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Date Created: 10/21/15
Phys 218 208 0105 January 2005 INTRODUCTION TO ERROR THEORY Sante R Scuro Physics Laboratory Texas AEJM University College Station TX 77843 ABSTRACT The measurement of a physical quantity can never be made with perfect accuracy there will always be some error or uncertainty present For any measurement there are an in nite number of factors that can cause a value obtained experimentally to deviate from the true theoretical value Most of these factors have a negligible effect on the outcome of an experiment and can usually be ignored However some effects can cause a signi cant alteration or error in the experimental result If a measurement is to be useful it is necessary to have some quantitative idea of the magnitude of the errors Therefore when experimental results are reported they are accompanied by an estimate of the experimental error called the uncertainty This uncertainty indicates how reliable the experimenter believes the results to be These notes are mainly a review of llErroT Theoi gj7 from an old Physics 218 Lab manual in TAMU Contents O CO 4 0 Types of Errors 11 Random errors 12 Systematic errors Statistical Methods Percent Errors Propagation of Errors 41 Addition and Subtraction 42 Multiplication and Division 43 Exponents and Roots Examples 51 Example The Estimate of the Table Top Surface Area 52 Example The Estimate of the Acceleration due to Gravity q T U U 1 Types of Errors In order to determine the uncertainty for a measurement the nature of the errors affecting the experiment must be examined There are many different types of errors that can occur in an experiment but they will generally fall into one of two categories random errors or systematic errors 11 Random errors Random errors usually result from human and from accidental errors Accidental errors are brought about by changing experimental conditions that are beyond the control of the experimenter examples are vibrations in the equipment changes in the humidity uctu ating temperatures etc Human errors involve such things as miscalculations in analyzing data the incorrect reading of an instrument or a personal bias in assuming that particular readings are more reliable than others By their very nature random errors cannot be quan ti ed exactly since the magnitude of the random errors and their effect on the experimental values is different for every repetition of the experiment So statistical methods are usually used to obtain an estimate of the random errors in the experiment 12 Systematic errors A systematic error is an error that will occur consistently in only one direction each time the experiment is performed ie the value of the measurement will always be greater or lesser than the real value Systematic errors most commonly arise from defects in the instrumentation or from using improper measuring techniques For example measuring a distance using the worn end of a meter stick using an instrument that is not calibrated or incorrectly neglecting the effects of viscosity air resistance and friction are all factors that can result in a systematic shift of the experimental outcome Although the nature and the magnitude of systematic errors are difficult to predict in practice some attempt should be made to quantify their effect whenever possible In any experiment care should be taken to eliminate as many of the systematic and random errors as possible Proper calibration and adjustment of the equipment will help reduce the systematic errors leaving only the accidental and human errors to cause any spread in the data Although there are statistical methods that will permit the reduction of random errors there is little use in reducing the random errors below the limit of the precision of the measuring instrument 2 Statistical Methods When several independent measurements of a quantity are made7 an expected result to report for that quantity is represented by the average of the measurements For a set of experimental data containing N elements7 or measurements7 given by 1 7 2 7 3 7 7SN7 the average 37 or expectation value ltSgt7 is calculated using the formula 577 21 0 W Elm MZ H l S1Sz53 SN N 22 The reason why it is more appropriately called expectation value lies beneath the fact that the average represents the closest approximation that is available to the true value of the quantity being measured It is sometimes referred to as the best estimate of the true value The data 51 752 7 3 7 7SN are dispersed around the mean7 or average A measure of this dispersion is called the standard deviation and is given by AS 71 N 52 N32 23 N71 gm 7 H 512S S SJZV7N 2 N71 39 24 The smaller the standard deviation is the more closely the data is grouped about the mean If there is a large number of normally distributed points ie Gaussian distribution7 statistical analysis shows that about 683 of them will fall within the interval between SiASandSAS If the systematic errors have been reduced as far as possible7 the random errors will dominate and hence will limit the accuracy of the nal result Clearly there are two ways to reduce the effect of random errors and improve the accuracy of an experimental result7 26 o eliminate the majority of the random errors inherent in the experiment 0 obtain as many data points as is reasonable7 thereby increasing N7 and therefore reducing 3 3 Percent Errors The standard deviation is a measure of the precision of an experiment the smaller the AS the greater the precision of the best estimate One way to report the precision of the experimental value is through the use of the percent standard deviation given by AS AS x 100 31 Unfortunately the average and standard deviation indicate nothing about the accuracy of the measurement Le how close the experimental or average value is to the true value In other words an experiment can yield extremely precise consistent values without generating a result that is close to the true value this type of result often occurs when the equipment has not been zeroed or calibrated properly and when other systematic errors have not been properly reduced In many experiments it is desirable to indicate the overall accuracy of the nal experimental value by reporting some type of percent error If the quantity measured has a standard or true theoretical value then the accuracy of the experimental value is given by a ratio of the error to the true value l Experimental Value 7 Theoretical Value l Error Theoretical Value X 100 39 3392 For many experiments the true value of the quantity being measured is unknown In this situation it is often useful to compare two results obtained by different methods so that a percent difference can be obtained For instance if the two experimental values are represented by 1 and SQ then the percent difference is de ned by 1 7 52 S S lt71 2gt If an experiment is performed properly with care taken to reduce the random and islism x 100 33 systematic errors as much as possible then the percent errors will be correspondingly small The magnitude of the percent errors will depend heavily on the overall precision of the measuring instruments This means that while in some cases an error of 5 or 10 might be acceptable in other cases such an error would indicate a very poorly run experiment Thus the success of an experiment in terms of a percent error can only be judged when the method and instrumentation of the experiment are taken into consideration 4 Propagation of Errors In many experiments the quantities measured are not the quantities of nal interest Since all measurements have uncertainties associated with them clearly any calculated quantity will have an uncertainty that is related to the uncertainties of the direct measurements The procedure used to estimate the error for the calculated quantities is called the propagation of errors Consider the general case rst Suppose the variables A B C represent independent measurable quantities that will be used to obtain a value for some calculated quantity U Since U is a function of A B 0 it can be written as U fABC The measurements of the quantities A B C yield estimates of the expectation values written as A B C39 and the associated uncertainties AA AB AC for each variable To nd the expectation value or best estimate for the quantity U the expectation value of each measured variable is substituted into the equation for U UfABC 41 If the errors for A B C are independent random and sufficiently small it can be shown that the uncertainty for U is given by AU 279244 ltgt2AB2 ltgt2AC2 7 42 where the partial derivatives are evaluated using the expectation values A B C39 as the values for the independent variables The correct notation to express the nal estimate for the calculated quantity U is given by U U i AU 43 In the next three sections 41 43 we lighten our notation removing the 77bar77 signs and considering A B C as the expectation values for calculation purposes only We restore the original notation in the examples given in 5 41 Addition and Subtraction Suppose two quantities A and B are added and the uncertainties associated with each variable are AA and AB It follows that UAB 1 44 where equation 42 gives AU AA2 AB2 45 If the two values are subtracted then 8U 8U UA B w M1 71 46 and equation 42 gives same result as in the case of addition7 239e AU AA2 AB 47 Hence these equations can be generalized for any combination of addition and subtraction of any number of variables7 Le U iAiBiCi 48 AU AA2 AB2 A02 49 42 Multiplication and Division Suppose two quantities A and B are multiplied and the uncertainties associated with each variable are AA and AB It follows that 8U 8U UAB 7B 7A 410 7 8A 7 QB 7 where equation 42 gives7 after some algebraic rearrangement7 AA 2 AB 2 AU U 7 7 411 4 lt B lt gt Likewise if the two values are divided then A 8U 1 8U A U E w M 43912 and equation 427 after some algebraic rearrangement7 gives same result as in the case of multiplication7 Le AA 2 AB 2 AU U 7 7 413 4 lt B lt gt Hence these equations can be generalized for any combination of multiplication and division of any number of variables7 Le 414 w Ultgt2ltA7gt2 lt gt2e gt2 43 Exponents and Roots Suppose that a calculation involves the use of exponents or roots of a quantity A Whose uncertainty is AA It follows in the most general case that Va 6 IR U kAD 416 AU kaAa lAA 417 5 Examples 51 Example The Estimate of the Table Top Surface Area The rectangular dimensions A and B ofa table top are determined directly by estimating the left and right reading of each dimension as shown in Fig1 where the error or uncertainty is given by half of the smallest reading unit on the meter stick The smallest unit in this example is 1mm and therefore the uncertainty in reading is 05 mm BR 55 04 Cm Meter Stick BL30 00 Cm AL2OU cm Metersnck AR9812cm Figure 1 The smallest unit on the meter stick is 1mm Let left and right readings of A and B be respectively see Fig1 AL 200 i 005 cm AR 9812 i 005 cm 51 BL 3000 i 005 cm BR 5504 i 005 cm 52 It is easy to see that the two dimensions A and B7 and their uncertainties AA and AB7 are determined by equations 48 and 49 applied to the left and right reading values lndeed7 we have A 9812 7 200 9612 cm AA 0052 005 z 007 cm 53 B 5504 7 3000 2504 cm AB 005 005 z 007 cm 54 We usually write the above results in a more appropriate form for an estimate as follows A 9612 i 007 cm7 B 2504 i 007 cm 55 The next step is to calculate the expectation value for the surface area S AB and the associate uncertainty AS as provided by equation 4157 76 S 9612x25042407cm2 56 007 2 007 2 2 AS 4 2407 m 77cm 57 And again7 this can be written in a more compact form7 76 S 2407 i 7 cm2 58 52 Example The Estimate of the Acceleration due to Gravity Suppose the acceleration due to gravity can be calculated from experimental data using the following equation QmZ 9 W 7 59 where the estimate measurements are given by 9592 i Ax7411j 005cm7 Z l7iA 1001iO2cm7 7113 i Ah110 iO1cm7 510 252 iAt3708 i0003s Hence g z 981 cm s lnstead7 the uncertainty here is compound and therefore given by the combination of equation 415 and 4177 76 4 a1lt5gt2lt4gt2lt5gt2ltgt27 alerwe erwe 005 2 02 2 01 2 0006 2 7 981ltmgt 7 513 z 9 crns 2 514 This can be written in a more compact form Le g 981 i 9 crns Z 515 Acknowledgements This work was supported in part by the TAMU Visual Physics Foundation References 1 Physics 218 Lab Manual 7 Ed by SA Ramirez RA Seidel and JC Hiebert Hayden McNeil Publishing Inc ISBN 0 7380 0532 0 TAMU College Station 2001 2 An Introduction to Error Analysis 2 Ed by JR Taylor University Science Books ISBN 0 935702 75 X April 1997
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