ANALYSIS OF EXPERIMENTAL ERRORS All physical measuremets ecoutered i the verificatio of physics theories ad cocepts are subject to ucertaities that deped o the measurig istrumets used ad the coditios uder which the eperimet was performed. A ucertaity i ay sigle measuremet will result i a ucertaity of the fial eperimetal result. These ucertaities are ofte referred to as EXPERIMENTAL ERRORS! This essay is to help the studet become aware of the potetial sources of error ad the bias that are iheret i ay process of measuremet. The studet should be able to apply the cocepts of arithmetic mea, stadard deviatio, ad stadard error i aalyzig errors likely to occur whe makig precisio measuremets i the laboratory. TYPES OF EXPERIMENTAL ERRORS There are three fudametal types of eperimetal errors: Systematic errors Persoal errors or Mistakes Radom Errors SYSTEMATIC ERRORS Systematic errors usually cause the results of a measuremet to be cosistetly too high or too low, below the true (or actual) value. These errors may be reduced by havig well desiged eperimets ad good equipmet. Geerally, the errors are due to: a. Faulty istrumet usually due to poor calibratio, adjustmets of the istrumets, or slight imperfectios i the costructio or desig of the istrumet. Cosequetly, the readigs or measuremets are off by the same amout each time. Checkig the zerooffset before measuremets ad applyig the appropriate adjustmets or correctios to the readigs reduces the systematic errors. b. Theoretical over-simplificatio of the system whe all the variables that affect a system are ot take ito accout. A eample is whe frictioal effects are eglected i the theoretical problem. Cosequetly, there is always a sigificat discrepacy betwee the epected theoretical results ad the eperimetal results obtaied whe these frictioal effects caot be igored. c. The evirometal coditios durig measuremets may differ from the stadard coditios used for the calibratio of the istrumet scales. These variatios have to be compesated for i order to miimize the systematic errors. Every eperimeter should strive to idetify, miimize, ad elimiate ay obvious systematic errors as much as possible. Good eperimetal desigs may utilize the advatages due to symmetry or repetitio of measuremets i reverse order to miimize subtle systematic errors. Ufortuately, it is very difficult to reliably idetify ad estimate systematic errors.
PERSONAL ERRORS (OR MISTAKES) Persoal errors ca be completely elimiated if the eperimeter eercises utmost care, cautio, ad skepticism while readig the data (observatioal mistake) ad while performig the eeded calculatios (arithmetic mistake). If the measuremet values are read icorrectly or if the calculatios are wrogly carried out, the etire result will be wrog! Therefore, the eperimeter is strogly ecouraged to cross-check the data ad calculatios. I a group, a oe-time careless mistake i measuremet by oe parter ca affect the etire result. Therefore, each parter should idepedetly read the data ad check ay calculatios for accuracy. RANDOM ERRORS Radom errors are usually due to ukow variatios i the eperimetal coditios. The sources of these radom errors caot always be idetified ad ca ever be totally elimiated i ay measuremet. This class of errors usually causes about half of the measuremets to be too high ad the other half of the measuremets to be too low. These radom errors may be: a. Observatioal - icosistecy of a observer i estimatig the last digit whe readig the scale of a measurig device betwee the smallest divisio. b. Evirometal - physical variatios that may affect the equipmet or the eperimet setup such as fluctuatios i the lie voltage, temperature chages, or mechaical vibratios. Fortuately, radom errors may be miimized by takig several careful observatios of the same quatity ad ca be determied by statistical aalysis. These radom errors are sometimes referred to as statistical errors. ESTIMATION OF STATISTICAL ERRORS IN MEASUREMENT We shall ow discuss methods of estimatig radom errors i measuremets. It is importat to state the precisio or the estimated error i a measuremet, based o the measurig istrumet. For a measured value,, that has a estimated error, δ, the measuremet would be reported as ± δ. Least Cout I all physical measuremets, the measurig istrumet must have a measuremet scale ad that scale must have a least cout. The least cout is the smallest marked scale divisio o that istrumet. The ucertaity (estimated error) itroduced i a sigle measuremet is kow to be approimately equal to the value of the least cout of the measuremet scale. 014 Marti Okafor Aalysis of Eperimetal Errors page
Arithmetic Mea or Average value ( ) Repeated measuremets of a certai sigle quatity () i the presece of oly radom errors will cluster about some average value ( ). This Arithmetic Mea (or Mea) is the most valid represetatio of the quatity beig measured. If measuremets of the quatity are made, the Mea is give by the equatio = i= 1 i (1.1) or + + + 1 = (1.) Stadard Deviatio (s or σ-1) If a very large umber of measuremets are made with oly radom errors preset, the plot of the data would be observed to be a ormal distributio curve cetered aroud the Mea value. I the presece of variatios caused by systematic errors, this ormal distributio curve would be skewed to the left or right. A useful cocept for determiig the scatter of the data about the Mea is the Stadard Deviatio. This Stadard Deviatio idicates a measure of the deviatios of the measuremets from the Mea ad is give by the equatio: s = = i= 1 σ 1 ( ) i 1 (1.3) or s = = σ 1 ( ) 1 (1.4) From statistical aalysis, for large umber of measuremets, with oly radom errors preset, approimately 68% of the measuremets will be withi ± 1s, i.e. ( s, + s) or oe STDDEV of the mea. Approimately 95 % of all the measuremets will be withi ± s. Essetially, all measuremets should fall withi ± 3s of the Mea. Ay measuremet that is ot withi ± 3s of the mea is likely a readig/measuremet error ad should be discarded. If the data poits are scattered over a arrow rage of values, σ will be small ad the distributio curve would be arrow, clusterig aroud the Mea. Otherwise, if the scatter of the data is large, the σ 014 Marti Okafor Aalysis of Eperimetal Errors page 3
will be large ad the distributio curve will be broad, spreadig over a wider rage of values o both sides of the Mea. Stadard Error (α) If several series of measuremets are made with oly radom errors preset, a measure of the scatter of the mea values of each series aroud the true value is described by the stadard error. The stadard error is thus the stadard deviatio of the mea values of each series from the true value. The Stadard Error is defied as: s α = (1.5) We apply the stadard error i statig the estimated ucertaity i a measuremet. This stadard error idicates the umber of sigificat figures i the mea of the quatity measured. This first digit of the stadard error, after roudig to oe figure, idicates the decimal place i which the ucertaity i the mesuremet eists. Therefore, the last digit retaied i the mea should be i the same decimal place as the first digit of the stadard error. For eample, if for a give set of measuremets the calculated mea, = 1.3465, ad the stadard error, α = 0.00134, the with the stadard error (rouded to oe figure), α = 0.001, the mea should be reported as = 1.35. Therefore the result is stated as: 1.35 ± 0.001. The stadard error idicates that the ucertaity i the measuremet eists i the third decimal place. The resultat umber of sigificat figures i the reported mea idicates the precisio of the eperimet. The followig eample will illustrate the cocepts of statig the measured quatity i terms of the mea value with the precisio idicated. EXAMPLE Nie repeated measuremets of the legth (i cetimeters) of a woode board are give as follows: 1.5, 1.4, 1.50, 1.7, 1.84, 1.00, 1.3, 1.68, 1.45. (a) (b) (c) Calculate the mea: = i= 1 i = 1.46444 cm Fid the stadard error: s α = = 0.5954/3 = 0.086508 cm State the legth of the block. = 1.46 ± 0.09 cm 014 Marti Okafor Aalysis of Eperimetal Errors page 4
Precisio ad Accuracy Based o the defiitio of stadard deviatio, for a set of measuremets, a low value of the stadard deviatio idicates a high precisio the data poits are closely clustered, with low scatter. Hece, the smaller the stadard error, the more precise are the set of measuremets. Scietists egaged i origial research usually seek high precisio of their data. This is a idicatio of the degree of reproducibility of their work. Thus high precisio implies that a series of measuremets may be repeated uder similar coditios to yield results which are i close agreemet with others. However, i the freshma college physics laboratory, the objective is ofte to perform the eperimets with such great care to obtai reasoably accurate values of a established quatity or physical costat such as the acceleratio due to gravity, or verify a kow physical law or priciple. The accuracy of a measuremet refers to how closely a measuremet compares with a kow stadard or accepted or theoretical value. High precisio measuremets do ot ecessarily imply high accuracy. Estimated ucertaity of a measuremet usually reflects both the precisio of the istrumet ad the accuracy of the measuremet. Accuracy is determied by calculatig the Percetage Error. Percetage Error The percetage error (or percet error) is calculated from the followig equatio: Eperimetal Accepted Percet Error = 100 (1.6) Accepted Percetage Differece For a quick compariso of two measuremets, it is ofte ecessary to calculate their percetage differece. I this case, oe of the measuremets beig compared is cosidered a accepted value. Istead the percetage differece (or percet differece) idicates how close the measuremets compare to each other relative to their average value. The percet differece is calculated as: value1 value Percet Differece = 100 (1.7) average value value ( 1, ) 014 Marti Okafor Aalysis of Eperimetal Errors page 5
PROPAGATION OF ERRORS Most calculatios performed i the laboratory work ivolve the derivatio of a ukow quatity by usig two or more measured values that have some error associated with them. Propagatio of errors is a method to determie how the error i the derived quatity relates to the errors i the measured values. Let us cosider two circumstaces: 1. If there are o repeated measuremets for the measured values ad if there is isufficiet kowledge of the estimated errors i the measured values, a approimate way of estimatig the error i a calculated quatity is by applyig the rules for sigificat figures. For ay measured value, the reliably kow digits are reported ad the last digit reported is estimated by iterpolatig betwee the smallest marked scale divisio. Do ot attempt to iterpolate to more tha oe estimated place. Correctly stated, measured values idicate the appropriate umber of sigificat figures that are applied to subsequet calculatios. It is, therefore, very importat that the proper sigificat figures be take ito accout for all calculatios performed i the laboratory! Review the Rules for Sigificat Figures.. The secod is a situatio where two or more measured values, with kow estimated errors, yield a calculated quatity. It ca be show by methods of statistical aalysis that certai rules ca be applied to hadle error propagatio. Additio ad Subtractio of measuremets Whe the estimated errors i measuremets are much less tha the measured values, a better approimatio of the error i the result of a additio or subtractio is give by the square root of the sum of the squares of the estimated errors. Cosider a eample where a quatity L is calculated from the measuremets ad y where ± α ad y ± α y The calculated quatity, L(,y), is defied as a fuctio of ad y. For additio: L = + y, or subtractio: L = - y Usig advaced statistical methods, the error i L, is give by ( ) ( ) α = α + α L y ad the result is stated as L ± α L. Multiplicatio ad Divisio of measuremets Whe measuremets are multiplied or divided, the fractioal error i the result is the square root of the sum of the squares of the fractioal errors of the measured values. 014 Marti Okafor Aalysis of Eperimetal Errors page 6
Cosider the equatios: for multiplicatio: L = y or divisio: the fractioal error i L is give by: L = y L α α y = + α L y After re-arragig terms, the error i L is give by: L = y + y α α α ad the result is agai stated as L ± α L. 014 Marti Okafor Aalysis of Eperimetal Errors page 7