Uncertainty means thatmeasurement errorThe degree of uncertainty about the measured value.Conversely, it also indicates the credibility of the results.It is a measurementResult qualityIndicators of.The smaller the uncertainty is, the higher the quality is, and the higher the level isuse valueHigher;The greater the uncertainty, the lower the quality and level of the measurement results, and the lower their use value.When reporting the results of physical quantity measurement, the corresponding uncertainty must be given, which is convenient for users to evaluate its reliability on the one hand, and also enhances themeasurement result BetweenComparability。
4. Generally, for a given set of information, the measurement uncertainty is corresponding to the givenMeasuredOf the value of.The change of this value will lead to the change of the corresponding uncertainty.[1]
effect
Measurement uncertaintyYes Forerror analysisThe latest understanding and explanation in, previously usedmeasurement errorBut they have completely different meanings.More accurately defined as measurement uncertainty.It indicates the degree of uncertainty of the measured value due to the existence of measurement error.
calculation
The value of uncertainty is the sum of each value andaverage valueThe maximum distance between.
Example: There is a column of numbers.A1,A2, ... , An，Their average value is A, then the uncertainty is:max{ |A - Ai|, i = 1, 2, ..., n}[2]
conceptual analysis
Announce
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Conceptual differences
Uncertainty and error
Statisticians and surveyors have been looking for appropriate terms to express correctlymeasurement result Reliability.For example, the commonly usedAccidental error, due to the imprecise expression of the word "accidental"random errorReplaced by.The meaning of the word "error" is vague, for example, "error is ± 1%", which makes people feel unclear.However, the meaning of "uncertainty is 1%" is clear.Therefore, the random error andsystematic error。Measurement uncertaintyAndmeasurement errorIt is a completely different concept. It is neither error nor equal to error.
Uncertainty and error
The error indicates that the deviation of the measurement result from the true value is a point, and the measurement uncertainty indicatesMeasured valueThe dispersion ofNumber axisThe upper represents an interval.
According to the information used, the non negative parameter that characterizes the dispersion of the measured value is called measurement uncertainty.This is the latest definition in General Measurement Terms and Definitions (JJF 1001-2011).Measurement uncertainty is an independent and closely related parameter that indicates the dispersion of measurement results.The measurement uncertainty should be included in the complete representation of the measurement.For measuring uncertaintystandard deviation When expressed, it is called standard uncertainty, and if the half width of the interval that describes the confidence level is used, it is called expanded uncertainty.
Classification difference
Class A and Class B evaluation and combination of uncertainty
Since the uncertainty of measurement results is often caused by a variety of reasonsstandard deviation , which is called standard uncertainty component and is represented by the symbol ui.
Use to performstatistical analysisThe method of evaluating standard uncertainty is called uncertainty A evaluation;The corresponding standard uncertainty obtained is called Type A uncertainty component, which is represented by the symbol uA.It is usedExperimental standard deviationTo characterize.
The standard uncertainty is evaluated by a method different from the statistical analysis of the observation column, which is called uncertainty B evaluation;The corresponding standard uncertainty obtained is called Type B uncertainty component, which is represented by the symbol uB.It is estimated by experiment or other information, and contains subjective identification components.For a certain uncertainty component, whether to use Class A method or Class B method should be selected by the surveyor according to the specific situation.Class B assessment methods are widely used.
When the measurement result is obtained from the value of several other quantities, the standard uncertainty calculated according to the variance and covariance of other quantities is called the composite standard uncertainty.It is the measurement resultstandard deviation OfEstimated value, denoted by the symbol uc.The variance is the square of the standard deviation,covarianceyesrelevanceThe resulting variance.Including covariance will expand the uncertainty of the composite standard.The combined standard uncertainty is still the standard deviation, which represents the dispersion of measurement results.The synthesis method used is often called the uncertainty propagation rate, and the propagation coefficient is also called the sensitivity coefficient, which is represented by Ci.The degree of freedom of the combined standard uncertainty is called the effective degree of freedom, which is expressed by uc and indicates the evaluated reliability.
Expanded uncertainty
The expanded uncertainty is the quantity that determines the measurement result range, and most of the value distribution reasonably assigned to the measured value can be expected to be included in this range.It is sometimes referred to as the range uncertainty.The expanded uncertainty is expressed as a multiple of the combined standard uncertaintyMeasurement uncertainty。It is usually represented by the symbol U: combined uncertainty andInclusion factorThe product of k is called the total uncertainty (U).Here, the value of k is generally 2, and sometimes 3.It depends on the measured importance, benefits and risks.The expanded uncertainty is the half width of the value interval of the measurement result, which can be expected to cover most of the measured value distribution.The value range of the measurement result is the measured valueprobability distributionIncluded inpercentage, calledConfidence probability, confidence level orconfidence level , denoted by.At this time, the expanded uncertainty is represented by the symbol U, which gives that the interval can contain most of the possible values to be measured (such as 95% or 99%).
The coverage factor is the numerical factor multiplied by the combined standard uncertainty in order to obtain the expanded uncertainty, sometimes called the coverage factor.The value of the coverage factor determines the confidence level of the expanded uncertainty.When=2, p=95%; when=3, p=99%.
Principle 2: The average number of digits is allowed, but only one digit can be reserved according to Principle 1. At this time, the uncertainty should be rounded off, and the number of digits of the average value should be determined again
Carry Principle 1: Only one significant number is reserved. If the second significant number is not 0, carry it;
According to the mean value, the uncertainty of rounding off is not required, and it should be written as
According to the uncertainty of mean rounding, it is found that rounding is required, and it should be written as
Carry principle 2: according to principle 3Two significant digits are reserved, and carry is also required if the third significant digit is not 0.
for example
First get 0.22 according to carry principle 2, and then re determine according to principle 2average, Last
Principle 3: Sometimes two digits can be reserved, this is because: the first significant digit of the uncertainty of 1 is less than 3;2 The number of digits of the average value is allowed.
For example:
Here: the first significant digit of the uncertainty of 1 is less than 3;2 The average value is accurate to 0.01, just allowing the uncertainty to remain 2 digits.Consider the carry principle 2, and finally write
Inconformity with Principle 1 - Digits are not allowed, Inconformity with Principle 3 - Carry Principle 1
Solution uncertainty
stayGB/TAppendix B of 601-2002 D definesTitrimetric analysisstandard solution The calculation method of uncertainty of.Namely, there are generally four calibration methods for standard titration solution:
(1) Working withReference reagentCalibrate the concentration of standard titration solution;
(2) Titrate with standardSolution calibrationConcentration of standard titration solution;
(3) Dissolve the working reference reagentConstant volumeCalibrate the concentration of standard titration solution after measuring;
(4) Standard titration solution directly prepared with working reference reagent.
C1 -- accurate value of concentration of standard titration solution, mol/L;
V -- Volume of calibrated standard titration solution, mL.
The third way
Including: EDTA disodium standard titration solution [c (EDTA)=0.02mol/L], calculate the concentration value of standard titration solution (mol/L), and express it as (3-15):
Including: potassium dichromate, potassium iodate and sodium chloride.Calculate the concentration value of standard titration solution (mol/L) as (3-16):
c=mw*1000/VM （3—16）
Where: m - accurate value of the quality of working reference reagent, g;
W - mass fraction of working reference reagent,%;
V -- volume of standard titration solution, mL;
M -- molar mass value of working reference reagent, g/mol.
Where: uA -- Class A standard uncertainty component of the average concentration of standard titration solution, mol/L;
UB -- Type B synthetic standard uncertainty component of the average concentration of standard titration solution, mol/L.
Mode (2)
workReference reagentCalibrate the concentration of standard titration solution (the first way)average valueCalculation of uncertainty.
Since there are four calibration methods for standard titration solution, the calculation of uncertainty is also divided into four types.
There are two methods to calculate the Class A uncertainty of the average concentration of standard titration solution.
a. Class A of average concentration of standard titration solutionrelative standard The estimation of the uncertainty component uArel (c -) is calculated according to Formula (3-19):
C - Eight for twoParallel determinationThe average concentration of standard titration solution, mol/L.
In formula (3-19):
σ(c) =[CrR95(8)]/[f(n)] （3—20）
In the formula: CrR95 (8) - two people eight measured in parallelRepeatabilityCritical difference, mol/L;
F (n) - Critical range coefficient (according to Table 1 in GB/T 11792-1989).
a. Concentration of standard titration solutionaverage valueCalculation of Class A relative standard uncertainty components of.
The experiment of calculating two people eight parallel measurement by bessel methodstandard deviationThen, calibrate the component of Type A relative standard uncertainty of the average concentration of titration solution, and calculate according to Formula (3-21):
uArel(c-)=[s(c)]/[(8^1/2)*c-]（3—21）
Where: s (c) - the experimental standard deviation of the parallel determination results of two people and eight people, mol/L;
C - the average concentration of standard titration solution measured in parallel by two people, mol/L.
Mode (3)
The calculation of Class B relative synthetic standard uncertainty component of the average concentration of standard titration solution, taking electronic balance weighing as an example, calculates the uncertainty.According to Formula (3-13), Type B relative synthetic standard uncertainty component of the average concentration of standard titration solution.
Of working reference reagentmass fraction The relative standard uncertainty component of the value of is calculated according to Formula (3-25):
urel(w)=[{u^2(w)+u^2(wr)]^1/2}/w（3—25）
Where: u (w) - standard uncertainty component of the mass fraction of working reference reagent,%;
U (wr) - standard uncertainty component of the numerical range of the mass fraction of the working reference reagent（Reference materialsDoes not include this item),%;
W - mass fraction of working reference reagent,%;
In formula (3-25):
u(w)U/k（3—26）
Where: U - expanded uncertainty (total uncertainty) of the mass fraction of the working reference reagent,%;
K -- Inclusion factor (generally, k=2)
In formula (3-25):
(u (wr)=a/k evenly distributed, k=3 ^ 1/3) (3-27)
Where: a - half width of the numerical range of the mass fraction of the working reference reagent,%.
The relative standard uncertainty component of the volume value of the solution to be calibrated shall be calculated according to Formula (3-28):
urel(V1-V2)={[U^2(v1)+U^2(v2)]^1/2}/(v1-v2)（3—28）
Where: U (v1) - standard uncertainty component of the volume value of the calibrated solution, mL;
u(V2)——Blank experimentStandard uncertainty component of the volume value of the calibrated solution, mL;
V1-v2 -- the volume of the calibrated solution actually consumed, mL.
After necessaryellipsis, the relative standard uncertainty component of the volume value of the calibrated solution is calculated according to Formula (3-29):
Where: U1 (v) - correction of weighing waterburetteStandard uncertainty component introduced in volume, mL;
U2 (v) - byInterpolation methodDetermine the volume of the solution to be calibratedCorrection valueStandard uncertainty component introduced at, mL;
U3 (V) - standard uncertainty component introduced by the rounding error of the calibrated solution volume correction value, mL;
U4 (V) - standard uncertainty component introduced by rounding error of temperature correction value, mL;
V1 -- Volume value of the calibrated solution, mL;
V2 -- Volume value of calibrated solution in blank test, mL
The standard uncertainty component introduced in the calibration of buret volume by weighing water isJJG196-1990.measurestayStandard temperatureThe actual volume value (V20) at 20 ℃, in milliliter (mL), is calculated according to Formula (3-30):
m——Measurement methodUse the table to find the value of pure water quality, g;
ρ W - density value of pure water at ℃, g/mL.
Then the volume correction value of the calibrated solution shall be:
V=(M0-M)/ρW （3—31）
Therefore, the component of relative standard uncertainty introduced when measuring water to correct the buret volume is calculated according to Formula (3-32):
Where: urel (m0-m) - the relative standard uncertainty component of the difference between the value of pure water quality weighed and the value of pure water quality found in the measurement table;
Urel (ρ w) - relative standard uncertainty component introduced by pure water density value.
Among them, it is a certain capacity, temperatureAir density, GlassCoefficient of volume expansionThe quality of pure water is regarded as the true value, and its standard uncertainty component is zero, but there is pure water qualityRounding off of valuesThe standard uncertainty component introduced.
In formula (3-32):
ur(m0-m)={[u^2(m0)+u^2(m)]^1/2}/(m0-m)（3—33）
Where: u (m0) - standard uncertainty component of the value of weighing pure water quality, g;
U (m) - standard uncertainty component of the value of pure water quality found in the table of measurement method, g;
M0 -- the value of the weight of pure water, g;
M -- the value of pure water quality obtained by the measurement method in the table, g.
In formula (3-33): (uniformly distributed, k=3 ^ 1/3) (3-34)
Where: a - maximum allowable error of electronic balance, g.
In formula (3-33): (u (m)=a/k uniformly distributed, k=3 ^ 1/3) (3-35)
Where: a - half width of the rounding error interval of the pure water quality value obtained by the measurement method in the table, g.
In equation (3-32): ur (ρ w)=u (ρ w)/ρ w (3-36)
Where: u (ρ w) - standard uncertainty component introduced by pure water density value, g/mL;
ρ w - density value of pure water at ℃, g/mL.
In formula (3-36): (u (ρ w)=a/k uniformly distributed, k=3 ^ 1/3) (3-37)
Where: a -- half width of rounding error interval of pure water density value, g/mL.
Substitute and into (3-32).Then the standard uncertainty component introduced when the buret volume is calibrated by weighing water is calculated according to Formula (3-38):
u2(V)=(m0-m)u1r(V)/ρw（3—38）
fromInterpolation methodThe standard uncertainty component introduced when determining the volume correction value of the calibrated solution is expressed in milliliters (mL) and calculated according to Formula (3-39):
(u2 (V)=a/k evenly distributed, k=6 ^ 1/3) (3-39)
Where: a - the value greater than the volume of the calibrated solution andlight rainHalf of the difference between the calibration values of the two calibration points of the volume of the calibrated solution, mL.
The standard uncertainty component introduced by the rounding error of the volume correction value of the calibrated solution, expressed in milliliters (mL), is calculated according to Formula (3-4 0):
Where: a - half width of the rounding error interval of the buret correction value, mL.
The standard uncertainty component introduced by the rounding error of the temperature correction value is expressed in milliliters (mL) and calculated according to Formula (3-41):
(uniformly distributed, k=3 ^ 1/3) (3-41)
Where: a - half width of rounding error interval of temperature correction value, mL/L;
V1 -- Volume value of the calibrated solution, mL.
Substitute the above u1, u2, u3 and u4 into Formula (3-29) to obtain the relative standard uncertainty component of the volume value of the calibrated solution.
workReference reagentThe relative standard uncertainty component of the molar mass value is calculated according to Formula (3-4 2):
ur(M)=u(M)/M（3—42）
Where: u (M) - standard uncertainty component of molar mass value of working reference reagent, g/moL;
M -- molar mass value of working reference reagent, g/moL.
In formula (3-42):
（3—43）
In the formula: u (M1) -Relative atomic massThe standard uncertainty component introduced by the standard uncertainty of the value of, g/moL;
U (M2) - standard uncertainty component introduced by rounding off error of molar mass value of working reference reagent, g/moL.
In formula (3-43):
（3—44）
Where: qi -- workReference reagentThe number of an element in the molecule;
UA1 - standard uncertainty of the value of the relative atomic mass of an element in the working reference reagent molecule, g/moL;
N -- the number of elements in the working reference reagent molecule.
In formula (3-43):
(uniformly distributed, k=3 ^ 1/3) (3-45)
Where: a - half width of the rounding error interval of the molar mass value of the working reference reagent, g/moL.
The component of the relative standard uncertainty introduced by the rounding error of the average concentration of the standard titration solution measured in parallel by two people is calculated according to Formula (3-46):
(uniformly distributed, k=3 ^ 1/3) (3-46)
Where: a -- half width of the rounding error interval of the average concentration of standard titration solution measured in parallel by two people, mol/L;
C - the average concentration of standard titration solution measured in parallel by two people, mol/L.
Substitute Formula (3-22) to obtain the component of Type B synthetic relative standard uncertainty of the average concentration of standard titration solution.
The concentration of standard titration solution obtained from (1) and (2) respectivelyaverage valueThe relative standard uncertainty components of Class A and Class B ofClass B standard uncertaintyThe component sum is then substituted into Formula (3-18) to obtain the synthetic standard uncertainty of the average concentration of the standard titration solution, and the expanded uncertainty (synthetic standard uncertainty) of the average concentration of the standard titration solution can be obtained by substituting into Formula (3-17)
Mode (4)
Example of expression of expanded uncertainty of average concentration of standard titration solution (according to JJF 1059-1999):
The synthetic standard uncertainty of the average concentration of standard titration solution=5.6 10-5 mol/L, the inclusion factor=2, and the expanded uncertainty of the average concentration of standard titration solution (mol/L)=25.6 10-5 mol/L=0.000112 mol/L.
In the form of concentration value:
① 000 moL/L， =0.0002 moL/L； =2。
② （0.1000±0.0002）moL/L； =2。
It is expressed in the relative form of concentration value as:
① 000（1±2 10-3）moL/L； =2 10-4； =2。
② 000 moL/L； =2 10-4； =2。
Choose one of the above four representations.
Concentration of standard titration solutionaverage valueNot included in the calculation of uncertainty ofEnd point errorThe relative standard uncertainty component introduced.Users can pressPrinciples of Analytical Chemistry, calculate the relative standard uncertainty component introduced by the end point error.
Other methods
Calculation of uncertainty in other three ways
With reference to the calculation of the uncertainty of the average concentration of standard titration solution in the first way, the uncertainty of the average concentration of standard titration solution in the second way, the third way and the fourth way can be calculated.
GB50325-2001: In indoor airFormaldehyde detection, usingOn site inspection methodThe uncertainty of measurement results within the measurement range of 0~0.6 mg/m3 should be less than or equal to 25%.
GB 50325-2001: For the detection of radon in indoor air, the uncertainty of the measurement results of the selected method should not exceed 25%（Confidence95%）。
GB 6566-2001 Requirements for Uncertainty in Measurement: When Radium-266, Thorium-232, and Potassium - 40 in the sample radiateSpecific activityWhen the sum is greater than 37Bq/kg, the test method requirements of this standardMeasurement uncertainty(Expansion factor=1) not more than 20%.
