Introduction
Given tutorial contains short theoretical information in the main sections of metrology: international system of units, errors of results and measuring instruments, random errors and processing of measurement results, estimation of the error of indirect measurements, methods for standardizing the errors of measuring instruments.
The basic definitions and formulas necessary to solve problems are given. Typical problems are provided with explanations and detailed solutions; the rest of the problems are provided with answers to check the correctness of the solution. All physical quantities are specified in the International System of Units (SI).
When solving problems, it is necessary to write out formulas in literal terms, substitute numerical values into them, and after calculations, provide the final result indicating the error and units of measurement.
The training manual is intended to conduct practical classes in the course “Metrology” and other disciplines containing sections of metrological support.
1. International System of Units (SI)
1.1. Basic information
On January 1, 1982, GOST 8.417-81 “GSI. Units physical quantities", in accordance with which the transition to the International System of Units (SI) was carried out in all fields of science, technology, National economy, as well as in educational process in all educational institutions.
The International SI System contains seven basic units for measuring the following quantities:
Length: meter (m),
Weight: kilogram (kg),
Time: second (s),
Electric current strength: ampere (A),
Thermodynamic temperature: kelvin (K),
Luminous intensity: candela (cd),
Amount of substance: mole (mol).
Derived units of the SI system (more than 130 in number) are formed using the simplest equations between quantities (defining equations), in which the numerical coefficients are equal to one. Along with basic and derived units, the SI system allows the use of decimal multiples and submultiples, formed by multiplying the original SI units by the number 10 n, where n can be a positive or negative integer.
1.2. Problems and examples
1.2.1. How will the unit of electrical voltage (volt, V) be expressed in terms of the SI base units?
Solution. Let us use the following equation for voltage, where R- power released in a section of a circuit when current flows through it I. Therefore, 1 V is the electrical voltage that causes electrical circuit direct current of 1 A with a power of 1 W. Further transformations:
Thus, we obtain a relationship in which all quantities are expressed through the basic units of the SI system. Hence, .
1.2.2. How is the unit of electrical capacitance (farad, F) expressed in terms of the SI base units?
Answer: p>
1.2.3. How is the unit of electrical conductivity (Siemens, cm) expressed in terms of the SI base units?
1.2.4. How is the unit of electrical resistivity () expressed in terms of the SI base units?
1.2.5. How is a unit of measurement expressed? electrical inductance(Henry, Gn) through SI base units?
where is the residual error.
Mean square error of the arithmetic mean
Estimates , , are called point estimates.
In practice, interval estimates are usually used in the form of confidence probability and confidence limits of error (confidence interval). For the normal law, the confidence probability P(t) determined using the probability integral Ф(t)(4.11) (function tabulated)
where is the multiplicity of the random error, and is the confidence interval.
Knowing the confidence limits and , we can determine the confidence probability
If the confidence limits are symmetrical, i.e. , then and .
For a small number of measurements in the series (), the Student distribution is used.
The probability density depends on the value of the random error and the number of measurements in the series n, i.e. . Trust boundaries E in this case are determined
where is the Student coefficient (determined from Table III of the Appendix).
The confidence limit and confidence probability also depend on the number of measurements.
4.1.5. When statistically processing observation results, the following operations are performed.
1. Elimination of systematic errors, introduction of amendments.
2. Calculation of the arithmetic mean of the corrected observation results, which is taken as an estimate of the true value of the measured quantity (formula 4.8).
3. Calculation of the assessment of the SKP measurements () and the arithmetic mean measurement () (formulas 4.9, 4.10).
4. Testing the hypothesis about the normal distribution of observation results.
5. Calculation of the confidence limits of the random error of the measurement result with a confidence probability of 0.95 or 0.99 (formula 4.14).
6. Determination of the limits of the non-excluded systematic error of the measurement result.
7. Calculation of confidence limits for the error of the measurement result.
8. Recording the measurement result.
4.1.6. The hypothesis about the normality of the distribution is tested using the (Pearson) or (Von Mises-Smirnov) criterion, if ; according to the composite criterion, if . When the normality of the distribution is not checked.
If the observation results are normally distributed, then the presence of misses is determined. Table IV of the Appendix shows the limit values of the coefficient for different meanings the theoretical probability of a large error occurring, usually called the significance level, given a certain sample size. The procedure for detecting misses is as follows. A variation series is constructed from the observation results. The arithmetic mean of the sample () and the UPC of the sample () are determined. Then the coefficients are calculated
The obtained values are compared with for a given significance level q for a given sample size. If or , then this result is a miss and must be discarded.
4.1.7. Checking the agreement of the experimental distribution to normal using a composite criterion is carried out as follows. Selecting the significance level q ranging from 0.02 to 0.1.
Criterion 1. A comparison is made of the value calculated from experimental data d with theoretical distribution points and (shown in Appendix Table V) and corresponding to the normal distribution law at a given significance level q 1 criterion 1.
Calculation of value d produced according to the formula:
The hypothesis that a given series of observation results belongs to the normal distribution law is correct if the calculated value d lies within
Criterion 2. Assessment according to criterion 2 is to determine the number of deviations m e experimental values t e i from theoretical value t t for a given significance level q 2. To do this, given q 2 and n the parameter is found according to data from table VI of the appendix.
parameter according to formula (4.18)
The calculated value is compared with the theoretical value and the number of deviations for which the inequality is satisfied is calculated. The value is compared with the theoretical number of deviations, which is found from Table VI of the Appendix. If , then the distribution of this series of observations does not contradict the normal one.
If both criteria are met, then this series is subject to a normal distribution. In this case, the significance level of the composite criterion is assumed to be equal to .
4.1.8. The limits of the non-excluded systematic error are determined using the formula:
where is the border i th non-excluded systematic error; - coefficient determined by the accepted confidence probability; at R = 0,95 = 1,1.
As the limits of non-excluded systematic error, we can take the limits of permissible main and additional errors of measuring instruments.
4.1.9. When calculating the confidence limit of the error of the result, the ratio is determined. If , then we neglect the random error and assume that . If , then the error limit is found by summing random and non-excluded systematic errors, considered as random variables:
Where TO- coefficient depending on the ratio of random and non-excluded systematic error;
Estimation of the SKP of the arithmetic mean.
The limits of random and systematic errors must be chosen at the same confidence level.
4.1.10. The measurement result is written in the form .
4.2. Problems and examples
4.2.1. The error in the voltage measurement result is distributed evenly in the range from V to V.
Find the systematic error of the measurement result, the mean square error and the probability that the error of the measurement result lies in the range from B to B (Fig. 4.1).
Solution. The systematic error is equal to the mathematical expectation, which for a uniform distribution law is determined by formulas (4.1, 4.5).
The root mean square error is determined by formulas (4.2, 4.3, 4.5).
The probability of an error falling into specified interval is determined from relation (4.4).
where is the height of the distribution law.
Hence, .
4.2.2. The error in the current measurement result is distributed evenly with the parameters mA, mA. Determine the limits of the error interval and (Fig. 4.1).
Answer: mA; mA.
4.2.3. The error in the voltage measurement result is distributed according to a uniform law with the parameters With= 0.25 1/V, mV. Determine the limits of the error interval and (Fig. 4.1).
Answer: B; IN.
4.2.4. The error in the current measurement result is distributed uniformly in the range from mA; mA. Find the systematic error of the measurement result, the mean square error and the probability R that the error of the measurement result lies in the range from mA to mA.
Answer: mA; mA; R = 0,5.
4.2.5. The power measurement error is distributed according to a triangular law in the range from W to W. Find the systematic error of the measurement result, the mean square error and the probability R that the error of the measurement result ranges from to W. (formulas 4.4, 4.6).
Answer: ; W; R = 0,28.
4.2.6. For the distribution law of voltage measurement errors shown in Fig. 4.2, determine the systematic error, mean square error, if B. Find the probability R that the error of the measurement result ranges from to W.
Answer: B; IN; R= 0.25.R mW. Systematic error. Hz, equal to (1- mA,
2. if there is a systematic error, we will use formula (4.12)
Therefore, the probability of the error exceeding the confidence interval is:
1. q = 1 - 0,988 = 0,012; 2. q = 1 - 0,894 = 0,106.
4.2.19. The resistance measurement error is distributed according to the normal law, with the mean square error being Ohm. Find the probability that the resistance measurement result differs from the true resistance value by no more than 0.07 ohms if:
1. Systematic error;
2. Systematic error Ohm.
Answer: R 1 = 0,92; R 2 = 0,882.
4.2.20. The error in the voltage measurement result is distributed according to the normal law with a mean square error of mV. Confidence limits of error 4.2.22. Write down the law of error distribution obtained by summing five independent components with parameters: mathematical expectation
Solution. Let's convert the values of the confidence interval limits into absolute values of kHz or kHz. Confidence probability
1.1. Definition of metrology.
1.2. Definition of measurement.
1.3. Types of measuring instruments.
1.4. Types and methods of measurements.
1.5. Accuracy of measurements.
1.6. Presentation of measurement results.
1.7. Rounding rules.
1.8. Unity of measurements.
1.9. Conclusion on the section.
2. Assessment of measurement errors based on the given metrological characteristics of measuring instruments.
2.1. Standardized metrological characteristics of measuring instruments.
2.1.1. Appointment of N.M.H.
2.1.2. Nomenclature of N.M.H., currently accepted.
2.1.2.1. N.M.H. necessary to determine the measurement result.
2.1.2.2. N.M.H., necessary to determine the measurement error.
2.1.3. The development trend of N.M.H. complexes
2.2. Estimates of errors in direct measurements with single observations.
2.2.1. Components of measurement error.
2.2.2. Summation of measurement error components.
2.2.3. Examples of estimating the error of direct measurements.
2.3. Estimation of errors of indirect measurements.
2.3.1. Components of errors in indirect measurements.
2.3.2. Summation of errors.
2.3.3. Examples of estimating errors of direct measurements.
2.4. Estimation of errors of indirect measurements.
2.4.1. Components of errors in indirect measurements.
2.4.2. Summation of direct measurement errors
2.4.3. Examples of estimating the error of indirect measurements.
3. Ways to reduce measurement errors.
3.1. Ways to reduce the influence of random errors.
3.1.1. Multiple observations with direct measurements.
3.1.2. Multiple observations with indirect measurements.
3.1.3. Smoothing of experimental dependencies using the least squares method for joint measurements.
3.2. Ways to reduce the influence of systematic errors.
4. Standardization.
Fundamentals of metrology and standardization.
Tyurin N.I. Introduction to metrology. - M.: Standards Publishing House, 1976.
1. Basic concepts of metrology.
Metrology cf.: biology, geology, meteorology.
Logos is a word, a relation (logometer).
"Logia" is the science of...
Subway metrology? metro - underground (French) - literally: capital (1863 - London; 1868 - New York; 1900 - Paris; 1935 - Moscow)
Metro policy- metropolis, main city.
Head waiter - head waiter, main, first - ratio, measure of primacy.
The meter is a measure of length, but: metrology is much older than the meter; meter was “born” in 1790, meter - from the Greek - measure.
Metrology - the study of measures (ancient dictionary).
“Russian metrology or a table comparing Russian measures, weights and coins with French ones.”
Linear and linear measures:
1 vershok=4.445 cm;
1 arshin=16 vershoks=28 inches - pipes
1 fathom = 3 arshins;
1 verst=500 fathoms
Capacity measures:
1 barrel=40 buckets;
1 bucket = 10 mugs (damask glasses);
1 mug=10 glasses=2 bottles=20 scales=1.229 l
Weights:
1 pood = 40 pounds = 16.380 kg;
1 pound=32 lots;
1 lot=3 spools;
1 spool=96 shares=4.266 g.
"Small spool but precious".
1 pound of medical weight = 12 ounces = 96 drams = 288 = 5760 grains = 84 spools.
Meticulous:not a grain.
Coins:
1 imperial=10 rubles (gold);
Silver: ruble, fifty dollars, quarter, two-kopeck piece, ten-kopeck piece, nickel.
Copper: three-kopek coin, penny (2 kopecks), 1 kopeck = 2 money = 4 half rubles.
The rich man fell in love with the poor woman,
A scientist fell in love with a stupid woman,
I fell in love with ruddy - pale,
Gold - copper half...
M. Tsvetaeva.
We are talking about concepts such as measures of length, measures of capacity, measures of weight...
Accordingly, there is a concept of length; capacity, or in modern language - volume; weight, or, as we now know, better to say mass, temperature, etc.
How to combine all these concepts?
Now we say that all these are physical quantities.
How to determine what a physical quantity is? How are definitions given in such an exact science as, for example, mathematics? For example, in geometry. What is an isosceles triangle? It is necessary to find a higher one in the hierarchical ladder of concepts; what concept stands above the concept of physical quantity? The superior concept is the property of an object.
Length, color, smell, taste, mass - these are different properties of an object, but not all of them are physical quantities. Length and mass are physical quantities, but color and smell are not. Why? What is the difference between these properties?
Length and mass are what we know how to measure. You can measure the length of the table and find out that it is so many meters. But you can't measure the smell, because... Units of measurement have not yet been established for it. However, smells can be compared: this flower smells stronger than this one, i.e. the concept applies to smell more less.
Comparing the properties of objects by type more or less is a more primitive procedure compared to measuring something. But this is also a way of knowing. There is an alternative representation when all parameters and relationships of objects and phenomena are designated as three classes of physical quantities.
The first class of physical quantities includes :
quantities, based on the number of sizes of which, are harder, softer, colder, etc. Hardness (the ability to resist penetration), temperature as the degree of heating of the body, the strength of the earthquake.
Second view: relations of order and equivalence not only between the sizes of quantities, but also between the differences in pairs of their sizes. Time, potential, energy, temperature associated with the thermometer scale.
Third type: additive physical quantities.
Additive physical quantities are quantities on the set of sizes of which not only the relations of order and equivalence, but also the operations of addition and subtraction are defined.
The operation is considered certain, if its result is also the size of the same physical quantity and there is a method for its technical implementation. For example: length, mass, thermodynamic temperature, current strength, emf, electrical resistance.
How does a child perceive the world? At first, of course, he doesn’t know how to measure anything. At the first stage, he develops the concepts of more and less. Then comes the stage that is closer to measurement - this is the counting of objects, events, etc. There is already something in common with measurement. What? That the result of counting and measuring is a number. Not relations like more - less, but a number. How do these numbers differ, i.e. number as a result of counting and number as a result of measurement?
The measurement result is a named number, for example 215m. The number 2.15 itself expresses how many units of length are contained in a given length of a table or other object. And the result of counting 38 pieces is something. Counting is counting, and measurement is measurement.
This is how the process of development of a child’s knowledge of the world proceeds, the same or approximately this is how the development of primitive man proceeded, i.e. at the first stage of comparing things by type more - less, then - counting.
Then comes the next stage, when you want to express in the form of a number something that cannot be counted by piece - the volume of liquid, the area of a piece of land, etc., i.e. something continuous rather than discrete.
So, various physical quantities are measured, and a physical quantity is a property of an object, which is qualitatively common to many objects, and quantitatively individual for each given object.
Are there many physical quantities? With development human society their list is constantly growing. At first there were only length, area, volume, spatial quantities and time, then mechanical quantities were added - mass, force, pressure, etc., thermal quantities - temperature, etc. In the last century, electrical and magnetic quantities were added - current strength, voltage, resistance, etc. Currently, there are more than 100 physical quantities. For brevity, in what follows, the word “physical” can be omitted and simply said size..
Concept magnitude contains qualitative sign, i.e. what is this quantity, for example length, and quantitative sign, for example, the length became 2.15m. But the same length of the same table can be expressed in other units, for example, in inches, and you get a different number. However, it is clear that the quantitative content of the concept “length of a given table” remains unchanged.
In this regard, the concept is introduced size quantities and concept meaning quantities. The size does not depend on the units in which the value is expressed, i.e. He invariant in relation to the choice of unit.
The “maximum-minimum” method is based on the assumption that when assembling a mechanism, it is possible to combine increasing links made to the largest maximum dimensions with decreasing links made to the smallest maximum dimensions, or vice versa.
This calculation method ensures complete interchangeability during the assembly and operation of products. However, the tolerances of the component dimensions calculated using this method, especially for dimensional chains containing many links, may turn out to be unjustifiably small in technical and economic terms, therefore this method used for designing dimensional chains with a small number of component links of low accuracy.
First task
The nominal size of the closing link can be determined by the formula (see example of the first problem).
If we take the total number of chain links n, then the number of components will be n – 1. Let's accept: m– number of increasing links, R – number of decreasing ones, then
n – 1 = m + p.
IN general view The formula for calculating the nominal size of the trailing link will be:
(8.1)
For example (see section 8.1)
A0 = A 2 – A1 = 64 – 28 = 36 mm.
Based on equality (8.1), we obtain:
; (8.2)
. (8.3)
Subtract term by term from equality (8.2) equality (8.3), we obtain:
.
Since the sum of increasing and decreasing links is all the constituent links of the chain, the resulting equality can be simplified:
. (8.4)
Thus, the tolerance of the trailing link equal to the sum tolerances of all component links in the chain.
To derive formulas for calculating the maximum deviations of the closing link, subtract term by term from equality (8.2) equality (8.1) and from equality (8.3) equality (8.1), we obtain:
; (8.5)
. (8.6)
Thus, the upper deviation of the closing dimension is equal to the difference between the sums of the upper deviations of the increasing and lower deviations of the decreasing dimensions; the lower deviation of the closing dimension is equal to the difference between the sums of the lower deviations of the increasing and upper deviations of the decreasing dimensions.
For the example of the first problem (see section 8.1) we get:
= 0.04 + 0.08 = 0.12 mm;
Thus,
Let us determine the tolerance of the closing link through the obtained maximum deviations:
This value coincides with the previously found tolerance value, which confirms the correctness of the problem solution.
Second task
When solving the second problem, the tolerances of the component dimensions are determined by the given tolerance of the closing dimension TA0 in one of the following ways: equal tolerances or tolerances of the same quality.
1. When deciding equal tolerance method – approximately equal tolerances are assigned to the component dimensions, guided by the average tolerance.
So, we assume that
then the sum of the tolerances of all component sizes is equal to the product of the number of component links and the average tolerance, i.e.:
.
Let's substitute this expression into equality (8.4): 
, from here
. (8.7)
By found value Tcp Ai establish tolerances for component sizes, taking into account the size and responsibility of each size.
In this case, the following conditions must be met: the accepted tolerances must correspond to standard tolerances, the sum of the tolerances of the component dimensions must be equal to the tolerance of the trailing dimension, i.e. equality (8.4) must be satisfied. If equality (8.4) cannot be ensured with standard tolerances, then a non-standard tolerance is established for one component size, determining its value using the formula
. (8.8)
The equal tolerance method is simple and gives good results, if the nominal sizes of the constituent links of the dimensional chain are in the same interval.
Let's solve the example of the second problem (see Section 8.1) using the equal tolerance method (8.7):
mm.
A1 = 215; TA1 = 0.04;
A2 = 60; TA2 = 0.04;
A3 = 155; TA3 = 0.04.
In this example, equality (8.4) is observed, and there is no need to adjust the tolerance of one of the component dimensions.
Let us write down equality (8.5) for this example:
0,12 = 0,06 – (-0,03 – 0,03).
(The numerical values of the maximum deviations of the component dimensions are chosen conditionally.)
TA1 = 0.04, which means Ei(A1) = +0.02;
Ei(A2) = -0.03; TA2 = 0.04, which means Es(A2) = +0.01;
Ei(A3) = -0.03; TA3 = 0.04, which means Es(A3) = +0.01.
Let's check that equality (8.6) is satisfied:
0 = 0,02 – (0,01 +0,01);
Thus, we get the answer:
; 
; 
.
2. A more universal and simplified selection of tolerances for any variety of sizes of component links is way tolerances of one qualification .
With this method, the dimensions of all component links (except for the corrective Aj) assign tolerances from one quality level, taking into account the nominal dimensions of the links.
To derive the formula, the initial dependence is equality (8.4):
.
However, the tolerance of any size can be calculated using the formula
Where A– the number of tolerance units, constant within one qualification (Table 8.1); - the tolerance unit depends on the nominal size of the component link (Table 8.2).
Table 8.1
Number of tolerance units
|
Quality |
Quality |
Quality |
Quality |
||||
Meaning of tolerance units
|
Size intervals, mm |
i, µm |
Size intervals, mm |
i, µm |
|
1,86.; conclusionsSince the tolerance of the closing link depends on the number of component dimensions, the basic rule for designing dimensional chains can be formulated as follows: when designing parts, assemblies of assembly units and mechanisms, it is necessary to strive to ensure that the number of dimensions forming the dimensional chain is minimal. This is the principle of the shortest dimensional chain. The drawings indicate only component dimensions with prescribed deviations. Closing dimensions are usually obtained automatically as a result of processing parts or assembly, so they are not controlled and are not indicated on the drawings. It is not recommended to put dimensions in closed chains on drawings. It is especially unacceptable to enter closing dimensions with deviations, since this causes defects during the manufacture of the part. The least critical dimensions, which may have large deviations, should be taken as closing dimensions. |
Metrology– the science of measurements, methods and means of ensuring their unity and methods of achieving the required accuracy.
The main areas of metrology include:
General theory of measurements;
Units of physical quantities and their systems;
Methods and means of measurement;
Methods for determining measurement accuracy;
Fundamentals of ensuring the uniformity of measurements and uniformity of measuring instruments;
Standards and exemplary measuring instruments;
Methods for transferring unit sizes from standards and reference measuring instruments to working measuring instruments.
The main subject of metrology is the extraction of quantitative information about the properties of objects and processes with a given accuracy and reliability.
A measuring instrument (MI) is a set of measuring instruments and metrological standards that ensure their rational use.
Structure of metrological support for measurements.
Scientific metrology, being the basis of measurement technology, deals with the study of measurement problems in general and the elements that form the measurement: measuring instruments (MI), physical quantities (PV) and their units, measurement methods, results, errors, etc.
The regulatory and technical foundations of metrological support are a complex of state ones. standards.
The organizational basis is metrological. ensuring our state is metrological. service of the Russian Federation.
State the system for ensuring the uniformity of measurements establishes a unified nomenclature of standard interconnected rules and regulations, requirements and norms related to the organization, methodology for assessing and ensuring measurement accuracy.
2. Physical properties and quantities.
Physical quantity(PV) is a property that is qualitatively common for many objects, but quantitatively individual for each of them.
PV is divided into measurable And assessed.
Measured PV can be expressed quantitatively by a certain number of established units of measurement.
For some reason, a unit of measurement cannot be entered for assessed PVs; they can only be estimated.
Based on the degree of conditional independence from any quantities, basic, derivative and additional PVs are distinguished.
By size they are divided into dimensional and dimensionless.
There are PVs true, valid, measured.
True PV value– a value that would ideally reflect, qualitatively and quantitatively, the corresponding properties of the object.
Actual PV value- a value found experimentally and so close to the true value that it can be used instead for a certain purpose.
Measured PV value– the value of the quantity measured by the indicator device of the measuring instrument.
A measurement condition is a set of influencing quantities that describe the state of the environment and the measuring instruments. 3 types: normal, working, extreme.
3. International system of units.
A set of basic and derived units of PV, formed in accordance with accepted principles, is called a system of PV units.
Main characteristics of the SI system:
1) versatility;
2) unification of all areas and types of measurements;
3) the ability to reproduce units with high accuracy in accordance with their definition with the smallest error.
Basic units of the SI system.
1. length (meter)
2. weight (kg)
3. time (sec)
4. electric current strength (amps)
5. temperature (Kelvin)
6. amount of substance (mol)
7. luminous intensity (condela)
2 additional: plane angle (radian)
solid angle (steradian)
VW derivatives can be coherent and incoherent.
Coherent they call a derived unit of quantity related to other units of the system by an equation in which the numerical factor is equal to 1. All other derived units are called incoherent.
PV units can be multiples or submultiples.
1.6.2 Processing observation results and estimating measurement errors
The error of the measurement result is assessed during the development of the MVI. Sources of errors are the OM model, measurement method, SI, operator, influencing factors of measurement conditions, algorithm for processing observation results. As a rule, the error of the measurement result is estimated using the confidence probability R= 0,95.
When choosing the P value, it is necessary to take into account the degree of importance (responsibility) of the measurement result. For example, if an error in a measurement could result in loss of life or serious environmental consequences, the P value should be increased.
1. Measurements with single observations. In this case, the result of a measurement is taken to be the result of a single observation x (with the introduction of a correction, if any), using previously obtained (for example, during the development of MVI) data on the sources that make up the error.
Confidence limits of the NSP measurement result Θ( R) is calculated using the formula
Where k(P) is the coefficient determined by the accepted R and number m 1 components of the NSP: Θ( R) - boundaries found by non-statistical methods j th component of the NSP (the boundaries of the interval within which this component is located, determined in the absence of information about the probability of its location in this interval). At P - 0.90 and P = 0.95 k(P) is equal to 0.95 and 1.1, respectively, for any number of terms m 1. At P=0.99 values k(P) the following (Table 3.3): Table 3.3
If the components of the NSP are distributed uniformly and are specified by confidence limits 0(P), then the confidence limit of the NSP of the measurement result is calculated using the formula
The standard deviation (RMS) of a measurement result with a single observation is calculated in one of the following ways:
2. Measurements with multiple observations. In this case, it is recommended to begin processing the results by checking for the absence of errors (gross errors). A miss is the result of x n an individual observation included in a series of n observations, which, for given measurement conditions, differs sharply from the other results of this series. If the operator during the measurement discovers such a result and reliably finds its cause, he has the right to discard it and carry out (if necessary) additional observation to replace the discarded one.
When processing existing observation results, individual results cannot be arbitrarily discarded, as this may lead to a fictitious increase in the accuracy of the measurement result. Therefore, the following procedure is used. Calculate the arithmetic mean x of the observation results x i using the formula
Then the estimate of the standard deviation of the observation result is calculated as
expected miss x n from x:
Based on the number of all observations n(including x n) and the value accepted for measurement R(usually 0.95) according to or any reference book, but probability theories find z( P, n)— normalized sample deviation of the normal distribution. If Vn< zS(x), then observation x n is not a miss; if V n > z S(x), then x n is a miss to be excluded. After eliminating x n, repeat the determination procedure X And S(x) for the remaining series of observation results and checking for a miss of the largest of the remaining series of deviations from the new value (calculated based on n - 1).
The arithmetic mean x is taken as the measurement result [see. formula (3.9)] of the observation results xh The error x contains random and systematic components. The random component, characterized by the standard deviation of the measurement result, is estimated using the formula
It is easy to check whether the observation results x i belong to the normal distribution for n ≥ 20 by applying the 3σ rule: if the deviation from X does not exceed 3σ, then the random variable is normally distributed. Confidence limits of random error of measurement result with confidence probability R find by formula
where t is the Student coefficient.
Confidence limits Θ( R) The NSP of a measurement result with multiple observations is determined in exactly the same way as in a measurement with a single observation - using formulas (3.3) or (3.4).
Summation of the systematic and random components of the error of the measurement result when calculating Δ( R) is recommended to be carried out using criteria and formulas (3.6-3.8), in which S(x) is replaced by S(X) = S(X)/√n;
3. . The value of the measured quantity A is found from the results of measurements of the arguments alf ait at associated with the desired quantity by the equation
The type of function ƒ is determined when establishing the OP model.
The desired value A is related to the measured arguments by the equation
Where b i are constant coefficients
It is assumed that there is no correlation between measurement errors a i. Measurement result A calculated by the formula
Where and i— measurement result and i with the amendments introduced. Estimation of the standard deviation of the measurement result S(A) calculated using the formula
Where S(a i)- assessment of the standard deviation of the measurement result a i.
Confidence limits ∈( R) random error A with a normal distribution of errors a i
Where t(P, neff)— Student’s coefficient corresponding to the confidence probability R(usually 0.95, in exceptional cases 0.99) and the effective number of observations n eff calculated by the formula
Where n i-number of observations during measurement a i.
Confidence limits Θ( R) NSP of the result of such a measurement, the sum Θ( R) and ∈( R) to obtain the final value Δ( R) is recommended to be calculated using criteria and formulas (3.3), (3.4), (3.6) - (3.8), in which m i ,Θ i, And S(x) are replaced accordingly by m, b i Θ i, And s(A)
Indirect measurements with nonlinear dependence. For uncorrelated measurement errors a i the linearization method is used by expanding the function ƒ(a 1 ,…,a m) into a Taylor series, that is
where Δ a i = a i - a— deviation of an individual observation result a i from a i ; R- remainder term.
The linearization method is acceptable if the increment of the function ƒ can be replaced by it full differential. Remaining member 
neglected if
Where S(a)— estimation of the standard deviation of random errors in the measurement result a i. In this case, deviations Δ a i(should be taken from the possible values of the errors and such that they maximize R.
Measurement result A calculated using the formula  = ƒ(â …â m).
Estimation of the standard deviation of the random component of the error in the result of such an indirect measurement s(Â) calculated by the formula
a ∈( P) - according to formula (3.13). Meaning n eff NSP boundary Θ( P) and error Δ( P) the result of indirect measurement with a linear dependence is calculated in the same way as with a linear dependence, but with the replacement of coefficients b i by δƒ/δa i
Casting method(for indirect measurements with nonlinear dependence) is used for unknown distributions of measurement errors and i and with correlation between errors and i to obtain the result of an indirect measurement and determine its error. This assumes the presence of a number n observation results and ij. measured arguments a i. Combinations and ij received in j experiment, substitute into formula (3.12) and calculate a series of values A j measured quantity A. The measurement result  is calculated using the formula
Estimation of the standard deviation s(Â)— the random component of the error  — is calculated using the formula
a ∈ ( R) -according to the formula(3.11). Boundaries of the NSP Θ( R) and error Δ( R) measurement result  is determined by the methods described above for a nonlinear relationship.
