Measurement errors. Absolute, relative errors

Measurement error

Measurement error- assessment of the deviation of the measured value of a quantity from its true value. Measurement error is a characteristic (measure) of measurement accuracy.

Reduced error- relative error, expressed as the ratio of the absolute error of the measuring instrument to the conventionally accepted value of a quantity, constant over the entire measurement range or in part of the range. Calculated by the formula

Where X n- normalizing value, which depends on the type of scale of the measuring device and is determined by its calibration:

If the instrument scale is one-sided, i.e. the lower measurement limit is zero, then X n determined equal to the upper limit of measurement;
- if the instrument scale is double-sided, then the normalizing value is equal to the width of the instrument’s measurement range.

The given error is a dimensionless quantity (can be measured as a percentage).

Due to the occurrence

Instrumental/instrumental errors- errors that are determined by the errors of the measuring instruments used and are caused by imperfections in the operating principle, inaccuracy of scale calibration, and lack of visibility of the device.
Methodological errors- errors due to the imperfection of the method, as well as simplifications underlying the methodology.
Subjective / operator / personal errors- errors caused by the degree of attentiveness, concentration, preparedness and other qualities of the operator.

In technology, instruments are used to measure only with a certain predetermined accuracy - the main error allowed by the normal under normal operating conditions for a given device.

If the device operates under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by a deviation of the ambient temperature from normal, installation, caused by a deviation of the device’s position from the normal operating position, etc. The normal ambient temperature is 20°C, and the normal atmospheric pressure is 01.325 kPa.

A generalized characteristic of measuring instruments is the accuracy class, determined by the maximum permissible main and additional errors, as well as other parameters affecting the accuracy of measuring instruments; the meaning of the parameters is established by standards for certain types of measuring instruments. The accuracy class of measuring instruments characterizes their precision properties, but is not a direct indicator of the accuracy of measurements performed using these instruments, since the accuracy also depends on the measurement method and the conditions for their implementation. Measuring instruments, the limits of the permissible basic error of which are specified in the form of the given basic (relative) errors, are assigned accuracy classes selected from the following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0 ; 5.0; 6.0)*10n, where n = 1; 0; -1; -2, etc.

By nature of manifestation

Random error- error that varies (in magnitude and sign) from measurement to measurement. Random errors may be associated with imperfection of instruments (friction in mechanical devices, etc.), shaking in urban conditions, with imperfection of the measurement object (for example, when measuring the diameter of a thin wire, which may not have a completely round cross-section as a result of imperfections in the manufacturing process ), with the characteristics of the measured quantity itself (for example, when measuring the number of elementary particles passing per minute through a Geiger counter).
Systematic error- an error that changes over time according to a certain law (a special case is a constant error that does not change over time). Systematic errors may be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
Progressive (drift) error- an unpredictable error that changes slowly over time. It is a non-stationary random process.
Gross error (miss)- an error resulting from an oversight by the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the division number on the instrument scale, if a short circuit occurred in the electrical circuit).

By measurement method

Direct measurement error
Error of indirect measurements- error of the calculated (not directly measured) quantity:

If F = F(x 1 ,x 2 ...x n) , Where x i- directly measured independent quantities with an error Δ x i, Then:

Literature

Nazarov N. G. Metrology. Basic concepts and mathematical models. M.: Higher School, 2002. 348 p.

Laboratory classes in physics. Textbook/Goldin L.L., Igoshin F.F., Kozel S.M. et al.; edited by Goldina L.L. - M.: Science. Main editorial office of physical and mathematical literature, 1983. - 704 p.

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The measurement of a quantity is an operation as a result of which we find out how many times the measured quantity is greater (or less) than the corresponding value taken as the standard (unit of measurement). All measurements can be divided into two types: direct and indirect.

DIRECT these are measurements in which the physical quantity of immediate interest to us is measured (mass, length, time intervals, temperature change, etc.).

INDIRECT these are measurements in which the quantity of interest to us is determined (calculated) from the results of direct measurements of other quantities associated with it by a certain functional relationship. For example, determining the speed of uniform motion by measuring the distance traveled over a period of time, measuring the density of a body by measuring the mass and volume of the body, etc.

A common feature of measurements is the impossibility of obtaining the true value of the measured value; the measurement result always contains some kind of error (inaccuracy). This is explained both by the fundamentally limited measurement accuracy and by the nature of the measured objects themselves. Therefore, to indicate how close the obtained result is to the true value, the measurement error is indicated along with the obtained result.

For example, we measured the focal length of a lens f and wrote that

f = (256 ± 2) mm (1)

This means that the focal length ranges from 254 to 258 mm. But in fact, this equality (1) has a probabilistic meaning. We cannot say with complete confidence that the value lies within the specified limits, there is only a certain probability of this, therefore equality (1) must be supplemented with an indication of the probability with which this relationship makes sense (we will formulate this statement more precisely below).

An assessment of errors is necessary because, without knowing what they are, it is impossible to draw certain conclusions from the experiment.

Typically, absolute and relative error are calculated. The absolute error Δx is the difference between the true value of the measured quantity μ and the measurement result x, i.e. Δx = μ - x

The ratio of the absolute error to the true value of the measured quantity ε = (μ - x)/μ is called the relative error.

The absolute error characterizes the error of the method that was chosen for measurement.

The relative error characterizes the quality of measurements. The measurement accuracy is the reciprocal of the relative error, i.e. 1/ε.

§ 2. Classification of errors

All measurement errors are divided into three classes: misses (gross errors), systematic and random errors.

A MISS is caused by a sharp violation of measurement conditions during individual observations. This is an error associated with a shock or breakdown of the device, a gross miscalculation by the experimenter, unforeseen intervention, etc. a gross error usually appears in no more than one or two dimensions and differs sharply in magnitude from other errors. The presence of a miss can greatly distort the result containing the miss. The easiest way is to establish the cause of the mistake and eliminate it during the measurement process. If a mistake was not excluded during the measurement process, then this should be done when processing the measurement results, using special criteria that make it possible to objectively identify a gross error, if any, in each series of observations.

SYSTEMATIC ERROR is a component of measurement error that remains constant and changes naturally with repeated measurements of the same quantity. Systematic errors arise if, for example, thermal expansion is not taken into account when measuring the volume of a liquid or gas produced at a slowly changing temperature; if, when measuring mass, one does not take into account the effect of the buoyant force of air on the body being weighed and on the weights, etc.

Systematic errors are observed if the ruler scale is applied inaccurately (unevenly); the capillary of the thermometer in different areas has a different cross-section; in the absence of electric current through the ammeter, the instrument needle is not at zero, etc.

As can be seen from the examples, a systematic error is caused by certain reasons, its value remains constant (the zero shift of the instrument scale, unequal-armed scales), or changes according to a certain (sometimes quite complex) law (unevenness of the scale, uneven cross-section of the thermometer capillary, etc.).

We can say that systematic error is a softened expression that replaces the words “experimenter error.”

Such errors occur because:

measuring instruments are inaccurate;
the actual installation differs in some way from the ideal;
The theory of the phenomenon is not entirely correct, i.e. some effects are not taken into account.

We know what to do in the first case; calibration or calibration is needed. In the other two cases there is no ready-made recipe. The better you know physics, the more experience you have, the more likely it is that you will discover such effects, and therefore eliminate them. There are no general rules or recipes for identifying and eliminating systematic errors, but some classification can be made. Let us distinguish four types of systematic errors.

Systematic errors, the nature of which is known to you, and the value can be found, therefore, eliminated by introducing corrections. Example. Weighing on unequal-arm scales. Let the difference in arm lengths be 0.001 mm. With a rocker length of 70 mm and weight of the weighed body 200 G systematic error will be 2.86 mg. The systematic error of this measurement can be eliminated by using special weighing methods (Gaussian method, Mendeleev method, etc.).
Systematic errors that are known to be less than a certain value. In this case, when recording the response, their maximum value can be indicated. Example. The data sheet supplied with the micrometer states: “the permissible error is ±0.004 mm. Temperature +20 ± 4° C. This means that when measuring the dimensions of any body with this micrometer at the temperatures indicated in the passport, we will have an absolute error not exceeding ± 0.004 mm for any measurement results.
Often the maximum absolute error given by a given device is indicated using the accuracy class of the device, which is depicted on the device scale by the corresponding number, most often circled.

The number indicating the accuracy class shows the maximum absolute error of the device, expressed as a percentage of the largest value of the measured value at the upper limit of the scale.

Let a voltmeter be used in the measurements, having a scale from 0 to 250 IN, its accuracy class is 1. This means that the maximum absolute error that can be made when measuring with this voltmeter will be no more than 1% of the highest voltage value that can be measured on this instrument scale, in other words:

δ = ±0.01·250 IN= ±2.5 IN.

The accuracy class of electrical measuring instruments determines the maximum error, the value of which does not change when moving from the beginning to the end of the scale. In this case, the relative error changes sharply, because the instruments provide good accuracy when the needle deflects almost the entire scale and does not provide it when measuring at the beginning of the scale. This is the recommendation: select a device (or the scale of a multi-range device) so that the arrow of the device goes beyond the middle of the scale during measurements.

If the accuracy class of the device is not specified and there is no passport data, then half the price of the smallest scale division of the device is taken as the maximum error of the device.

A few words about the accuracy of the rulers. Metal rulers are very accurate: millimeter divisions are marked with an error of no more than ±0.05 mm, and centimeter ones are no worse than with an accuracy of 0.1 mm. The error of measurements made with the accuracy of such rulers is almost equal to the error of reading by eye (≤0.5 mm). It is better not to use wooden and plastic rulers; their errors can be unexpectedly large.

A working micrometer provides an accuracy of 0.01 mm, and the measurement error with a caliper is determined by the accuracy with which the reading can be made, i.e. vernier accuracy (usually 0.1 mm or 0.05 mm).
Systematic errors caused by the properties of the measured object. These errors can often be reduced to chance. Example.. The electrical conductivity of a certain material is determined. If for such a measurement a piece of wire is taken that has some kind of defect (thickening, crack, inhomogeneity), then an error will be made in determining the electrical conductivity. Repeating the measurements gives the same value, i.e. some systematic error was made. Let's measure the resistance of several pieces of such wire and find the average value of the electrical conductivity of this material, which may be greater or less than the electrical conductivity of individual measurements; therefore, the errors made in these measurements can be attributed to the so-called random errors.
Systematic errors that are not known to exist. Example.. Determine the density of any metal. First, we find the volume and mass of the sample. There is a void inside the sample that we know nothing about. An error will be made in determining the density, which will be repeated for any number of measurements. The example given is simple; the source of the error and its magnitude can be determined without much difficulty. Errors of this type can be identified with the help of additional research, by taking measurements using a completely different method and under different conditions.

RANDOM is the component of measurement error that changes randomly during repeated measurements of the same quantity.

When repeated measurements of the same constant, unchanging quantity are carried out with the same care and under the same conditions, we obtain measurement results - some of them differ from each other, and some of them coincide. Such discrepancies in measurement results indicate the presence of random error components in them.

Random error arises from the simultaneous influence of many sources, each of which in itself has an imperceptible effect on the measurement result, but the total influence of all sources can be quite strong.

A random error can take on different absolute values, which are impossible to predict for a given measurement. This error can be equally positive or negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause scatter of repeated measurements relative to the true value ( Fig.14).

If, in addition, there is a systematic error, then the measurement results will be scattered relative to not the true, but the biased value ( Fig.15).

Rice. 14 Fig. 15

Let us assume that the period of oscillation of a pendulum is measured using a stopwatch, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the reading value, a slight unevenness in the movement of the pendulum all this causes scattering in the results of repeated measurements and therefore can be classified as random errors.

If there are no other errors, then some results will be somewhat overestimated, while others will be somewhat underestimated. But if, in addition to this, the clock is also behind, then all the results will be underestimated. This is already a systematic error.

Some factors can cause both systematic and random errors at the same time. So, by turning the stopwatch on and off, we can create a small irregular spread in the starting and stopping times of the clock relative to the movement of the pendulum and thereby introduce a random error. But if, moreover, we are in a hurry to turn on the stopwatch every time and are somewhat late to turn it off, then this will lead to a systematic error.

Random errors are caused by parallax error when counting instrument scale divisions, shaking of the foundation of a building, the influence of slight air movement, etc.

Although it is impossible to eliminate random errors in individual measurements, the mathematical theory of random phenomena allows us to reduce the influence of these errors on the final measurement result. It will be shown below that for this it is necessary to make not one, but several measurements, and the smaller the error value we want to obtain, the more measurements need to be made.

It should be borne in mind that if the random error obtained from the measurement data turns out to be significantly less than the error determined by the accuracy of the device, then, obviously, there is no point in trying to further reduce the value of the random error; anyway, the measurement results will not become more accurate.

On the contrary, if the random error is greater than the instrumental (systematic) error, then the measurement should be carried out several times in order to reduce the error value for a given series of measurements and make this error less than or of the same order of magnitude as the instrument error.

Absolute and relative errors are used to assess the inaccuracy in highly complex calculations. They are also used in various measurements and for rounding calculation results. Let's look at how to determine absolute and relative error.

Absolute error

Absolute error of the number call the difference between this number and its exact value.
Let's look at an example : There are 374 students in the school. If we round this number to 400, then the absolute measurement error is 400-374=26.

To calculate the absolute error, you need to subtract the smaller number from the larger number.

There is a formula for absolute error. Let us denote the exact number by the letter A, and the letter a – the approximation to the exact number. An approximate number is a number that differs slightly from the exact one and usually replaces it in calculations. Then the formula will look like this:

Δa=A-a. We discussed above how to find the absolute error using the formula.

In practice, absolute error is not sufficient to accurately evaluate a measurement. It is rarely possible to know the exact value of the measured quantity in order to calculate the absolute error. Measuring a book 20 cm long and allowing an error of 1 cm, one can consider the measurement to be with a large error. But if an error of 1 cm was made when measuring a wall of 20 meters, this measurement can be considered as accurate as possible. Therefore, in practice, determining the relative measurement error is more important.

Record the absolute error of the number using the ± sign. For example , the length of a roll of wallpaper is 30 m ± 3 cm. The absolute error limit is called the maximum absolute error.

Relative error

Relative error They call the ratio of the absolute error of a number to the number itself. To calculate the relative error in the example with students, we divide 26 by 374. We get the number 0.0695, convert it to a percentage and get 6%. The relative error is denoted as a percentage because it is a dimensionless quantity. Relative error is an accurate estimate of measurement error. If we take an absolute error of 1 cm when measuring the length of segments of 10 cm and 10 m, then the relative errors will be equal to 10% and 0.1%, respectively. For a segment 10 cm long, an error of 1 cm is very large, this is an error of 10%. But for a ten-meter segment, 1 cm does not matter, only 0.1%.

There are systematic and random errors. Systematic is the error that remains unchanged during repeated measurements. Random error arises as a result of the influence of external factors on the measurement process and can change its value.

Rules for calculating errors

There are several rules for the nominal estimation of errors:

when adding and subtracting numbers, it is necessary to add up their absolute errors;
when dividing and multiplying numbers, it is necessary to add relative errors;
When raised to a power, the relative error is multiplied by the exponent.

Approximate and exact numbers are written using decimal fractions. Only the average value is taken, since the exact value can be infinitely long. To understand how to write these numbers, you need to learn about true and dubious numbers.

True numbers are those numbers whose rank exceeds the absolute error of the number. If the digit of a figure is less than the absolute error, it is called doubtful. For example , for the fraction 3.6714 with an error of 0.002, the correct numbers will be 3,6,7, and the doubtful ones will be 1 and 4. Only the correct numbers are left in the recording of the approximate number. The fraction in this case will look like this - 3.67.

Absolute measurement error is a quantity determined by the difference between the measurement result x and the true value of the measured quantity x 0:

Δ x = |x - x 0 |.

The value δ, equal to the ratio of the absolute measurement error to the measurement result, is called the relative error:

Example 2.1. The approximate value of π is 3.14. Then its error is 0.00159. The absolute error can be considered equal to 0.0016, and the relative error equal to 0.0016/3.14 = 0.00051 = 0.051%.

Significant figures. If the absolute error of the value a does not exceed one place unit of the last digit of the number a, then the number is said to have all the correct signs. Approximate numbers should be written down, keeping only the correct signs. If, for example, the absolute error of the number 52400 is 100, then this number should be written, for example, as 524·10 2 or 0.524·10 5. You can estimate the error of an approximate number by indicating how many correct significant digits it contains. When counting significant figures, the zeros on the left side of the number are not counted.

For example, the number 0.0283 has three valid significant figures, and 2.5400 has five valid significant figures.

Rules for rounding numbers. If the approximate number contains extra (or incorrect) digits, then it should be rounded. When rounding, an additional error occurs that does not exceed half a unit of the place of the last significant digit ( d) rounded number. When rounding, only the correct digits are retained; extra characters are discarded, and if the first discarded digit is greater than or equal to d/2, then the last digit stored is increased by one.

Extra digits in integers are replaced by zeros, and in decimals they are discarded (as are extra zeros). For example, if the measurement error is 0.001 mm, then the result 1.07005 is rounded to 1.070. If the first of the digits modified by zeros and discarded is less than 5, the remaining digits are not changed. For example, the number 148935 with a measurement precision of 50 has a rounding value of 148900. If the first of the digits replaced by zeros or discarded is 5, and there are no digits or zeros following it, then it is rounded to the nearest even number. For example, the number 123.50 is rounded to 124. If the first digit to be replaced by zeros or discarded is greater than or equal to 5 but followed by a significant digit, then the last remaining digit is increased by one. For example, the number 6783.6 is rounded to 6784.

Example 2.2. When rounding 1284 to 1300, the absolute error is 1300 - 1284 = 16, and when rounding to 1280, the absolute error is 1280 - 1284 = 4.

Example 2.3. When rounding the number 197 to 200, the absolute error is 200 - 197 = 3. The relative error is 3/197 ≈ 0.01523 or approximately 3/200 ≈ 1.5%.

Example 2.4. A seller weighs a watermelon on a scale. The smallest weight in the set is 50 g. Weighing gave 3600 g. This number is approximate. The exact weight of the watermelon is unknown. But the absolute error does not exceed 50 g. The relative error does not exceed 50/3600 = 1.4%.

Errors in solving the problem on PC

Three types of errors are usually considered as the main sources of error. These are called truncation errors, rounding errors, and propagation errors. For example, when using iterative methods for searching for the roots of nonlinear equations, the results are approximate, in contrast to direct methods that provide an exact solution.

Truncation errors

This type of error is associated with the error inherent in the task itself. It may be due to inaccuracy in determining the source data. For example, if any dimensions are specified in the problem statement, then in practice for real objects these dimensions are always known with some accuracy. The same applies to any other physical parameters. This also includes the inaccuracy of the calculation formulas and the numerical coefficients included in them.

Propagation errors

This type of error is associated with the use of one or another method of solving a problem. During calculations, error accumulation or, in other words, propagation inevitably occurs. In addition to the fact that the original data themselves are not accurate, a new error arises when they are multiplied, added, etc. The accumulation of error depends on the nature and number of arithmetic operations used in the calculation.

Rounding errors

This type of error occurs because the true value of a number is not always accurately stored by the computer. When a real number is stored in computer memory, it is written as a mantissa and exponent in much the same way as a number is displayed on a calculator.

In physics and other sciences, it is very common to make measurements of various quantities (for example, length, mass, time, temperature, electrical resistance, etc.).

Measurement– the process of finding the value of a physical quantity using special technical means – measuring instruments.

Measuring instrument is a device that is used to compare a measured quantity with a physical quantity of the same kind, taken as a unit of measurement.

There are direct and indirect measurement methods.

Direct measurement methods – methods in which the values of the quantities being determined are found by direct comparison of the measured object with the unit of measurement (standard). For example, the length of a body measured by a ruler is compared with a unit of length - a meter, the mass of a body measured by a scale is compared with a unit of mass - a kilogram, etc. Thus, as a result of direct measurement, the determined value is obtained immediately, directly.

Indirect measurement methods– methods in which the values of the quantities being determined are calculated from the results of direct measurements of other quantities with which they are related by a known functional relationship. For example, determining the circumference from the results of measuring the diameter or determining the volume of a body from the results of measuring its linear dimensions.

Due to the imperfection of measuring instruments, our senses, the influence of external influences on the measuring equipment and the object being measured, as well as other factors, all measurements can only be made with a certain degree of accuracy; therefore, the measurement results do not give the true value of the measured value, but only an approximate one. If, for example, body weight is determined with an accuracy of 0.1 mg, this means that the found weight differs from the true body weight by less than 0.1 mg.

Measurement accuracy – characteristic of measurement quality, reflecting the closeness of measurement results to the true value of the measured quantity.

The smaller the measurement errors, the greater the measurement accuracy. The accuracy of measurements depends on the instruments used in the measurements and on the general measurement methods. It is completely useless to strive to go beyond this limit of accuracy when making measurements under these conditions. It is possible to minimize the impact of reasons that reduce the accuracy of measurements, but it is impossible to completely get rid of them, that is, more or less significant errors (errors) are always made during measurements. To increase the accuracy of the final result, any physical measurement must be done not once, but several times under the same experimental conditions.

As a result of the i-th measurement (i – measurement number) of the value “X”, an approximate number X i is obtained, which differs from the true value of Xist by a certain amount ∆X i = |X i – X|, which is an error made or, in other words , error. The true error is not known to us, since we do not know the true value of the measured quantity. The true value of the measured physical quantity lies in the interval.

Х i – ∆Х< Х i – ∆Х < Х i + ∆Х

where X i is the value of X obtained during the measurement (that is, the measured value); ∆X – absolute error in determining the value of X.

Absolute mistake (error) of measurement ∆Х is the absolute value of the difference between the true value of the measured quantity Hist and the measurement result X i: ∆Х = |Х source – X i |.

Relative error (error) of measurement δ (characterizing the accuracy of measurement) is numerically equal to the ratio of the absolute measurement error ∆X to the true value of the measured value X source (often expressed as a percentage): δ = (∆X / X source) 100%.

Errors or measurement errors can be divided into three classes: systematic, random and gross (misses).

Systematic they call an error that remains constant or changes naturally (according to some functional dependence) with repeated measurements of the same quantity. Such errors arise as a result of the design features of measuring instruments, shortcomings of the adopted measurement method, any omissions of the experimenter, the influence of external conditions, or a defect in the measurement object itself.

Any measuring instrument contains one or another systematic error, which cannot be eliminated, but the order of which can be taken into account. Systematic errors either increase or decrease the measurement results, that is, these errors are characterized by a constant sign. For example, if during weighing one of the weights has a mass 0.01 g greater than indicated on it, then the found value of body mass will be overestimated by this amount, no matter how many measurements are made. Sometimes systematic errors can be taken into account or eliminated, sometimes this cannot be done. For example, fatal errors include instrument errors, about which we can only say that they do not exceed a certain value.

Random errors are called errors that change their magnitude and sign in an unpredictable way from experiment to experiment. The appearance of random errors is due to many diverse and uncontrollable reasons.

For example, when weighing with scales, these reasons may be air vibrations, settled dust particles, different friction in the left and right suspension of cups, etc. Random errors manifest themselves in the fact that, having made measurements of the same value X under the same experimental conditions, we get several differing values: X1, X2, X3,..., Xi,..., Xn, where Xi is the result of the i-th measurement. It is not possible to establish any pattern between the results, therefore the result of the i -th measurement of X is considered a random variable. Random errors can have a certain impact on a single measurement, but with repeated measurements they obey statistical laws and their influence on the measurement results can be taken into account or significantly reduced.

Mistakes and gross errors– excessively large errors that clearly distort the measurement result. This class of errors is most often caused by incorrect actions of the experimenter (for example, due to inattention, instead of the instrument reading “212”, a completely different number is recorded - “221”). Measurements containing misses and gross errors should be discarded.

Measurements can be carried out in terms of their accuracy using technical and laboratory methods.

When using technical methods, the measurement is carried out once. In this case, they are satisfied with such accuracy that the error does not exceed a certain, predetermined value determined by the error of the measuring equipment used.

With laboratory measurement methods, it is necessary to more accurately indicate the value of the measured quantity than is allowed by its single measurement using a technical method. In this case, several measurements are made and the arithmetic mean of the obtained values is calculated, which is taken as the most reliable (true) value of the measured value. Then the accuracy of the measurement result is assessed (taking into account random errors).

From the possibility of carrying out measurements using two methods, it follows that there are two methods for assessing the accuracy of measurements: technical and laboratory.