad Radom Error, subjects that are of paramout mportace to the physcal scetst. There are prmarly two kds of errors that affect epermetal results: ) Systematc Error Systematc error refers a collectve sese to a broad rage of errors that arse due to mperfect equpmet desg ad/or operator error. By operator error we also clude eact computatoal procedures used durg the data reducto process. We do ot, however, clude accdetal bluders ad arthmetc mstakes whch are remedable. Propertes of Systematc Errors Systematc errors are characterzed by the followg propertes: ) They are repeatable from epermet to epermet, wth respect to both the sg of the error ad also ts appromate magtude ) They are ot ameable to ay kd of statstcal aalyss ) They caot be reduced or detected by repeatg a epermet Eamples: ) Defectve meter zero readg a strumet ) Icorrect postog of the fducal mark the maufacture of a ppet 3) Igorg the stem correcto whe usg a thermometer or the temperature correcto for the metal scale whe usg a barometer 4) Usg overlarge tres o a car thus creatg a costat error the speedometer readg 5) Usg a mpure sample as a calbrato stadard (e.g., KHT) whe determg the ormalty of a acd or base 6) Usg a out-of-date value for a physcal costat such as the gas costat R or the gravtatoal costat whch vares wth locato 7) Use of a ucalbrated balace 8) Early trucato of a fte seres or falure to take to accout computer roudoff error the evaluato of a fte seres etc. Each of these eamples wll cause a error a epermetal result, ether drectly durg the course of the epermet or later durg the data reducto procedure. Furthermore, every tme the epermet s repeated, these errors wll cotue to eter, usually to about the same etet ad wth the same postve or egatve sg.

2 Detecto ad elmato of systematc error s very dffcult because oe s rarely certa of ts org or, for that matter, ts estece. Oe mght, for eample, rely o comparso measuremets betwee dfferet laboratores of the quatty questo, especally f the measuremets are teded to be hghly accurate. More routely, the use of blak determatos or the use of certfed stadards (e.g., the use of a NBS stadard Bezoc Acd sample for the calbrato of a bomb calormeter) are obvous strateges to reduce systematc error but o procedure, however etreme, wll guaratee a error free measuremet. I fact t s mpossble to ever kow for certa that all systematc error has bee elmated from a epermet! Measures of Systematc Error There are o formal measures. The best you ca do s estmate systematc error based o comparsos agast kow stadards or kow strumet shortcomgs. Relatoshp to Accuracy SYSTEMATIC ERROR IS THE LIMITING FACTOR IN DETERMINING ACCURACY Thus systematc error, whch s always preset to some etet, wll ultmately determe the accuracy of a measuremet; a cocept we ow formalze. Defto of Accuracy The oto of accuracy s based o the cocept of the 'true value' of a measuremet,.e., the quaestum. We deote ths quatty, whch s the holy gral of ay measuremet, as. For reasos already dscussed, oe ca ever kow but we ca at least hope that the mea of our measuremets wll appromate t. If we defe the dfferece betwee the mea of a fte umber of measuremets ad the true value, as the bas, bas = - assumg a fte umber of measuremets) the we ca assert the followg: 'whe the bas s zero, the measuremets have o systematc error ad are therefore sad to be 'accurate'. Ths smply formalzes the relatoshp betwee systematc error ad accuracy. Of course we caot kow (ever!), so of course we ca ever really kow f the bas s zero ad therefore f our results are truly accurate. ) Radom Error Repeated measuremets of some physcal quatty wll always ehbt fluctuatos of the measured values about some mea value. Ideed, t s a fudametal assumpto (sometmes called the Postulate of Measuremet) that, as the umber

3 3 of measuremets ted to fty *, the rug mea wll evetually settle dow to some fed quatty called the 'lmtg mea'. If ths assumpto were ot true, t would be mpossble to meagfully assg a value to ay varable, ad t would certaly vtate the whole cocept of measuremet. Furthermore, the greater the resoluto of the measurg apparatus, the greater the etet of these fluctuatos. These fluctuatos are assumed to arse because of a umber of small heretly ucotrollable processes, each of whose effect s dvdually small. These errors are thus called Radom Errors ad ca be reckoed as the dfferece betwee the value of the th measuremet ad the mea value,.e., where s the th radom error assocated wth the th measured value. Note that ca be postve or egatve. Propertes of Radom Error ) The radom errors assocated wth a specfc measurg apparatus are a characterstc of the desg of that partcular apparatus. ) The sg of the dvdual errors,, s assumed to be postve or egatve wth equal probablty. 3) These errors, whe tabulated for a very large umber of measuremets, wll be assumed to occur wth a frequecy that follows a ormal dstrbuto, that s, f oe plots f() vs., the resultg curve wll be appromately Gaussa ature. Ths meas that radom error ca be aalyzed statstcally. 4) Ulke systematc errors, radom error s detected by smply observg that the measured values fluctuate about the mea ad therefore, by chagg the desg of the measurg apparatus, they ca be reduced. However, they ca ever be totally elmated. Eamples of Radom Error ) Nose effects electrcal crcuts (always preset) ) Browa moto 3) Ay 'o/off' cyclc evet (e.g., a electrcal heater) that s part of the measurg apparatus provded that the evet behaves radomly relatve to the tme whch a measuremet s made. If the tmg of the measuremet ad the duty cycle of the 'o/off' evet are sychroous, the error s more lkely to be systematc ature. Ths s a eample showg that the characterzato of what s radom ad what s systematc s ot always clear-cut. Measures of Radom Error Because radom error s ameable to statstcal aalyss, t s possble to derve a measure of ts fluece. We dstgush betwee radom error the dvdual * Symbolcally, lm () where s usually deoted as the populato mea,.e., the mea value for the fte epermet.

4 4 measuremets,,..., ad the radom error assocated wth the mea value, computed from the dvdual values. The most commo measures of the precso the orgal data pots are the sample VARIANCE ad the correspodg STANDARD DEVIATION, vz., or S Sample Varace S Sample Stadard Devato where s the sample mea defed by: For the mea tself, the preferred measure of ucertaty s the STANDARD ERROR, defed as: (the deotes the sample estmate) S S S.E. ( ) ( ) Notce that repeatg epermets reduces the stadard error by the factor. Thus whle the stadard devato (for the fte epermet) s a characterstc of a partcular apparatus ad therefore ca be reduced oly by redesg of the apparatus, the stadard error ca be reduced by repetto of epermets. Whe dealg wth the mea of a seres of epermets, the stadard error s actually the best estmate of radom error so epermetal repetto s ofte well worthwhle. O the other had, the reducto the stadard error s ot lear wth respect to so at some pot t may be more ecoomcal to look at mprovg the apparatus stead of just makg more measuremets. Relatoshp to Precso RANDOM ERROR IS THE LIMITING FACTOR IN DETERMINING PRECISION Defto of Precso Precso reflects how close our measuremets are to oe aother. If we make a seres of measuremets ad the calculate the stadard devato for these results we wll fd that the smaller the stadard devato the better our oto of precso, whereas coversely, the more spread out the results, the larger the

5 5 stadard devato ad hece the poorer the precso. The phrase 'precse', lke the word 'accurate' s etrely depedet o what the epermetalst s wllg to accept ad has o quattatve meag beyod that. Icdetally, the word 'precse' s sometmes used mprecsely to deote 'accurate' as well as 'hgh resoluto' (presumably a lot of sgfcat dgts). I summary the, accuracy measures how close the mea value s to the true value whereas precso measures the closeess of the dvdual results. Repetto wll crease precso by reducg the stadard error, but wll have o effect o accuracy. The followg dagrams llustrate these pots. I the case of good precso ad poor accuracy, a mproperly adjusted sght would epla the systematc error causg a o-zero bas. Oce the sght s adjusted, the bas goes to zero. I the case of poor accuracy ad poor precso we have both systematc ad radom error to coted wth ad t s dffcult to kow whch s whch. The systematc error the horzotal drecto could be eplaed by a strog ad varable wd whle the vertcal error could be eplaed by defectve ammuto. Of course the guy could also be just a lousy shot. Good Precso, Poor Accuracy Good Precso, Good Accuracy = Bas 0 Bas = 0 Poor Precso, Poor Accuracy Bas 0 These dagrams also make the pot that just because a seres of measuremets s precse, o cocluso whatsoever ca be draw about the accuracy of the measuremets uless oe ca also be certa that there s lttle lkelhood of systematc error. Method of Statg Error a Measuremet As already dcated, the stadard error s the approprate measure of the ucertaty whe the mea value s used to epress a epermetal result. It s recommeded that ucertaty be epressed as follows:

6 6 X X s.e. where the mea ad the stadard error were defed prevously. Furthermore, the stadard error, s.e., should be epressed to o more tha two sgfcat dgts ad should agree wth X as to the umber of sgfcat dgts after the decmal. It s also mportat to state the umber of measuremets, sce the stadard error depeds o, as well as make some statemet as to the epermetalst s belef the magtude of the systematc error. Numercal Eample Suppose we wsh to measure the desty of a ewly sytheszed lqud compoud at a fed temperature of 5 o C. We obta the followg four values:.37,.3303,.300 ad.30 where the results are epressed the uts of gms cm -3. Usg the equatos gve above, we compute the followg statstcs: d = 4 S 4 4 d.3 S (d d) S S s.e Note that we have carred oe etra sgfcat fgure whe a computed value eceeds fve dgts. Roudg s doe at the ed, NOT at each step of the calculato. We ow epress our measured result the form: Measured Desty d s.e ( gms cm 3 ) where the epressed ucertaty the desty s the sample estmate of the stadard error computed for a total of four measuremets assumg a eglgble systematc error. Ths last statemet s a tegral part of the error statemet ad should ot be omtted (but usually s, ufortuately). How do we kow the systematc error s eglgble? I fact we do't for reasos already dscussed, but t probably s very small IF we have made a coscetous effort to accout for each possble error the epermetal process. Oe techque would be to make a seres of measuremets usg ths apparatus o a lqud of

7 7 kow desty, such as water, ad the use whatever dscrepacy ested betwee accepted values the lterature ad our measuremets as a dcato of possble systematc error. Eve so, systematc error s very elusve

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best

best estimate (mean) for X uncertainty or error in the measurement (systematic, random or statistical) best Error Aalyss Preamble Wheever a measuremet s made, the result followg from that measuremet s always subject to ucertaty The ucertaty ca be reduced by makg several measuremets of the same quatty or by mprovg