The new concepts of measurement error s regularities an effect characteristics Ye Xiaoming[1,] Liu Haibo [3,,] Ling Mo[3] Xiao Xuebin [5] [1] School of Geoesy an Geomatics, Wuhan University, Wuhan, Hubei, China, 379. [] Key Laboratory of Precision Engineering & Inustry Surveying, State Bureau of Surveying an Mapping, Wuhan, Hubei, China, 379 [3] Institute of Seismology, China Earthquake Aministration, Wuhan, Hubei China, 371. [] Wuhan Institute of earthquake metrological verification an measurement engineering, Wuhan, Hubei China, 371. [5]Wuhan University Library. Wuhan, Hubei, China 379. Abstract: In several literatures, the authors give a kin of thinking of measurement theory system base on error non-classification philosophy, which completely overthrows the existing measurement concepts system of precision, trueness an accuracy. In this paper, aiming at the issues of error s regularities an effect characteristics, the authors will o a thematic explanation, an prove that the error s regularities actually come from ifferent cognitive perspectives, is also unable to be use for classifying errors, an that the error s effect characteristics actually epen on artificial conition rules of repeate measurement, an is still unable to be use for classifying errors. Thus, from the perspectives of error s regularities an effect characteristics, the existing error classification philosophy is still a mistake; an uncertainty concept system, which must be interprete by the error non-classification philosophy, naturally become the only way out of measurement theory. Key wors: measurement error; precision; trueness; accuracy; uncertainty. 1. Introuction In several literatures [1] [] [3], the authors give a kin of thinking of measurement theory system base on error non-classification philosophy. The main logic of this thinking is briefly introuce as follows: The concept of error is efine as the ifference between the measurement result an its true value. Because the measurement result is unique, an the true value is also unique, so the error of the measurement result is the only unknown an constant eviation. For a final measurement result, this constant eviation consists of two parts: 1, the eviation between the final measurement result an mathematical expectation, which is the so-calle ranom error in existing theory;, the eviation between mathematical expectation an true value, which is the so-calle systematic error in existing theory. The total eviation is equal to the aitive synthesis of them. Because both of the so-calle ranom error an the so-calle systematic error are unknown an constant eviation, an have no any ifference in characteristics, therefore, having no characteristic ifference must cause having no the classification ifference basing characteristic! The stanar eviation of the eviation between final measurement result an mathematical expectation (so-calle ranom error) is given by the analysis of current measurement ata statistic; The eviation between mathematical expectation an true value (so-calle systematic error) is also prouce by measurement, its formation principle is actually the same as the current measurement, an its stanar eviation can be obtaine by tracing back to its upstream measurement; Thus, the total stanar eviation of final measurement result is equal to the synthesize of the two stanar eviations accoring to the probability laws. This total stanar eviation is uncertainty (this give a more clear meaning to the uncertainty concept). This constant eviation theory is completely opposite to the ranom variation theory of existing measurement theory, that is, in the opinion of the authors, the existing measurement error classification theory is wrong, an the error classification efinition an all the concepts of precision, trueness an accuracy shoul be abolishe. For example: in 5, the Chinese surveying an Mapping Bureau gave that the elevation result of Mount Everest is 88.3 meters with stanar eviation of ±.1 meters. This result's error (the ifference between the result an the true value at implementing measurement) is an yeses7@163.com 1
unknown constant, the stanar eviation of ±.1 is only the probability interval evaluation of the unknown constant, an it is actually a wrong unerstaning to mathematical concept that existing measurement theory uses ranom error (precision) concept to explain the unknown constant as ranom variation (ispersion). The ifference between the new theory an the existing theory is shown in Fig1. Existing measurement theory New concept theory After ajustment, the ifference between the measure result an the mathematical expectation is in ranom variation, is iscrete. After ajustment, the ifference between the measure result an the mathematical expectation is constant, is not iscrete. The systematic error is certain regularity, the ranom error is ranom regularity, an the two kins of errors have complete ifferent characteristics. Both so-calle systematic error an so-calle ranom error are constant eviation, have no ifference in characteristic, can not be classifie. The total error can only be evaluate with precision an trueness, an the precision an trueness can not be synthesize. Precision, trueness an accuracy are abanone, an the total error is evaluate by uncertainty. Fig1. The comparison of two theory s logic The central ifference between the two theories is, the existing theory consiers that the error has the systematic / ranom classification, but the new concept theory hols that the error has no systematic / ranom classification. Because people's brain is washe for a long time by error classification theory, some people still tangle for error s variation regularity an effect characteristics, even if the concept logic of new theory is rigorous. Then, what are the error s variation regularity an effect characteristics? An what is the relationship between these issues an constant eviation theory? Therefore, the authors o a thematic explanation here.. Error s regularity The concept of error is the ifference between the measurement result an its true value, so the error must be a constant eviation, that is to say, any single error is a constant. The purpose of measurement is to reuce an evaluate error. From the unknown an constant characteristics of single error, this purpose naturally faces ifficulty. However, before the final measurement result is forme, our measurement is usually to measure repeately, an there will be a lot of error samples. When we observe a group of error samples, the error can show some regularity incluing certain regularity an ranom regularity. This provies paths for reucing an evaluating error: by certain regularity we can esign some methos for compensating an correcting error; by ranom regularity we can esign the statistics metho for reucing error an obtain error s statistic evaluation metho. That is, the error s variation regularity issue is actually aime at a group of error samples before the final measurement result is forme, instea of single error after the final measurement result is forme. However, it is important, the error s certain regularity an ranom regularity is actually from ifferent perspectives, is ifferent error s processing methos, an naturally still cannot be use to achieve error classification. The same kin of error can be processe accoring to certain regularity, also can be processe accoring to ranom regularity, an still can be processe accoring to both certain regularity an ranom regularity. There is still not error s classification issue accoring to certain regularity an ranom regularity. These are also the knowlege that the
new concept theory is totally ifferent from the existing measurement theory. For example: the frequency error values of a quartz crystal at ifferent temperature are shown in Table 1. Table1 Accoring to table 1, if corresponing to the temperature Error value Temperature -6 values to observe the error values, we can get the certain R f / f (1 1 ) regularity as shown in Fig; but if ignoring the temperature values to only o the statistic of error values, we can get ranom regularity as shown in Fig3. That is to say, corresponing to the temperature values to observe the error values, we see the certain regularity; viewing the temperatures as arbitrary an only observing the error s istribution, we see the ranom regularity. Naturally, there are two ways to eal with it in practice. 1, Ranom moel processing: By oing frequency - -3 - -1 1 3-3 -15-7 13 1-3 -8 values statistics, we can obtain an average frequency value an 5-11 use it as the nominal frequency value of the quartz crystal, an 6-11 obtain the maximum error range (in the temperature between -~1 egrees) is about ±37 1-6 7-8 an use it as the frequency error evaluation value. This expresses that the error of nominal frequency exists in a probability interval of ±37 8 9-1 15 1-6 at arbitrary temperature between -~1 egrees. It 1 37 belongs to this processing moe that temperature correction is not consiere in esigning an using some electronic equipment which uses quartz crystal., Function moel processing: By frequency error values, the frequency error s function 3 moel is fitte as R 9.98351-.13518T -.1861T.1T (Fig). In this way, temperature-frequency error can be correcte by the measurement value of temperature sensor, an a more accurate frequency value can be calculate. Resiual error (as shown in Fig5) is still processe by statistical rules, an the maximum error range of the resiual error was reuce to ± 3.8 1-6. This error processing metho has been wiely use in the manufacture of photoelectric geoimeter. 5 3 1 δ - - 6 8 1-1 - f(δ) -3 - Fig.The temperature-frequency error of quartz crystal Fig3.The frequency error s istribution 5 3 1 - - 6 8 1-1 - -3 - Fig.The function moel fitting of frequency error 5 3 1 - - -1 6 8 1 - -3 - Fig5.The resiual error s curve 3
Visible, from the perspectives of error regularity, there is no essential ifference between Fig an Fig 5! It is wrong that people use to only unerstan the Fig as certain regularity an only unerstan the Fig 5 as ranom regularity. Another example, the cycle error of the phase type photoelectric geoimeter [] [5] shows perioic function regularity with istance. Table is the measurement ata of an instrument. By the ata of table, the cycle error s function moel is fitte as y 5.7 sin( 36 5.1 )( mm). Fig6 is the curve compare of error an fitte sine function. However, for the error moel y 5.7 sin( 36 5.1 )( mm), its error obviously istributes in the range of ±5.7mm. When the istance is regare as arbitrary an the statistic of error values are one only, we can see that it is also a ranom istribution. As shown in Fig7. 1 Table Stanar istance Measure istance Error value 1 6.37 6.3.5 7.39 7.8 1.1 3 8.3 8. 3.9 9.6 9.187 5.9 5 1.5 1.183 6.7 6 11.53 11.178 7.5 7 1.56 1.196 6. 8 13.58 13.13.5 9 1.63 1.3 3.1 1 15.66 15.61.5 11 16.69 16.7 -.1 1 17.69 17.8-1.5 13 18.69 18.87-1.8 1 19.68 19.37-3.9 15.67.35-5.8 16 1.68 1.3-5. 17.69.3-5.1 18 3.69 3.35-3.6 19.7.9 -. 5.71 5.77 -.6 1 6.7 6.7 8 6 5 1 15 5 - - -6-8 8 6 -.5.1.15..5 - -6-8 Fig6.The function moel fitting of cycle error That is to say, relating the istance an the error value together to observe, we see the cycle regularity; viewing the istance as arbitrary an only observing error istribution, we see a ranom regularity. Naturally, in practice, there are also two methos to eal with it. 1, Function moel processing: Direct correct measurement result base on the error function moel y 5.7 sin( 36 5.1 )( mm). This processing metho is wiely recognize in the fiel of geoesy an geomatics., Ranom moel processing: Direct use measurement result without correction of the error function moel, thus, istance measurement result s uncertainty, which is contribute by the cycle error, will be ± 5.7mm(confience probability 1%); If we want to reuce cycle error by statistical metho, we may arbitrarily change the range to repeately measure the same istance by Table3 AB AC AB= AB +y AB AC= AC +y AC BC= AC-AB 1 1. 18. 1.55 17.9965 7.991 1.. 1.53 19.995 7.989 3 33. 1. 33.5.993 7.9898 7. 35. 7. 35.15 7.9995 5. 3. 1.997 3.55 8.18 6 8. 36. 8.35 35.9998 7.996 7 3. 38. 3.55 37.9965 7.991 8 36.. 35.9998 3.9968 7.9971 9 38. 6. 37.9965 6. 8.38 1 6. 3. 6. 3.3 8.9 11 3.. 3.3 1.997 7.9915 1 16.. 15.9998 3.9968 7.9971 13 18. 6. 17.9965 6. 8.38 1 19. 7. 18.9953 7. 8.67 15. 5. 1.997 5.55 8.18 AC - AB = 8. Fig7.The cycle error s istribution Average value =7.9988
ifferential metho, an achieve the self reuction of cycle errors by obtaining average value. Table 3 is the statistical reucing process of cycle errors simulate by using the cycle error y 5.7 sin( 36 5.1 )( mm) : set the measure istance BC=8.m, ranomly an arbitrarily simulate the 15 istance ifference ata, an finally give the average istance is 7.9988m. Final result is only smaller 1.mm than stanar istance, an the cycle error is significantly reuce without using function moel. In short, the errors variation regularity is a observation effects obtaine through observing a group of error samples instea single error; error s variation is certainly associate with measurement conition s variation, an these measurement conitions may be temperature, measurement range, instrument, time, location, leveling, sighting, electronic noise an so on; error s certain regularity an ranom regularity are observation results from ifferent perspectives, they have no mutual exclusion, an taking regularities to achieve error classification is similarly impossible. It is obviously inappropriate that VIM [6] [7] takes preictable manner an unpreictable manner to efine the error classification. 3. Error's effect characteristics The new concept theory emphasizes error has no systematic / ranom classification. It refers to the error has no the ifference whether it follows ranom istribution, but oes not negative error can prouce systematic / ranom effects. The error s effect characteristics an error s ranom istribution are two completely ifferent things: following ranom istribution refers to the error exists in a finite probability interval, systematic / ranom effects refer to the error sources contribute eviation / ispersion to subsequent repeate measurement. See Table. Table Current theory Error is classifie as systematic error an ranom error. The systematic error oes not follow ranom istribution, an the ranom error follows ranom istribution. Ranom istribution is ranom variation. Systematic error contributes systematic effects (contribute eviation), an ranom error contributes ranom effects (contribute ispersion). The systematic error is certain regularity, an the ranom error is ranom regularity. The new concepts theory Error cannot be classifie accoring to systematic an ranom. Any error follows a ranom istribution. Ranom istribution is that the error is in a finite probability interval instea of ranom variation. The error s systematic / ranom effects epen on the variation rules of measuring conitions in repeate measurements. The error s regularity epens on the perspectives of observation, an the error can show various regularities. It can be seen, the core of the existing systematic error concept is that it oes not follow ranom istribution, but the new theory stresses that any error follows a ranom istribution an that the error s systematic effects is completely ifferent from the existing systematic error concept. Just as important, the error s systematic or ranom effects epen on the variation rule of the measuring conitions in the repeate measurements, which is actually another angle of the error s regularity issue. In the case of quartz crystal s frequency, the frequency error varies with the temperature, so the temperature is relate measurement conition. If the repeate measurements are in constant temperature, temperature - frequency error will remain unchange, prouce systematic effects, an not rive the measurement results sequence ispersion; if the repeate measurements are in ifferent temperature, temperature - frequency error will change, prouce ranom effects, an rive the measurement results sequence ispersion. In the case of the photoelectric geoimeter, the cycle error is the perioic function of the measuring istance, so the istance is relate measurement conition. If repeate measurements are in the same istance conition, the cycle error will remain unchange, prouce systematic 5
effects, an not rive the measurement results sequence ispersion; if repeate measurement is in ifferent istance conition, the cycle error will change, prouce the ranom effects, an rive the measurement results sequence ispersion. (Such as Table 3). Moreover, besies systematic / ranom effect characteristics, error also has non-effect characteristic. For example, using the ifferential metho to measure istance (as shown in Table 3), the aitive constant error of photoelectric geoimeter has no effect to the measurement result of istance. Also, all the errors, which have no intrinsic physical relation with the measurement result, are unable to affect the measurement result. For example: the error of instrument A cannot affect the measurement result of instrument B. Visible, the error s systematic / ranom effects or measurement results sequence s eviation / iscrete epen on change rules of repeate measurement conitions. Temperature, measurement range, instrument, time, locations, leveling, even circuit noise, an so on, are measurement conitions. The same error can prouce systematic effects in a repeate measurements, also can prouce ranom effects in another repeate measurement, even cannot prouce effect to measurement result, naturally, using effect characteristics to classify error is still impossible, an it is a mistake that existing theory equate the error s effect characteristics with the error s classifications.. The new interpretation of uncertainty concept Because the existing uncertainty [8] [9] [1] concept system accepts the error classification philosophy an uncertainty s efinition clearly expresses the meaning of "ispersion", many people naturally unerstan it as being similar to precision. This kin of uncertainty is the neither fish nor fowl of course, naturally cause controversy [11]. Now, the theory of error classification is overthrown, the concepts of precision, trueness an accuracy must be abolishe, naturally, an uncertainty concept system, which must be interprete by the error non-classification philosophy, has become the only way out of measurement theory. The total error of measurement result is constant eviation, has no classification. Its numerical value is unknown, an uncertain. The unknown an uncertain egree of error s numerical value is the uncertainty, which use its probability interval evaluation value stanar eviation or times stanar eviation to express. Because the total error comes from the synthesis of many error sources accoring to algebraic law, the total stanar eviation is equal to the synthesis of stanar eviations of all the source errors accoring to the variance propagation law. Further, because the numerical value of measurement result is certain an the numerical value of error is uncertain, the uncertainty also expresses the uncertain egree that the true value cannot be etermine. Uncertainty is the evaluation value of the probability interval of measurement result s error, an expresses the egree that the true value cannot be etermine or the probable egree that the measurements result is close to the true value. This is the new explanation of the uncertainty concept. In some cases, the variation of the true value in the future an the ambiguity of the true value efinition also shoul be consiere as the error problem, thus, a broa unerstaning of uncertainty is given. A simple example for comparing: A igital caliper is use to measure the iameter of a steel ball. The maximum permissible error (MPE) of the igital caliper is: ±.mm. Continuous repeate to measure 1 times, each time is the same reaing 5.mm. In this way, the final average measurement result is 5.mm, an its stanar eviation is ±.mm. Processing accoring to existing error classification philosophy: 1.Ranom error is ±.mm, namely precision is ±.mm.. The output error of caliper oes not contribute ispersion, is systematic error, on t follow istribution, has no stanar eviation, an is evaluate by trueness. 3. The trueness is a qualitative concept, cannot use numerical inicator to express.. The accuracy, which is the comprehensive evaluation of total error, is also a qualitative concept, also cannot use numerical inicator to express. Processing accoring to the error non-classification philosophy: 1.The total error = the eviation between final result an expectation + the eviation between expectation an true value, an they are all constant eviation..the stanar eviation of the eviation between final result an mathematical expectation is ±.mm. 3. The eviation between mathematical expectation an true value is from the igital caliper s output error which is also to follow the ranom 6
istribution from caliper maker s perspective, an its stanar eviation can be obtaine by maximum permissible error (MPE) ±.mm.. The stanar eviation of total error is equal to the synthesis of the two stanar eviations accoring to probability principle. It is very easy to be obtaine that the total expansion uncertainty of final result 5.mm is ±.mm. 5. Conclusion The single error of any final measurement result is constant regularity, error classification cannot be achieve by the same constant regularity; although a group of error samples can show some certain regularity an ranom regularity, certain regularity an ranom regularity are observation result from ifferent perspectives an also cannot be use for classifying error; because error s effect characteristics only epen on the variation rule of the measuring conitions in repeate measurements, an the same kin of error can show various kins of effect characteristics, so using effect characteristics to classify error is still impossible. From all the perspectives incluing single error s constant characteristic, the variation regularity of a group of error samples, an error s effect characteristics, classifying error cannot be realize. Naturally, the concepts logic system of precision, trueness an accuracy of base on error classification theory must completely collapse, an an uncertainty concept system, which must be interprete by the error non-classification philosophy, has become the only way out of measurement theory. Taking the same conitions in repeate measurements will make all the source errors to keep constant, cannot make error to be reuce. By appropriately changing the relevant measurement conitions in repeate measurements, any error can be mae to contribute ispersion. Thus, reucing error can be realize by function moel or ranom moel processing, an the error s probability interval evaluation also can be obtaine. Uncertainty is the evaluation value of the probability interval of measurement result s error, an expresses the probable egree that the final measurement result is close to the true value. This is the new explanation of the uncertainty concept. Reference: [1].Ye Xiao-ming, Xiao Xue-bin, Shi Jun-bo, Ling Mo. The new concepts of measurement error theory, Measurement, Volume 83, April 16, Pages 96 15 [].Ye Xiao-ming, Ling Mo, Zhou Qiang, Wang Wei-nong, Xiao Xue-bin. The New Philosophical View about Measurement Error Theory. Acta Metrologica Sinica, 15, 36(6): 666-67. [3].Ye Xiao-ming. Errors Classification Philosophy Critique [C]// Proceeings of National Doctoral Forum on Surveying an Mapping. 11 []. JJG73-3, Electro-optical Distance Meter (EDM instruments) [5]. ISO 1713-:1,Optics an optical instruments -- Fiel proceures for testing geoetic an surveying instruments -- Part : Electro-optical istance meters (EDM measurements to reflectors) [6]. International vocabulary of metrology Basic an general concepts an associate terms (VIM) JCGM :1 [7]. JJF11-11 General Terms in Metrology an Their Definitions [8]. Churchill Eisenhart. Expression of the Uncertainties of Final Results [J] Science 1 June 1968: 11-1. [9]. Guie to the Expression of Uncertainty in Measurement, International Organization for Stanar, First eition correcte an reprinte,8,isbn9-67-1188-9 [1]. JJF159-1 Evaluation an Expression of Uncertainty in Measurement [11]. Schmit,H.Warum GUM?-Kritische Anmerkungen zur Normefinition er Messunsicherheit un zu verzerrten Elementarfehlermoellen [EB/OL]. http://www.gia.rwth-aachen.e/forschung/angwstatistik/warum_gum/warum_gum_zfv.pf 7