Part one

Estimation of measurement errors. Recording and processing results

In the exact sciences, in particular in physics, particular importance is attached to the problem of assessing the accuracy of measurements. That no measurement can be absolutely accurate is a fact of general philosophical significance. Those. during the experiment we always obtain an approximate value physical quantity, only approaching to one degree or another its true meaning.

Measurements, measurement accuracy indicators

Physics is one of the natural sciences, studies the material world around us, using physical method research, the most important component of which is the comparison of data obtained by theoretical calculations with experimental (measured) data.

The most important part of the process of teaching physics at a university is to carry out laboratory work. In the process of performing them, students carry out measurements of various physical quantities.

When measuring, physical quantities are expressed in the form of numbers that indicate how many times the measured quantity is greater or less than another quantity whose value is taken as a unit. Those. measurement is understood as “a cognitive process consisting of comparison, through a physical experiment, of a given physical quantity with a known physical quantity taken as a unit of measurement.”

Measurements are made using measures and measuring instruments.

Measure called a real reproduction of a unit of measurement, a fraction or multiple of its value (weight, measuring flask, magazines electrical resistance, containers, etc.).

Measuring instrument called a measuring instrument that makes it possible to directly read the value of the measured quantity.

Regardless of the purpose and principle of operation, any measuring device can be characterized by four parameters:

1) Measurement limits indicate the range of the measured value available to this device. For example, a caliper measures linear dimensions in the range from 0 to 18 cm, and a milliammeter measures currents from -50 to +50 mA, etc. On some devices you can change (switch) the measurement limits. Multi-range instruments can have several scales with different numbers of divisions. The reading should be carried out on a scale in which the number of divisions is a multiple of the upper limit of the device.

2) Division price C determines how many units of measurement (or their fractions) are contained in one (smallest) division of the instrument scale. For example, micrometer division value C = 0.01 mm/division(or 10 µm/div), and for a voltmeter C = 2 V/div etc. If C is the same across the entire scale (uniform scale), then to determine the division value you need the measurement limit of the device x nom divide by the number of divisions of the instrument scale N:

3) Sensitivity instrument α shows how many minimum scale divisions are per unit of the measured value or any fraction of it. From this definition it follows that the sensitivity of the device is the reciprocal of the division price: α = 1/C. For example, the sensitivity of a micrometer can be estimated as α = 1/0.01 = 100 divisions/mm(or α = 0.1 div/µm), and for a voltmeter α = 1/2 = 0.5 div/V etc.

4) Accuracy of a device characterizes the magnitude of the absolute error that is obtained during the measurement process with this device.

The accuracy of measuring instruments is characterized by the maximum calibration error Δ x deg. The maximum absolute or relative calibration error is indicated on the scale or in the instrument passport, or the accuracy class is indicated, which determines the systematic error of the instrument.

In order of increasing accuracy, electrical measuring instruments are divided into eight classes: 4.0; 2.5; 1.5; 1.0; 0.5; 0.2; 0.1 and 0.05. The number indicating the accuracy class is marked on the instrument scale and shows the maximum permissible value of the main error as a percentage of the measurement limit x nom

Cl. accuracy = ε pr = .(2)

There are instruments (mostly of high accuracy), the accuracy class of which determines the relative error of the device in relation to the measured value.

If there is no data on the accuracy class on the instruments and in their passports and no formula for calculating the error is indicated, then the instrumental error should be considered equal to half the scale of the instrument.

Measurements are divided into straight And indirect. In direct measurements, the desired physical quantity is determined directly from experience. The value of the measured quantity is counted off on the scale of the device or the number and value of measures, weights, etc. are calculated. Direct measurements are, for example, weighing on scales, determining the linear dimensions of the body correct form using a caliper, determining time using a stopwatch, etc.

In indirect measurements, the measured quantity is determined (calculated) from the results of direct measurements of other quantities that are related to the measured quantity by a certain functional relationship. Examples of indirect measurements are determining the area of ​​a table by its length and width, the density of a body by measuring the mass and volume of the body, etc.

The quality of measurements is determined by their accuracy. In direct measurements, the accuracy of experiments is established from an analysis of the accuracy of the method and instruments, as well as from the repeatability of measurement results. The accuracy of indirect measurements depends both on the reliability of the data used for calculation and on the structure of the formulas connecting these data with the desired value.

The accuracy of measurements is characterized by their error. Absolute measurement error call the difference between what is found experimentally x change and the true value of the physical quantity x ist

To assess the accuracy of any measurements, the concept is also introduced relative error.

Relative error measurement is the ratio of the absolute measurement error to the true value of the measured value (can be expressed as a percentage).

As follows from (3) and (4), in order to find the absolute and relative measurement error, we need to know not only the measured, but also the true value of the quantity of interest to us. But if the true value is known, then there is no need to make measurements. The purpose of measurements is always to find out the previously unknown value of a physical quantity and to find, if not its true value, then at least a value that differs quite slightly from it. Therefore, formulas (3) and (4), which determine the magnitude of errors, are unsuitable for practice. Often instead x ist use the arithmetic average of several measurements

Where x i– the result of a separate measurement.

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Measurement accuracy. Basic concept. Criteria for choosing measurement accuracy. Accuracy classes of measuring instruments. Examples of measuring instruments different classes accuracy.

Measurement is a set of operations for the use of a technical means that stores a unit of quantity, ensuring that the relationship of the measured quantity with its unit is found in an explicit or implicit form and the value of this quantity is obtained.

In general, metrology is the science of measurements, methods and means of ensuring their unity and methods of achieving the required accuracy.

Improvements in measurement accuracy have stimulated the development of science, providing more reliable and sensitive research tools.

The efficiency of various functions depends on the accuracy of measuring instruments: errors in energy meters lead to uncertainty in electricity metering; Scale errors lead to deception of buyers or to large volumes of unaccounted for goods.

Improving measurement accuracy allows you to identify deficiencies technological processes and eliminate these shortcomings, which leads to improved product quality, savings in energy and heat resources, raw materials.

Measurements can be classified according to their accuracy characteristics into:

Equally accurate - a series of measurements of any quantity performed with measuring instruments of equal accuracy and under the same conditions;

Non-equivalent - a series of measurements of any quantity performed by several measuring instruments of varying accuracy and (or) under several different conditions.

TO different types measuring instruments have specific requirements: for example, laboratory instruments must have increased accuracy and sensitivity. High-precision measuring instruments are, for example, standards.

A standard of a unit of quantity is a measuring instrument intended for reproducing and storing a unit of quantity, multiples or fractions of its values ​​in order to transfer its size to other means of measuring a given quantity. Standards are highly accurate measuring instruments and are therefore used for metrological measurements as a means of transmitting information about the size of a unit. The size of the unit is transferred “top to bottom” from more accurate measuring instruments to less accurate ones “along the chain”: primary standard ® secondary standard ® working standard of the 0th category ® working standard of the 1st category ... ® working measuring instrument.

Metrological properties of measuring instruments are properties that influence the measurement result and its error. Indicators of metrological properties are their quantitative characteristics and are called metrological characteristics. All metrological properties of measuring instruments can be divided into two groups:

· Properties that determine the scope of SI

· Properties that determine the quality of measurement. These properties include accuracy, precision and reproducibility.

The property of measurement accuracy, which is determined by error, is most widely used in metrological practice.

Measurement error is the difference between the measurement result and the true value of the measured value.

SI measurement accuracy is the quality of measurements, reflecting the closeness of their results to the actual (true) value of the measured quantity. Accuracy is determined by absolute and relative error indicators.

The absolute error is determined by the formula: Xn = Xn - X0,

where: Хп – error of the verified measuring instrument; Хп – the value of the same quantity found using the verified measuring instrument; X0 is the SI value taken as the basis for comparison, i.e. actual value.

However, the accuracy of measuring instruments is characterized to a greater extent by relative error, i.e. expressed as a percentage, the ratio of the absolute error to the actual value of the quantity measured or reproduced by SI data.

The standards normalize accuracy characteristics associated with other errors:

Systematic error is a component of the error of a measurement result that remains constant or changes naturally with repeated measurements of the same value. Such an error may occur if the center of gravity of the SI is shifted or the SI is not installed on a horizontal surface.

Random error is a component of the error of a measurement result that changes randomly in a series of repeated measurements of the same size quantity with the same care. Such errors are not natural, but are inevitable and are present in the measurement results.

The measurement error must not exceed the established limits, which are specified in the technical documentation for the device or in the standards for control methods (tests, measurements, analysis).

To eliminate significant errors, regular verification of measuring instruments is carried out, which includes a set of operations performed by state metrological service bodies or other authorized bodies in order to determine and confirm the compliance of the measuring instrument with the established technical requirements.

In everyday life production practice A generalized characteristic, the accuracy class, is widely used.

The accuracy class of measuring instruments is a generalized characteristic expressed by the limits of permissible errors, as well as other characteristics that affect the accuracy. Accuracy classes of a specific type of SI are set in regulatory documents. At the same time, for each accuracy class, specific requirements for metrological characteristics are established, which together reflect the level of accuracy of the measuring instruments of this class. The accuracy class allows you to judge the limits within which the measurement error of this class lies. This is important to know when choosing SI depending on the specified measurement accuracy.

Accuracy classes are designated as follows:

s If the limits of permissible basic error are expressed in the form of absolute SI error, then the accuracy class is designated in capital letters Roman alphabet. Accuracy classes, which correspond to smaller limits of permissible errors, are assigned letters located closer to the beginning of the alphabet.

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§ 32. ACCURACY AND ERROR OF MEASUREMENT.

No measurement can be made absolutely accurately. There is always some difference between the measured value of a quantity and its actual value, which is called measurement error. The smaller the measurement error, the naturally higher the measurement accuracy.

Measurement accuracy characterizes the error that is inevitable when working with the most precise measuring instrument or a device of a certain type. The measurement accuracy is influenced by the properties of the measuring tool material and the design of the tool. Accuracy of measurement can only be achieved if the measurement is carried out according to the rules.

The main reasons that reduce measurement accuracy may be:

1) unsatisfactory condition of the tool: damaged edges, dirt, incorrect position zero mark, malfunction;

2) careless handling of the tool (impacts, heat, etc.);

3) inaccuracy of installation of the tool or the part being measured relative to the tool;

4) the temperature difference at which the measurement is made (the normal temperature at which the measurement should be made is 20°);

5) poor knowledge of the device or inability to use a measuring instrument. Incorrect choice of measurement tool.

The degree of measurement accuracy of any device depends on its care, as well as its correct use.

Increased measurement accuracy can be achieved by repeating measurements and then determining the arithmetic mean obtained as a result of several measurements.

When starting to measure, you need to have a good knowledge of the measuring instruments, the rules for handling the instrument and master the techniques for using it.

The result of a measurement is the value of a quantity found by measuring it. The result obtained always contains some error.

Thus, the measurement task includes not only finding the value itself, but also estimating the error allowed during the measurement.

The absolute measurement error D refers to the deviation of the measurement result of a given value A from its true meaning A x

D= A – Ax. (B.1)

In practice, instead of the true value which is unknown, the actual value is usually used.

The error calculated using formula (B.1) is called the absolute error and is expressed in units of the measured value.

The quality of measurement results is usually conveniently characterized not by the absolute error D, but by its ratio to the measured value, which is called the relative error and is usually expressed as a percentage:

ε = (D / A) 100%. (B.2)

The relative error ε is the ratio of the absolute error to the measured value.

The relative error ε is directly related to the measurement accuracy.

Measurement accuracy is the quality of a measurement, reflecting the closeness of its results to the true value of the measured value. Measurement accuracy is the reciprocal of its relative error. High measurement accuracy corresponds to small relative errors.

The magnitude and sign of the error D depends on the quality of the measuring instruments, the nature and conditions of the measurements, and the experience of the observer.

All errors, depending on the reasons for their occurrence, are divided into three types: A) systematic; b) random; V) misses.

Systematic errors are errors whose magnitude is the same in all measurements carried out by the same method using the same measuring instruments.

Systematic errors can be divided into three groups.

1. Errors, the nature of which is known and the magnitude can be determined quite accurately. Such errors are called corrections. For example, A) when determining the length, the elongation of the measured body and the measuring ruler due to temperature changes; b) when determining weight - an error caused by “weight loss” in the air, the magnitude of which depends on temperature, humidity and atmospheric pressure air, etc.

The sources of such errors are carefully analyzed, the magnitude of the corrections is determined and taken into account in the final result.

2. Errors of measuring instruments δ cl t. For the convenience of comparing devices with each other, the concept of reduced error d pr (%) has been introduced

Where A k– some normalized value, for example, the final value of the scale, the sum of the values ​​of a two-sided scale, etc.

The accuracy class of a device d class t is a physical quantity that is numerically equal to the greatest permissible reduced error, expressed
as a percentage, i.e.

d cl p = d pr max

Electrical measuring instruments are usually characterized by an accuracy class ranging from 0.05 to 4.

If an accuracy class of 0.5 is indicated on the device, this means that the device readings have an error of up to 0.5% of the entire operating scale of the device. Errors in measuring instruments cannot be excluded, but they highest value Dmax can be determined.

The value of the maximum absolute error of a given device is calculated according to its accuracy class

(B.4)

When measuring with a device whose accuracy class is not specified, the absolute measurement error is usually equal to half the value of the smallest scale division.

3. The third type includes errors whose existence is not suspected. For example: it is necessary to measure the density of some metal; for this, the volume and mass of the sample are measured.

If the sample being measured contains voids inside, for example, air bubbles trapped during casting, then the density measurement is carried out with systematic errors, the magnitude of which is unknown.

Random errors are those errors whose nature and magnitude are unknown.

Random measurement errors arise due to the simultaneous influence on the measurement object of several independent quantities, the changes of which are of a fluctuation nature. It is impossible to exclude random errors from measurement results. It is only possible, on the basis of the theory of random errors, to indicate the limits between which the true value of the measured quantity lies, the probability of the true value being within these limits, and its most probable value.

Misses are observational errors. The source of errors is the lack of attention of the experimenter.

You should understand and remember:

1) if the systematic error is decisive, that is, its value is significantly greater than the random error inherent this method, then it is enough to perform the measurement once;

2) if random error is decisive, then the measurement should be performed several times;

3) if the systematic Dsi and random Dcl errors are comparable, then the total D total measurement error is calculated based on the law of adding errors, as their geometric sum

At practical use of certain measurement results, it is important to evaluate their accuracy. The term “measurement accuracy,” i.e., the degree of approximation of measurement results to a certain true value, does not have a strict definition and is used for qualitative comparison of measurement operations. For quantitative assessment, the concept of “measurement error” is used (the smaller the error, the higher the accuracy).

Error is the deviation of a measurement result from the true (actual) value of the measured quantity. It should be borne in mind that the true value of a physical quantity is considered unknown and is used in theoretical research. The actual value of a physical quantity is established experimentally under the assumption that the result of the experiment (measurement) is as close as possible to the true value. Estimation of measurement error is one of the important measures to ensure the uniformity of measurements.

The measurement error depends primarily on the SI errors, as well as on the conditions under which the measurement is carried out, on the experimental error of the technique and the subjective characteristics of a person in cases where he is directly involved in the measurements. Therefore, we can talk about several components of the measurement error or its total error.

The number of factors influencing the measurement accuracy is quite large, and any classification of measurement errors (Fig. 15) is to a certain extent arbitrary, since different errors depend on conditions measuring process appear in different groups.

Rice. 15. Classification of measurement errors

Types of errors

As mentioned above, measurement error is the deviation of the measurement result X from the true X and the value of the measured quantity. In this case, instead of the true value of the physical quantity X, its actual value X d is used.

Depending on the form of the expression, absolute, relative and reduced measurement errors are distinguished.

Absolute error is the error of a measuring instrument, expressed in units of the physical quantity being measured. It is defined as the difference Δ"= X i - X and or Δ = X - X d., where X i is the measurement result.

Relative error is the error of a measuring instrument, expressed as the ratio of the absolute error of the measuring instrument to the measurement result or the actual value of the physical quantity being measured. It is defined as the ratio δ = ±(Δ/Х d)·100%.

Reduced error is the error of the measuring instrument, expressed as the ratio of the absolute error of the measuring instrument to the conventionally accepted value of the quantity, constant over the entire measurement range Χ N.

Depending on the nature of the manifestation, the causes of occurrence and the possibilities of elimination, there are systematic and random measurement errors, as well as gross errors (misses ).

Systematic error– this is the error component taken as constant or changing naturally during repeated measurements of the same parameter. It is generally believed that systematic errors can be detected and eliminated. However, in real conditions it is impossible to completely eliminate these errors. There are always some non-excluded residuals that need to be taken into account in order to estimate their boundaries. This will be the systematic measurement error.

Random error is a component of the error that changes randomly under the same measurement conditions. The value of the random error is unknown in advance; it arises due to many unspecified factors. Random errors cannot be excluded from the results, but their influence can be reduced by statistical processing of measurement results.

The random and systematic components of the measurement error appear simultaneously, so that if they are independent, their total error is equal to the sum of the errors. In principle, the systematic error is also random and the indicated division is due only to the established traditions of processing and presenting measurement results.

Unlike random error, which is detected as a whole, regardless of its sources, systematic error is considered in its components depending on the sources of its occurrence. There are subjective, methodological and instrumental components of systematic error.

The subjective component of the error is associated with individual characteristics operator. Typically, this error occurs due to errors in readings and incorrect operator skills. Basically, systematic error arises due to methodological and instrumental components.

The methodological component of the error is due to the imperfection of the measurement method, methods of using measuring instruments, incorrect calculation formulas and rounding of results.

The instrumental component arises due to the actual error of the measuring instruments, determined by the class of its accuracy, the influence of the measuring instruments on the object of measurement and the limited resolution of the measuring instruments.

The expediency of dividing the systematic error into methodological and instrumental components is explained by the following:

· to improve the accuracy of measurements, limiting factors can be identified and, therefore, a decision can be made either to improve the methodology or to select more accurate measuring instruments;

· it becomes possible to determine the component of the total error, which increases either with time or under the influence of external factors, and, therefore, purposefully carry out periodic verification and certification;

· the instrumental component can be assessed after further development of the methodology, and the potential accuracy capabilities of the selected method will be determined only by the methodological component.

Gross errors (misses) arise due to erroneous operator actions, malfunction of measuring instruments or sudden changes in measurement conditions. As a rule, gross errors are identified as a result of statistical processing of measurement results using special criteria.