# A New Theoretical Interpretation of Measurement Error and Its Uncertainty

Authors: Huisheng Shi; Xiaoming Ye; Cheng Xing; Shijun Ding
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

The traditional measurement theory interprets the variance as the dispersion of a measured value, which is actually contrary to a general mathematical concept that the variance of a constant is 0. This paper will fully demonstrate that the variance in measurement theory is actually the evaluation of probability interval of an error instead of the dispersion of a measured value, point out the key point of mistake in the traditional interpretation, and fully interpret a series of changes in conceptual logic and processing method brought about by this new concept.


---

## Body

## 1. Introduction

Human scientific research begins with the measurement of various physical quantities, and measurement is the basis of modern scientific research. Especially for the research of artificial intelligence, it is an essential process to obtain natural information and evaluate its authenticity, which itself is the research area of measurement. Therefore, a rigorous measurement theory should be an important part of the modern scientific theory system. This paper points out the main problems existing in the traditional measurement theory and gives the correct interpretation method for the measurement theory.In traditional measurement theory, the measured value (or observed value) is considered as a random variable, and variance is interpreted as the dispersion of the measured value, both the precision and uncertainty are defined as the dispersion concept of measured value (or observed value) [1–5], so that people can hardly make clear the conceptual difference between them.However, in any measurement, both the measured value and every observed value are numerical values. According to probability theory, the variance of a numerical value (constant) is zero. So, how does a numerical value show dispersion? Next, we illustrate the contradictory expression of “variance” in traditional theory.For example, the measured value of Mount Everest elevation in 2005 isx=8844.43m, and its precision is σx=0.21m. But in fact, this mathematical expression gives an equation σ8844.43m=0.21m, which violates the basic mathematical concept, because the equation x=8844.43m inevitably leads to the equation σx=σ8844.43, and according to the concept of σ2C=0 in probability theory, there must be σ8844.43=0. Although many other measured values x1, x2, …, can be obtained by repeatedly measuring the height of the Mount Everest, and there can be x≠x1≠x2≠⋯, the equations x=8844.43m and σx=σ8844.43 still exist. Therefore, the equation σx=σ8844.43=0.21 can never be consistent with mathematical concepts.It can be seen that in traditional theories, the definition of precision and uncertainty is the dispersion of all possible measured values, but their mathematical expression is the dispersion of a single measured value (a single numerical value). This approach is actually a stealth change of concept and will inevitably lead to a series of conceptual logic troubles.A question arises. All measured values that diverge from each other can definitely be used to describe a random variable, but a measured value must be a numerical value and belong to a constant. So, how should the measurement theory be interpreted?So far, in the measurement industry, there is no literature questioning the conceptual category of measured values, such as the recent literature [6].In references [7–10], the authors proposed some new concepts to reinterpret measurement theory. Ye et al. [7] proposed a new error epistemology that all errors follow a random distribution and cannot be classified as systematic error and random error. Ye et al. [8] pointed out that the standard deviation (variance) is the evaluation value of the probability interval of error, any error has a variance for evaluating its uncertainty, and so on. Ye et al. [9] pointed out that the dispersion and deviation of repeated observations are determined by the changing rules of repeated measurement conditions and it is possible and correct to handle errors according to the function model or the random model. Ye and Ding [10] pointed out that the measured value is a numerical value whose variance is zero and the dispersion of a measured value is an incorrect concept. According to these new concepts, the mathematical expression of the Mount Everest elevation case should be x=8844.43m and σΔx=0.21m, where x represents the measured value and Δx represents its error.Although these new concepts have been proposed to reinterpret measurement theory, the root of these concepts and the interpretation process have not been fully described mathematically. Therefore, in this paper, the authors follow strict mathematical concept to point out the misunderstanding of the traditional concepts, give a clear interpretation for the origin of these new measurement concepts, and systematically explain a series of changes in theoretical logic and mathematical processing.


## 2. Constant and Random Variable

In probability theory, a constant is a numerical value, such as 100, 150,x = 100, x = 8844.43, and so on.Unlike constants, a random variable is an unknown quantity whose actual value cannot be given. Because the random variable is unknown or uncertain, we can only describe the probability range of its value. In order to study its probability range, it is necessary to study the distribution range of all its possible values (sample space), while all possible values refer to the set of test values of random variables under all permitted possible test conditions (the random test does not have the same conditions). Mathematical expectation and variance are the numerical expressions of probability range of the random variable.For a random variableL with all possible values Li, L∈Li, Pi is the probability that each Li is L (continuous random variables correspond to the probability density function PL), and its mathematical expectation is defined as follows:(1)EL=∑i=1nPiLi,orEL=∫−∞+∞LPLdL,and its variance is defined as the dispersion of its possible values Li:(2)σ2L=EL−EL2.This means that the random variableL exists within a probability interval with mathematical expectation EL and variance σ2L or that mathematical expectation and variance are the evaluation values of its probability interval. Note that describing a random variable requires two parameters: mathematical expectation and variance, both of which are indispensable.Now, suppose that there is a constantC, and EL=C and σ2L=0, then(3)EL−EL2=0.By substitutingEL=C, we obtain EL−C2=0.Therefore,L=C.That is, when the variance of a random variableL is reduced to zero, it becomes a constant C. In other words, for a constant C, because all its possible values are itself, we obtain(4)EC=C,(5)σ2C=EC−EC2=0.That is to say, a constant is a special random variable, and its mathematical expectation and variance are itself and 0, respectively. Of course, a constant is a known quantity and usually does not need to be expressed in terms of probability.It can be seen that both constant and random variable have their own mathematical expectation and variance. Therefore, if we can give the variance of a quantity but cannot give its mathematical expectation, there must be a conceptual mistake.It should be noted that for the random variableL∈Li, its basic feature is that its value is unknown, but for a sample Lk∈Li, because it is a numerical value, it is still a constant rather than a random variable, and ELk=Lk and σ2Lk=0. Obviously, EL≠ELk and σ2L≠σ2Lk. That is, constant and random variable are distinguished by whether they have a numerical value, and the sample Lk is a numerical value, which is a constant and cannot be described by the entire set Li. The conceptual differences between random variable and sample are shown in Table 1.Table 1
The conceptual differences between the random variable and the sample.
Random variableLSampleLkL∈LiLk∈LiThe value ofL is unknownLk is a numerical value, which is a constantIt has many possible valuesAll its possible values are itselfEvery sampleLi in Li is a possible value of L, that is, Pi≠0%Any other samplesLj in Li are definitely not Lk, that is, Pj=0% and Pk=100%We can only useLi to describe its probability rangeIt is a numerical value, and no probability range needs to be describedEL=∑PiLi,σ2L=EL−EL2ELk=Lk,σ2Lk=0EL≠ELk,σ2L≠σ2LkExample 1.
A dice has six faces corresponding to the values 1, 2, 3, 4, 5, and 6, respectively. After the dice is thrown, we do not know its display value, which is a random variableL∈1,2,3,4,5,6. Then, according to definitions (1) and (2), EL=3.5 and σ2L=2.92. Its meaning is that although the display value L is unknown, it exists within a probability interval with 3.5 as the center and 2.92 as the width evaluation. Obviously,(6)E1=1E2=2⋮E6=6≠EL=3.5,σ21=0σ22=0⋮σ26=0≠σ2L=2.92.Example 2.
The exam scores of all students in a school arexi, in which the exam score of a student A is x0. It is reasonable to express an unknown score x with the mathematical expectation Ex and variance σ2x of xi, but it is illogical to impose σ2x on x0 because x0 is a known constant and has no need to be expressed by variance and mathematical expectation. Moreover, σ2x is obviously not the dispersion of future exam scores of student A. That is, Ex≠Ex0andσ2x≠σ2x0.Example 3.
Someone’s salary isx0=10000RMB, and the salaries of all the employees in his company form a sample sequence xi. By making the statistics of xi, we can obtain the mathematical expectation Ex and variance σ2x. In exactly the same way, although x0∈xi, we cannot force x0 to belong to a random variable at all because x0=10000RMB is known.
That is to say, it is reasonable to evaluate the probability of an unknown event with the statistic values of a group of known events, but it is illogical to use it to evaluate the “probability” of a known event.
Now, we suppose there is a random variableL with mathematical expectation EL=C and ΔC=L−EL, and then, there is(7)L=EL+L−EL=C+ΔC.
For the constantC, there are(8)EC=C,σ2C=0.
For the random variableΔC=L−EL, there are(9)EΔC=EL−EL=EL−EL=0,(10)σ2ΔC=EΔC−EΔC2=EΔC2=EL−EL2=σ2L.
This is to say, a random variableL with the mathematical expectation C can be viewed as the superposition of a constant C and a random variable ΔC taking 0 as its mathematical expectation. And it should be noted that the variance σ2L or σ2ΔC always has nothing to do with constant C.


## 3. Origin of Conceptual Troubles in Traditional Theory

Figure1 shows the schematic diagram of the measurement concept in traditional measurement theory. Because people notice that a measured value is in a state of random change in repeated measurement, the measured value and the random error are considered as random variables, and the variance is the dispersion of the measured value or random error. Besides, the systematic error and the true value are constant in repeated measurement, so the systematic error and the true value are considered as constants which have no variances (or the variance is zero). In this way, according to formula (2), σ2x=Ex−Ex2. Therefore, traditional textbooks [11–13] usually use the form of σ2x or σx2 to express the variance. However, these are obviously inconsistent with the meanings of random variables and constants described in Section 2.Figure 1
Schematic diagram in traditional measurement theory.Besides, in actual measurement, we always have to give a numerical valuex0 as the final measured value. Therefore, the actual schematic diagram is shown in Figure 2. According to the concepts in Section 2, although the measured value x0 is a sample within a random distribution, because the measured value x0 is a numerical value and σ2x0=0 and Ex0=x0, it is illogical to replace σ2x0 with σ2x.Figure 2
Schematic diagram in the new concept theory.In addition, simply replacingσ2x0 with σ2x cannot express a complete mathematical meaning because the traditional theory cannot submit the mathematical expectation Ex.On the contrary, the systematic error and the true value are unknown and are regarded as constants by traditional theory. However, according to formula (4), the mathematical expectation of a constant is itself, so it is impossible to give the numerical values of mathematical expectations of the systematic error and true value. Therefore, this so-called constant is obviously not the same concept as the constant in the probability theory, and it is also a conceptual trouble in the traditional measurement theory.In short, in the traditional theory, except for the conceptual trouble of violating the concept that the variance of a constant is zero, the conceptual trouble of missing mathematical expectations is shown in Table2.Table 2
The conceptual trouble of missing mathematical expectations.
Measured valuexRandom errorx−ExSystematic errorEx−xTTrue valuexTMathematical expectationAbsent0AbsentAbsentVarianceσ2xσ2x00Because random errors have variance but systematic errors cannot be quantitatively evaluated, traditional theories believe that trueness and accuracy are both qualitative concepts, and the relationship between the uncertainty concept and them is of course very difficult to explain.


## 4. Probability Expression of Basic Measurement Concepts

It can be seen that because of the wrong understanding of the concept of the random variable, the conceptual logic of traditional theory actually has systematic troubles. Therefore, we need to reorganize the basic measurement conceptual logic according to the concepts in Section2.


### 4.1. Measured Value

In Figure2, the measured value x0 is an observed value within all possible observed values xi and is a numerical value. According to formulas (4) and (5), there are(11)Ex0=x0,σ2x0=0.Please note that any quantity with a numerical value, including the error sample, the measured value of error, the detected value of instrument error, the value of mathematical expectation, and the value of variance, is a constant.


### 4.2. Error

As shown in Figure2, the true value of the measurand is xT, and the error of the final measured value x0, which is an unknown deviation Δ=x0−xT, can be divided into ΔA=x0−Ex and ΔB=Ex−xT.First of all, the errorΔA is a random variable, and ΔA∈xi−Ex. Moreover, the sample space of error x−Ex is also xi−Ex, so error x−Ex can be used to represent ΔA, that is, ΔA=x−Ex. According to the formulas (1) and (2), there are(12)EΔA=Ex−Ex=0,(13)σ2ΔA=Ex−Ex−Ex−Ex2=Ex−Ex2=EΔA2.That is, although deviationΔA is unknown, it exists within a probability interval with 0 as center and σ2ΔA as width evaluation. In other words, σ2ΔA is the evaluation of probability interval of deviation ΔA, which expresses the degree that a surveyor cannot determine the value of deviation ΔA.Taking the normal distribution as an example, the varianceσ2ΔA expresses that the deviation ΔA is within the interval of −σΔA+σΔA under the confidence probability of 68%. Variance is actually a concept of error range with probability meaning and expresses an error’s possible deviation degree.In formula (13), the single deviation ΔA is a member within all its possible values, and the dispersion interval of all its possible values is the probability interval of this deviation ΔA. An unknown deviation follows a random distribution, which means that all possible values of the deviation follow a random distribution.This principle obviously can be extended to the errorΔB. In fact, when we trace back to the upstream measurement of forming error ΔB, we will find that the formation principle of error ΔB is similar to that of current error ΔA and that the error ΔB is also a member within all its possible values. Therefore, there is also variance σ2ΔB to evaluate the probability interval of error ΔB, and also EΔB=0.For example, the multiplicative constant errorR of a geodimeter [14, 15] comes from the frequency error of the quartz crystal and is always viewed as a systematic error without variance by traditional measurement theory. However, it is the output error in the field of instrument manufacturing, and the submission process of its variance will be demonstrated in a case in Section 6.Obviously, according to the principle of formulas (7)–(10), if the mathematical expectation of an error is C rather than 0, then C must be corrected to the final measured value, and the mathematical expectation of the remaining unknown error is still 0. That is, for any unknown error Δx, there is always(14)EΔx=0.Thus, the error’s variance is expressed as follows:(15)σ2Δx=Ex−EΔx2=EΔx2.Δx in formulas (14) and (15) can express not only the deviation ΔA between the measured value and the mathematical expectation, but also the deviation ΔB between the mathematical expectation and the true value. It can even express the deviation Δ=ΔA+ΔB between the measured value and the true value.In this way, according to formulas (1) and (2), there are(16)Δ=ΔA+ΔB,EΔ=EΔA+EΔB=0,σ2Δ=EΔ−EΔ2=EΔ2,=EΔA+ΔB2,=σ2ΔA+σ2ΔB.Because the final measured value is unique and constant, bothΔA and ΔB are unknown deviations. In addition, both of them have their own variance; hence, it is incorrect that the traditional measurement theory considers ΔA as the random error and considers ΔB as the systematic error. Moreover, the corresponding concepts of precision and trueness are also incorrect.It should be emphasized that formula (14) means that the mean value of all possible values of an unknown error is 0, which expresses the probabilities that an unknown error takes a positive value and a negative value are equal in our subjective cognition. From a statistical perspective, all possible values of an error refer to the set of all error values under all possible measurement conditions permitted by measurement specification, so the traditional concept of “repeated measurement under the same conditions” must be abandoned [8, 9]; otherwise, a unique error value obtained under a particular condition is only one sample within all possible values and does not represent all possible values, which is very easy to cause the illusion of EΔx≠0.In short, being different from the measured value, the error is unknown; because the error is unknown, we can only study its probability range; because of studying its probability range, we must study all possible values of error; because of studying all possible values of error, error samples must come from all the possible measurement conditions permitted by measurement specification, and the traditional concept of “repeated measurement under the same measurement conditions” must be abandoned.


### 4.3. Variance of Regular Error

Any error has its variance, including the regular error, because the regular error also has all its possible values.For example, the periodic error of a phase photoelectric distance meter [14, 15] conforms to the sine regularity, and its function model is δ=AsinD/λ×2π+φ. However, when we only observe the density distribution of all its possible values, this cyclic error’s probability density function fδ can be derived as(17)fδ=1πA2−δ2,δ≤A,0,δ>A.As shown in Figure3, furthermore, its variance can be derived as σ2δ=A2/2, and its mathematical expectation can be derived as Eδ=0.Figure 3
Regularity and randomness of periodic error.Another example, the rounding errorδ is a sawtooth cycle regularity function of a true value w. However, when we only observe the density distribution of all possible values, the error also follows a random distribution, as shown in Figure 4, and its probability density function is(18)fδ=12a,δ≤a,0,δ>a.Figure 4
Regularity and randomness of rounding error.Its variance can be derived asσ2δ=a2/3, and its mathematical expectation can be derived as Eδ=0.That is to say, the regularity and the randomness are the effect of observing all possible values of error from different perspectives, there is no opposition between them, and there is actually no need to dwell on the error’s regularity in the discussion of error evaluation. In other words, when a regular error is unknown, we can still use the mathematical expectation and the variance to describe its probability range. Furthermore, traditional measurement theories use the regularity and the randomness to classify errors into systematic errors and random errors, which is also proved to be incorrect.In addition, when the measurement conditions vary with the repeated measurement, the corresponding regular errors will cause the dispersion of the repeated observations, which is exactly the same as the dispersion caused by the noise which varies with time conditions [9]. This dispersion is exactly the means by which we obtain its variance. It can be seen that, in order to obtain the variance of an error, we need to collect error samples under all possible measurement conditions.


### 4.4. Probability Expression of True Value

With previous statements, we already know that for the measured valuex0, there are Ex0=x0 and σ2x0=0; for the error Δ, there are EΔ=0 and σ2Δ=EΔ2. Because the error is the difference between the measured value and the true value, that is, Δ=x0−xT, there are(19)xT=x0−Δ,ExT=Ex0−Δ=Ex0−EΔ=x0,σ2xT=ExT−ExT2=Ex0−Δ−x02=σ2Δ.The probability expressions of the true valuexT, the measured value x0, and the error Δ are summarized in Table 3 [10].Table 3
The probability expression of the true value, measured value, and error.
Measured valuex0ErrorΔTrue valuexTMathematical expectationx00x0Variance0σ2Δσ2ΔThe above is the case where an observed value is used as the final measured value. If the mean value ofn observations is taken as the final measured value, it can be inferred that the variance σ2ΔA will decrease by n times (Section 6.1).


## 4.1. Measured Value

In Figure2, the measured value x0 is an observed value within all possible observed values xi and is a numerical value. According to formulas (4) and (5), there are(11)Ex0=x0,σ2x0=0.Please note that any quantity with a numerical value, including the error sample, the measured value of error, the detected value of instrument error, the value of mathematical expectation, and the value of variance, is a constant.


## 4.2. Error

As shown in Figure2, the true value of the measurand is xT, and the error of the final measured value x0, which is an unknown deviation Δ=x0−xT, can be divided into ΔA=x0−Ex and ΔB=Ex−xT.First of all, the errorΔA is a random variable, and ΔA∈xi−Ex. Moreover, the sample space of error x−Ex is also xi−Ex, so error x−Ex can be used to represent ΔA, that is, ΔA=x−Ex. According to the formulas (1) and (2), there are(12)EΔA=Ex−Ex=0,(13)σ2ΔA=Ex−Ex−Ex−Ex2=Ex−Ex2=EΔA2.That is, although deviationΔA is unknown, it exists within a probability interval with 0 as center and σ2ΔA as width evaluation. In other words, σ2ΔA is the evaluation of probability interval of deviation ΔA, which expresses the degree that a surveyor cannot determine the value of deviation ΔA.Taking the normal distribution as an example, the varianceσ2ΔA expresses that the deviation ΔA is within the interval of −σΔA+σΔA under the confidence probability of 68%. Variance is actually a concept of error range with probability meaning and expresses an error’s possible deviation degree.In formula (13), the single deviation ΔA is a member within all its possible values, and the dispersion interval of all its possible values is the probability interval of this deviation ΔA. An unknown deviation follows a random distribution, which means that all possible values of the deviation follow a random distribution.This principle obviously can be extended to the errorΔB. In fact, when we trace back to the upstream measurement of forming error ΔB, we will find that the formation principle of error ΔB is similar to that of current error ΔA and that the error ΔB is also a member within all its possible values. Therefore, there is also variance σ2ΔB to evaluate the probability interval of error ΔB, and also EΔB=0.For example, the multiplicative constant errorR of a geodimeter [14, 15] comes from the frequency error of the quartz crystal and is always viewed as a systematic error without variance by traditional measurement theory. However, it is the output error in the field of instrument manufacturing, and the submission process of its variance will be demonstrated in a case in Section 6.Obviously, according to the principle of formulas (7)–(10), if the mathematical expectation of an error is C rather than 0, then C must be corrected to the final measured value, and the mathematical expectation of the remaining unknown error is still 0. That is, for any unknown error Δx, there is always(14)EΔx=0.Thus, the error’s variance is expressed as follows:(15)σ2Δx=Ex−EΔx2=EΔx2.Δx in formulas (14) and (15) can express not only the deviation ΔA between the measured value and the mathematical expectation, but also the deviation ΔB between the mathematical expectation and the true value. It can even express the deviation Δ=ΔA+ΔB between the measured value and the true value.In this way, according to formulas (1) and (2), there are(16)Δ=ΔA+ΔB,EΔ=EΔA+EΔB=0,σ2Δ=EΔ−EΔ2=EΔ2,=EΔA+ΔB2,=σ2ΔA+σ2ΔB.Because the final measured value is unique and constant, bothΔA and ΔB are unknown deviations. In addition, both of them have their own variance; hence, it is incorrect that the traditional measurement theory considers ΔA as the random error and considers ΔB as the systematic error. Moreover, the corresponding concepts of precision and trueness are also incorrect.It should be emphasized that formula (14) means that the mean value of all possible values of an unknown error is 0, which expresses the probabilities that an unknown error takes a positive value and a negative value are equal in our subjective cognition. From a statistical perspective, all possible values of an error refer to the set of all error values under all possible measurement conditions permitted by measurement specification, so the traditional concept of “repeated measurement under the same conditions” must be abandoned [8, 9]; otherwise, a unique error value obtained under a particular condition is only one sample within all possible values and does not represent all possible values, which is very easy to cause the illusion of EΔx≠0.In short, being different from the measured value, the error is unknown; because the error is unknown, we can only study its probability range; because of studying its probability range, we must study all possible values of error; because of studying all possible values of error, error samples must come from all the possible measurement conditions permitted by measurement specification, and the traditional concept of “repeated measurement under the same measurement conditions” must be abandoned.


## 4.3. Variance of Regular Error

Any error has its variance, including the regular error, because the regular error also has all its possible values.For example, the periodic error of a phase photoelectric distance meter [14, 15] conforms to the sine regularity, and its function model is δ=AsinD/λ×2π+φ. However, when we only observe the density distribution of all its possible values, this cyclic error’s probability density function fδ can be derived as(17)fδ=1πA2−δ2,δ≤A,0,δ>A.As shown in Figure3, furthermore, its variance can be derived as σ2δ=A2/2, and its mathematical expectation can be derived as Eδ=0.Figure 3
Regularity and randomness of periodic error.Another example, the rounding errorδ is a sawtooth cycle regularity function of a true value w. However, when we only observe the density distribution of all possible values, the error also follows a random distribution, as shown in Figure 4, and its probability density function is(18)fδ=12a,δ≤a,0,δ>a.Figure 4
Regularity and randomness of rounding error.Its variance can be derived asσ2δ=a2/3, and its mathematical expectation can be derived as Eδ=0.That is to say, the regularity and the randomness are the effect of observing all possible values of error from different perspectives, there is no opposition between them, and there is actually no need to dwell on the error’s regularity in the discussion of error evaluation. In other words, when a regular error is unknown, we can still use the mathematical expectation and the variance to describe its probability range. Furthermore, traditional measurement theories use the regularity and the randomness to classify errors into systematic errors and random errors, which is also proved to be incorrect.In addition, when the measurement conditions vary with the repeated measurement, the corresponding regular errors will cause the dispersion of the repeated observations, which is exactly the same as the dispersion caused by the noise which varies with time conditions [9]. This dispersion is exactly the means by which we obtain its variance. It can be seen that, in order to obtain the variance of an error, we need to collect error samples under all possible measurement conditions.


## 4.4. Probability Expression of True Value

With previous statements, we already know that for the measured valuex0, there are Ex0=x0 and σ2x0=0; for the error Δ, there are EΔ=0 and σ2Δ=EΔ2. Because the error is the difference between the measured value and the true value, that is, Δ=x0−xT, there are(19)xT=x0−Δ,ExT=Ex0−Δ=Ex0−EΔ=x0,σ2xT=ExT−ExT2=Ex0−Δ−x02=σ2Δ.The probability expressions of the true valuexT, the measured value x0, and the error Δ are summarized in Table 3 [10].Table 3
The probability expression of the true value, measured value, and error.
Measured valuex0ErrorΔTrue valuexTMathematical expectationx00x0Variance0σ2Δσ2ΔThe above is the case where an observed value is used as the final measured value. If the mean value ofn observations is taken as the final measured value, it can be inferred that the variance σ2ΔA will decrease by n times (Section 6.1).


## 5. Covariance Propagation

It can be seen from Table3 that after a measured value x0 is given, only variance σ2Δ needs to be studied.


### 5.1. Covariance

Considering that two correlated errors have common component, formula (15) can be extended to any two errors. That is,(20)σΔxiΔxj=EΔxiΔxj.Thus, for the error sequenceΔX=Δx1Δx2⋯ΔxtT, the definition of variance is(21)DΔX=EΔXΔXT.That is,(22)DΔX=EΔx1Δx2⋮ΔxtΔx1Δx2⋯Δxt=σ112σ12⋯σ1tσ21σ222⋯σ12⋮⋮⋮σt1σt2⋯σtt2.Obviously, the true definition of the variance is formula (21). Formula (15) is only a special case of formula (21) when t = 1, and formula (13) is only a special case of formula (15) when interpreting Δx as x−Ex.So, what is the meaning of covariance?It is assumed that the errorsk, p, and q are uncorrelated with each other, and that their variance are σk2,σp2, and σq2, respectively. Now, there are two errors δ=k+p and ε=k+q, and they contain a communal error component k. Therefore, we can obtain(23)σδ2=σk2+σp2,σε2=σk2+σq2.According to the definition of covariance,(24)σδε=Eδε=Ek+pk+q=Ek2+Ekp+Ekq+Epq,with the assumption that the errors k, p, and q are irrelevant from each other, we can obtain Ekp=0, Ekq=0, and Epq=0, and equation (24) becomes(25)σδε=Ek2=σk2.The covarianceσδε is actually the variance of their communal error component k. That is to say, the mathematical meaning of covariance is the probability evaluation of the communal error component contained in two errors. As long as there are communal error components among different errors, there must be a covariance between them. Of course, the symbol and coefficient of communal error component should be considered in the actual measurement.For example, the two measured value’s errors measured by the same instrument are correlated, and the errors of two instruments calibrated by the same benchmark are also correlated.Moreover, like the above principle, when two errors are associated with the same measurement condition, there is also a covariance between them. For example, both the error of light speed in atmosphere and the thermal expansion error of metal are functions of temperature, and there is a correlation between all possible values of the two errors.


### 5.2. Law of Covariance Propagation

Because any error has all its possible value and has its variance, the law of covariance propagation is extended to any error. In addition, the law of covariance propagation can only be interpreted as the propagation law of error’s probability interval and cannot be interpreted as propagation law of measured value’s dispersion.Here is a measurement equation:(26)Z=FX.We can derive the total differential of equation (26):(27)ΔZ=K⋅ΔX,where(28)ΔZ=ΔZ1ΔZ2⋮ΔZt,K=k11k12⋅⋅⋅k1nk21k22⋅⋅⋅k2n⋮⋮⋮⋮kt1kt2⋅⋅⋅ktn,ΔX=Δx1Δx2⋮Δxn.According to formula (21), the covariance matrix of the error sequence ΔZ is(29)DΔZ=EΔZΔZT=EK⋅ΔXK⋅ΔXT=K⋅DΔX⋅KT.Equation (29) is the law of covariance propagation. The relationship between equations (27) and (29) is as follows:(1)
In the error equation (27), the direct participants of synthesis are the error itself, each error is a deviation, and error synthesis always follows the algebra rule.(2)
In the variance equation (29), the participants of synthesis are all possible values of each error instead of each error itself. It expresses the propagation relation of dispersion of all possible values of errors and also the propagation relation of probability intervals between errors.


## 5.1. Covariance

Considering that two correlated errors have common component, formula (15) can be extended to any two errors. That is,(20)σΔxiΔxj=EΔxiΔxj.Thus, for the error sequenceΔX=Δx1Δx2⋯ΔxtT, the definition of variance is(21)DΔX=EΔXΔXT.That is,(22)DΔX=EΔx1Δx2⋮ΔxtΔx1Δx2⋯Δxt=σ112σ12⋯σ1tσ21σ222⋯σ12⋮⋮⋮σt1σt2⋯σtt2.Obviously, the true definition of the variance is formula (21). Formula (15) is only a special case of formula (21) when t = 1, and formula (13) is only a special case of formula (15) when interpreting Δx as x−Ex.So, what is the meaning of covariance?It is assumed that the errorsk, p, and q are uncorrelated with each other, and that their variance are σk2,σp2, and σq2, respectively. Now, there are two errors δ=k+p and ε=k+q, and they contain a communal error component k. Therefore, we can obtain(23)σδ2=σk2+σp2,σε2=σk2+σq2.According to the definition of covariance,(24)σδε=Eδε=Ek+pk+q=Ek2+Ekp+Ekq+Epq,with the assumption that the errors k, p, and q are irrelevant from each other, we can obtain Ekp=0, Ekq=0, and Epq=0, and equation (24) becomes(25)σδε=Ek2=σk2.The covarianceσδε is actually the variance of their communal error component k. That is to say, the mathematical meaning of covariance is the probability evaluation of the communal error component contained in two errors. As long as there are communal error components among different errors, there must be a covariance between them. Of course, the symbol and coefficient of communal error component should be considered in the actual measurement.For example, the two measured value’s errors measured by the same instrument are correlated, and the errors of two instruments calibrated by the same benchmark are also correlated.Moreover, like the above principle, when two errors are associated with the same measurement condition, there is also a covariance between them. For example, both the error of light speed in atmosphere and the thermal expansion error of metal are functions of temperature, and there is a correlation between all possible values of the two errors.


## 5.2. Law of Covariance Propagation

Because any error has all its possible value and has its variance, the law of covariance propagation is extended to any error. In addition, the law of covariance propagation can only be interpreted as the propagation law of error’s probability interval and cannot be interpreted as propagation law of measured value’s dispersion.Here is a measurement equation:(26)Z=FX.We can derive the total differential of equation (26):(27)ΔZ=K⋅ΔX,where(28)ΔZ=ΔZ1ΔZ2⋮ΔZt,K=k11k12⋅⋅⋅k1nk21k22⋅⋅⋅k2n⋮⋮⋮⋮kt1kt2⋅⋅⋅ktn,ΔX=Δx1Δx2⋮Δxn.According to formula (21), the covariance matrix of the error sequence ΔZ is(29)DΔZ=EΔZΔZT=EK⋅ΔXK⋅ΔXT=K⋅DΔX⋅KT.Equation (29) is the law of covariance propagation. The relationship between equations (27) and (29) is as follows:(1)
In the error equation (27), the direct participants of synthesis are the error itself, each error is a deviation, and error synthesis always follows the algebra rule.(2)
In the variance equation (29), the participants of synthesis are all possible values of each error instead of each error itself. It expresses the propagation relation of dispersion of all possible values of errors and also the propagation relation of probability intervals between errors.


## 6. Statistical Calculation of Variance

Since the number of error samples is always limited in actual measurement, formula (15) can be approximated as(30)σ2Δx≈∑i=1nΔxi2n.Formula (30) is also the source of least squares principle. That is to say, from the perspective of the new concept, only the concept of error evaluation changes, while the principle of the least square method used to obtain the best measured values does not change.In actual measurement, in order to achieve the reduction and evaluation of the measurement error, a large number of observations should be carried out. Because errors make a large number of observations contradict from each other, the optimal measured values must be given by adjustment process, and the errors of the measured values should also be evaluated. We only discuss the case of the least square adjustment in this section.


### 6.1. Direct Measurement for Single Measurand

A measurand is directly measured byn times, and n observations xi are obtained. In this way, using y0 to represent the best measured value, the error equations are(31)v1=x1−y0v2=x2−y0⋮vn=xn−y0.According to the least squares method, the final measured value is(32)y0=∑i=1nxin.The measurement model of this measurement method isV=X−Y, and vi, xi, and y0 are the samples of random variables V, X, and Y, respectively.Taking the total differentiation of equation (32), the error propagation equation is(33)Δy=∑i=1nΔxin.Now, we only discuss the variances of the error componentsxi−EX and y0−EY and make Δxi=xi−EX and Δy=y0−EY.BecauseΔxi=xi−EX is unknown, is a random variable, and has the same sample space xi−EX as X−EX, Δxi=xi−EX can be represented by Δx=X−EX, that is, Δxi=Δx=X−EX. Similarly, Δy=Y−EY.By applying the law of covariance propagation to formula (33), there is(34)σ2Δy=σ2Δxn.According to the measurement modelV=X−Y, there is(35)Δx=X−EX=V+Y−EV+Y=V+Δy.Therefore, according to the definition of variance, there is(36)σ2Δx=EΔx2=EV+Δy2=EV2+EΔy2=EV2+σ2Δy.SubstitutingEV2≈1/n∑i=1nvi2 and formula (34) into equation (36), we obtain(37)σ2Δx≈∑i=1nvi2n+σ2Δxn.Therefore,(38)σΔx≈∑i=1nvi2n−1.


### 6.2. Indirect Measurement for Single Measurand

Different from the direct measurement, each observationxi in the indirect measurement is the measured data of ai times of measurand. The error equations of the repeated measurement are(39)v1=x1−a1y0v2=x2−a2y0⋮vn=xn−any0.According to the least squares method, the final measured value is(40)y0=∑i=1naixi∑i=1nai2.The measurement model isVi=Xi−aiY, and vi, xi, and y0 are the samples of random variables Vi,Xi, and Y, respectively.Similarly, for the errorsΔx=Xi−EXi and Δy=Y−EY, the covariance propagation relationship is(41)σ2Δy=σ2Δx∑i=1nai2.Similarly, according to the measurement modelVi=Xi−aiY, there is(42)Xi−EXi=Vi+aiY−EVi+aiY=Vi+aiΔy.According to the definition of variance, there is(43)σ2Δx=EΔx2=limn⟶∞1nX1−EX12+X2−EX22+⋯=limn⟶∞1nV1+a1Δy2+V2+a2Δy2+⋯≈1n∑i=1nvi2+∑i=1nai2nσ2Δy=1n∑i=1nvi2+σ2Δxn.Therefore,(44)σΔx≈∑i=1nvi2n−1.


### 6.3. Indirect Measurement for Multiple Measurands

In this measurement mode, there aret different measurands, and each observation value xi is obtained by measuring the linear superposition value of multiple measurands. The error equations of the repeated measurements are(45)v1v2⋮vn=x1x2⋮xn−a11a12⋯a1ta21a22⋯a2t⋮⋮⋮⋮an1an2⋯anty1y2⋮yt.That is,(46)V=X−AY.According to the principle of the least squares, its measured values are(47)Y=ATA−1ATX.The measurement model isVi=Xi−ai1ai2⋯aitY1Y2⋮Yt, and vi, xi, and yj are the samples of random variables Vi,Xi, and Yj, respectively.The error propagation equation is(48)ΔY=ATA−1ATΔX.Similarly, for the errorsΔx=Xi−EXi and Δyj=Yj−EYj, the covariance propagation relationship is(49)DΔY=σ2ΔxATA−1.Similarly, according to the measurement model, there is(50)Xi−EXi=Vi+ai1ai2⋯aitΔy1Δy2⋮Δyt.According to the definition of variance, there is(51)σ2Δx=EΔx2=limn⟶∞1nX1−EX12+X2−EX22+⋯=limn⟶∞1n∑i=1nVi+ai1+ai2+⋯+aitΔy1Δy2⋮Δyt2≈1n∑i=1nvi2+1n∑i=1nΔy1Δy2⋯ΔytATAΔy1Δy2⋮Δyt.Omitting the tedious algebraic calculation process, the final result is(52)σΔx≈∑i=1nvi2n−t.As you can see, the change in Bessel’s formula is thatσx is written as σΔx, and σΔx represents the dispersion of all possible values of the deviation Δxi=xi−EX. Also, the standard deviation σΔy or σΔyj, which is given by formula (34), (41), or (49), is also the evaluation of probability interval of the deviation Δy=y0−EY or Δyj=yj−EYj. Obviously, it is incorrect to express Bessel’s formula as σ2xk≈1/n−1∑j=1nxj−x¯2 in the literatures [1, 4] because xk is a numerical value.For example, the measured frequency values of a quartz crystal at different temperatures are shown in Table4, and the nominal value of frequency is f0=5.000050MHz. Now, we need to give a temperature correction model for the frequency error and evaluate the standard deviation of the residual error after correction.Table 4
Observed values.
Temperature (°C)Frequency (MHz)Error value,Ri=Δfi/f01×10−6−40°4.999900−30−30°4.999975−15−20°5.000040−2−10°5.00008570°5.0001151310°5.0001101220°5.000070430°5.000035−340°5.000010−850°4.999995−1160°4.999995−1170°5.000010−880°5.000045−190°5.00012515100°5.00023537We use the first 4 terms of the Taylor series as the temperature model of the frequency error, that is,R=a+bT+cT2+dT3.In this way, the error equation set is(53)v1v2⋮vn=R1R2⋮Rn−1T1T12T131T2T22T23⋮⋮⋮⋮1TnTn2Tn3abcd.According to the least squares method, there is(54)n∑Ti∑Ti2∑Ti3∑Ti∑Ti2∑Ti3∑Ti4∑Ti2∑Ti3∑Ti4∑Ti5∑Ti3∑Ti4∑Ti5∑Ti6abcd=∑Ri∑RiTi∑RiTi2∑RiTi3.Substituting the values in Table4 into the above equation, there are(55)154504150029250045041500292500025687000041500292500025687000021952500000292500256870000219525000001983295000000abcd=−1461030450042713000.Solving the equations, the following equation is obtained:(56)a=9.983251,b=−0.013518,c=−0.018601,d=0.000214.Therefore, the frequency error’s function model is fitted as(57)R=9.983251−0.013518T−0.018601T2+0.000214T3.Figure5 is the comparison curve between the model and the actual error. According the formula (52), the standard deviation of residual error is(58)σΔR=∑i=1nvi2n−4=±2.3×10−6.Figure 5
The function model fitting of frequency error.Finally, the frequency of quartz crystal is given as follows:(59)f=f01+R×10−6.That is, temperature-frequency error can be corrected by the measured value of a temperature sensor, and a more accurate frequency value can be calculated. Residual error (as shown in Figure6), which is still a regular error, is also processed by statistical rules, and the standard deviation of the residual error is ±2.3 × 10−6. This error processing method has been widely used in the manufacture of photoelectric geodimeter [14, 15].Figure 6
The residual error’s curve.


## 6.1. Direct Measurement for Single Measurand

A measurand is directly measured byn times, and n observations xi are obtained. In this way, using y0 to represent the best measured value, the error equations are(31)v1=x1−y0v2=x2−y0⋮vn=xn−y0.According to the least squares method, the final measured value is(32)y0=∑i=1nxin.The measurement model of this measurement method isV=X−Y, and vi, xi, and y0 are the samples of random variables V, X, and Y, respectively.Taking the total differentiation of equation (32), the error propagation equation is(33)Δy=∑i=1nΔxin.Now, we only discuss the variances of the error componentsxi−EX and y0−EY and make Δxi=xi−EX and Δy=y0−EY.BecauseΔxi=xi−EX is unknown, is a random variable, and has the same sample space xi−EX as X−EX, Δxi=xi−EX can be represented by Δx=X−EX, that is, Δxi=Δx=X−EX. Similarly, Δy=Y−EY.By applying the law of covariance propagation to formula (33), there is(34)σ2Δy=σ2Δxn.According to the measurement modelV=X−Y, there is(35)Δx=X−EX=V+Y−EV+Y=V+Δy.Therefore, according to the definition of variance, there is(36)σ2Δx=EΔx2=EV+Δy2=EV2+EΔy2=EV2+σ2Δy.SubstitutingEV2≈1/n∑i=1nvi2 and formula (34) into equation (36), we obtain(37)σ2Δx≈∑i=1nvi2n+σ2Δxn.Therefore,(38)σΔx≈∑i=1nvi2n−1.


## 6.2. Indirect Measurement for Single Measurand

Different from the direct measurement, each observationxi in the indirect measurement is the measured data of ai times of measurand. The error equations of the repeated measurement are(39)v1=x1−a1y0v2=x2−a2y0⋮vn=xn−any0.According to the least squares method, the final measured value is(40)y0=∑i=1naixi∑i=1nai2.The measurement model isVi=Xi−aiY, and vi, xi, and y0 are the samples of random variables Vi,Xi, and Y, respectively.Similarly, for the errorsΔx=Xi−EXi and Δy=Y−EY, the covariance propagation relationship is(41)σ2Δy=σ2Δx∑i=1nai2.Similarly, according to the measurement modelVi=Xi−aiY, there is(42)Xi−EXi=Vi+aiY−EVi+aiY=Vi+aiΔy.According to the definition of variance, there is(43)σ2Δx=EΔx2=limn⟶∞1nX1−EX12+X2−EX22+⋯=limn⟶∞1nV1+a1Δy2+V2+a2Δy2+⋯≈1n∑i=1nvi2+∑i=1nai2nσ2Δy=1n∑i=1nvi2+σ2Δxn.Therefore,(44)σΔx≈∑i=1nvi2n−1.


## 6.3. Indirect Measurement for Multiple Measurands

In this measurement mode, there aret different measurands, and each observation value xi is obtained by measuring the linear superposition value of multiple measurands. The error equations of the repeated measurements are(45)v1v2⋮vn=x1x2⋮xn−a11a12⋯a1ta21a22⋯a2t⋮⋮⋮⋮an1an2⋯anty1y2⋮yt.That is,(46)V=X−AY.According to the principle of the least squares, its measured values are(47)Y=ATA−1ATX.The measurement model isVi=Xi−ai1ai2⋯aitY1Y2⋮Yt, and vi, xi, and yj are the samples of random variables Vi,Xi, and Yj, respectively.The error propagation equation is(48)ΔY=ATA−1ATΔX.Similarly, for the errorsΔx=Xi−EXi and Δyj=Yj−EYj, the covariance propagation relationship is(49)DΔY=σ2ΔxATA−1.Similarly, according to the measurement model, there is(50)Xi−EXi=Vi+ai1ai2⋯aitΔy1Δy2⋮Δyt.According to the definition of variance, there is(51)σ2Δx=EΔx2=limn⟶∞1nX1−EX12+X2−EX22+⋯=limn⟶∞1n∑i=1nVi+ai1+ai2+⋯+aitΔy1Δy2⋮Δyt2≈1n∑i=1nvi2+1n∑i=1nΔy1Δy2⋯ΔytATAΔy1Δy2⋮Δyt.Omitting the tedious algebraic calculation process, the final result is(52)σΔx≈∑i=1nvi2n−t.As you can see, the change in Bessel’s formula is thatσx is written as σΔx, and σΔx represents the dispersion of all possible values of the deviation Δxi=xi−EX. Also, the standard deviation σΔy or σΔyj, which is given by formula (34), (41), or (49), is also the evaluation of probability interval of the deviation Δy=y0−EY or Δyj=yj−EYj. Obviously, it is incorrect to express Bessel’s formula as σ2xk≈1/n−1∑j=1nxj−x¯2 in the literatures [1, 4] because xk is a numerical value.For example, the measured frequency values of a quartz crystal at different temperatures are shown in Table4, and the nominal value of frequency is f0=5.000050MHz. Now, we need to give a temperature correction model for the frequency error and evaluate the standard deviation of the residual error after correction.Table 4
Observed values.
Temperature (°C)Frequency (MHz)Error value,Ri=Δfi/f01×10−6−40°4.999900−30−30°4.999975−15−20°5.000040−2−10°5.00008570°5.0001151310°5.0001101220°5.000070430°5.000035−340°5.000010−850°4.999995−1160°4.999995−1170°5.000010−880°5.000045−190°5.00012515100°5.00023537We use the first 4 terms of the Taylor series as the temperature model of the frequency error, that is,R=a+bT+cT2+dT3.In this way, the error equation set is(53)v1v2⋮vn=R1R2⋮Rn−1T1T12T131T2T22T23⋮⋮⋮⋮1TnTn2Tn3abcd.According to the least squares method, there is(54)n∑Ti∑Ti2∑Ti3∑Ti∑Ti2∑Ti3∑Ti4∑Ti2∑Ti3∑Ti4∑Ti5∑Ti3∑Ti4∑Ti5∑Ti6abcd=∑Ri∑RiTi∑RiTi2∑RiTi3.Substituting the values in Table4 into the above equation, there are(55)154504150029250045041500292500025687000041500292500025687000021952500000292500256870000219525000001983295000000abcd=−1461030450042713000.Solving the equations, the following equation is obtained:(56)a=9.983251,b=−0.013518,c=−0.018601,d=0.000214.Therefore, the frequency error’s function model is fitted as(57)R=9.983251−0.013518T−0.018601T2+0.000214T3.Figure5 is the comparison curve between the model and the actual error. According the formula (52), the standard deviation of residual error is(58)σΔR=∑i=1nvi2n−4=±2.3×10−6.Figure 5
The function model fitting of frequency error.Finally, the frequency of quartz crystal is given as follows:(59)f=f01+R×10−6.That is, temperature-frequency error can be corrected by the measured value of a temperature sensor, and a more accurate frequency value can be calculated. Residual error (as shown in Figure6), which is still a regular error, is also processed by statistical rules, and the standard deviation of the residual error is ±2.3 × 10−6. This error processing method has been widely used in the manufacture of photoelectric geodimeter [14, 15].Figure 6
The residual error’s curve.


## 7. Uncertainty

According to Figure2, the total error of the final measured value is(60)Δ=ΔA+ΔB,where ΔA is the deviation between measured value and expectation and ΔB is the deviation between expectation and true value.Because the two errors are usually irrelevant, according to the law of covariance propagation (29), there is(61)σΔ=σ2ΔA+σ2ΔB.This total standard deviationσΔ is the evaluation of probability interval of total error Δ (the dispersion of all possible values of total error Δ). It can be seen that formula (61) is consistent with the traditional uncertainty evaluation (but the expression is changed from σx to σΔ). Therefore, this total standard deviation σΔ is actually the uncertainty, which expresses the probability range of the total error of final measured value. And the uncertainty concept, which is interpreted as the dispersion of the measured value (constant) in the traditional measurement theory, is also proved to be incorrect.It can be seen from Table3 that the uncertainty is also the possible degree that the true value deviates from the measured value. That is, the uncertainty is not only the uncertainty of the error but also the uncertainty of the true value but is not the uncertainty of the measured value. The measured value, which is a certain value, has no uncertainty.According to the interpretation of the traditional theory,σΔA and σΔB are referred as the uncertainty of Type A and the uncertainty of Type B, respectively. However, the current σΔB is actually σΔ of historical upstream measurement, and the current σΔ can also be used as σΔB in future downstream measurement. This kind of interpretation with A/B classification of the uncertainty evaluation is obviously too rigid.Moreover, currently widely used formula (61) is only applicable to the direct repeated measurement model in Section 6.1 but has no use at all for the indirect repeated measurement in Sections 6.2 and 6.3 because in indirect repeated measurement, there are usually some error sources which not only contribute to dispersion of repeated observations but also contribute to their deviation, and it is difficult to distinguish them with A/B classification method. Therefore, the A/B classification method is not universal in practice.Formula (61) comes from the covariance propagation law (29). Thus, the basic principle of uncertainty synthesis is covariance propagation law (29), and the uncertainty synthesis does not need to apply the interpretation of A/B classification mechanically. Here is a simple example to illustrate this principle, which is also a comparison with the traditional practice.For example, four points A, B, C, and D are located on a straight line (Figure7), and the observation data of distances obtained by geodimeter [14, 15] are shown in Table 5. Please solve the final measured values of each line segment and the uncertainty of each error.Figure 7
Distance measurement.Table 5
Observed values.
Line segmentObserved values1ABx1=39.8538m2BCx2=159.957m3CDx3=320.0015m4ACx4=199.8117m5BDx5=479.9601m6ADx6=519.8149mUsingy1, y2, and y3 to express the final measured values of AB, BC, and CD, respectively, and using k to express the measured value of the additive constant error of geodimeter, the observation error equation is(62)v1v2v3v4v5v6=x1x2x3x4x5x6−100101010011110101111111y1y2y3k.According to the least squares method, there are(63)y1y2y3k=100101010111001011111111100101010111001011111111T−1100101010111001011111111x1x2⋮x6=14×1−2−11−12−21−2111−1−21−11222200−2x1x2⋮x6.Substituting the numerical values of all observed values into equation (63), we obtain(64)y1y2y3k=39.8549159.9583320.0030−0.0013m.For the traditional measurement theory, the next step is to substitute (64) into (62), and six residual vi are obtained. Then σx is obtained by using the Bessel formula σx=∑i=1nvi2/n−t, and σy1, σy2, σy3, and σk are obtained by using the covariance propagation law. Finally, σy1, σy2, σy3, and σk are called as precision or uncertainty of Type A, but the uncertainty of Type B is almost impossible to discuss.However, from the perspective of the new conceptual theory, there are three conceptual troubles in the above variance submission process: (1) The degree of the freedomn−t is too small, so it is meaningless to apply the Bessel formula. (2) In Table 5, each observed value xi is a numerical value, and according to equation (64), each measured value yj is also a numerical value, so their variances should be 0. (3) The contribution of the covariance between the errors of each observation value xi has not been taken into account at all (uncertainty synthesis issue).The following is the variance submission process of the new conceptual theory for this case.Taking the total differential of equation (63), the error propagation equation is obtained as follows:(65)Δy1Δy2Δy3Δk=14×1−2−11−12−21−2111−1−21−11222200−2Δx1Δx2⋮Δx6.Applying covariance propagation law (29) to equation (65), the covariance propagation equation is obtained as follows:(66)σΔy12σΔy1Δy2σΔy1Δy3σΔy1ΔkσΔy2Δy1σΔy22σΔy2Δy3σΔy2ΔkσΔy3Δy1σΔy3Δy2σΔy32σΔy3ΔkσΔkΔy1σΔkΔy2σΔkΔy3σΔk2=116×1−2−11−12−21−2111−1−21−11222200−2DΔX1−2−12−21−22−1−21211−10−1110212−2.The acquisition process of the covariance matrixDΔX is as follows.For the observed valuexi, its error Δxi is composed of three parts: additive constant error K, multiplication constant error R, and uneven indexing error ci. That is,(67)Δxi=K+R⋅xi+ci.Its variance is(68)σΔxi2=σK2+xi2⋅σR2+σc2,where σK, σR, and σc are obtained by consulting instrument instructions or the tolerance standard in instrument specification. Furthermore, according to formula (21), DΔX can be deduced as follows:(69)DΔX=EΔx1Δx2⋮Δx6Δx1Δx2⋯Δx6=σK2+x12⋅σR2+σc2σK2+x2x1⋅σR2⋯σK2+x6x1⋅σR2σK2+x1x2⋅σR2σK2+x22⋅σR2+σc2⋯σK2+x6x2⋅σR2⋮⋮⋮⋮σK2+x1x6⋅σR2σK2+x2x6⋅σR2⋯σK2+x62⋅σR2+σc2.Finally, the covariance matrixDΔY is obtained by substituting equation (69) into equation (66), where σΔyj is called as uncertainty.Assuming thatσK=±2mm, σR=±1×10−6, and σc=±1mm, we can obtain(70)σΔy12σΔy1Δy2σΔy1Δy3σΔy1ΔkσΔy2Δy1σΔy22σΔy2Δy3σΔy2ΔkσΔy3Δy1σΔy3Δy2σΔy32σΔy3ΔkσΔkΔy1σΔkΔy2σΔkΔy3σΔk2=0.750.010.26−0.500.010.780.05−0.500.260.050.85−0.50−0.50−0.50−0.505.00mm2.Therefore, the uncertainties are(71)σΔy1≈±0.9mm,σΔy2≈±0.9mm,σΔy3≈±0.9mm,σΔk≈±2.2mm.It can be seen thatσΔyj<σΔxi. Besides, it can be seen that the indication errors Δxi not only lead to the dispersion of repeated observations x1∼x6, but also lead to their overall deviation. If we entangle in the influence characteristics (A/B classification) of indication error on repeated observations, it will not only be unable to express, but also will not help to solve the problem. Moreover, whether errors K,Rxi,ci,andΔxi or error Δyj are all deviations and have variances used to evaluate their probability intervals, the systematic error without variance does not exist.


## 8. Conclusions

In short, different from the traditional theory, the new theory follows rigorous mathematical concepts and regards both the observed value and measured value as constants and the error and true value as random variables so that the conceptual logic of new theory has changed in an overall way (shown in Table6). First, the variance (standard deviation) or the uncertainty is the evaluation of probability interval of a single error (deviation), but not the dispersion of a measured value, and any regular error’s size degree can be evaluated by them. Second, any error follows a random distribution, has variance which can be used to evaluate its size, and cannot be classified as the systematic error or the random error. Third, the error synthesis follows the algebraic rule, the variance synthesis follows the probability principle, and there is no need to use those old concepts such as systematic error, random error, precision, trueness, and accuracy. As a result, the revision of measurement textbooks and metrological terminology will become issues that need to be rediscussed in the future.Table 6
Conceptual logic difference between the two theories.
Traditional measurement theoryNew conceptual measurement theoryThe measured value is a random variable because it changes randomly in repeated measurementsThe measured value is a numeric value and is a constantThe true value is a constant because it remains constant during repeated measurementsThe true value is unknown and is a random variable that needs to be described by a probability rangeThe best measured value is given by random error analysisThe best measured value is given by analyzing the randomness of errorsWith the true value as the reference center, submit the evaluation of the deviation and dispersion (reproducibility) of the measured valueWith the measured value as the reference center, submit the evaluation value of the probability range that the true value deviates from the measured valueThe errors are divided into systematic and random classificationsThe errors are not divided into systematic and random classifications



---

*Source: 3864578-2020-09-17.xml*