Control questions and exercises. Absolute and relative errors

Physical quantities are characterized by the concept of "error accuracy". There is a saying that by taking measurements one can come to knowledge. So it will be possible to find out what is the height of the house or the length of the street, like many others.

Introduction

Let's understand the meaning of the concept of "measure the value." The measurement process is to compare it with homogeneous quantities, which are taken as a unit.

Liters are used to determine volume, grams are used to calculate mass. To make it more convenient to make calculations, we introduced the SI system of the international classification of units.

For measuring the length of the bog in meters, mass - kilograms, volume - cubic liters, time - seconds, speed - meters per second.

When calculating physical quantities, it is not always necessary to use the traditional method; it is enough to apply the calculation using a formula. For example, to calculate indicators such as average speed, you need to divide the distance traveled by the time spent on the road. This is how the average speed is calculated.

Using units of measurement that are ten, one hundred, one thousand times higher than the indicators of the accepted measuring units, they are called multiples.

The name of each prefix corresponds to its multiplier number:

1. Deca.
2. Hecto.
3. Kilo.
4. Mega.
5. Giga.
6. Tera.

In physical science, a power of 10 is used to write such factors. For example, a million is denoted as 10 6 .

In a simple ruler, the length has a unit of measure - a centimeter. It is 100 times smaller than a meter. A 15 cm ruler is 0.15 m long.

A ruler is the simplest type of measuring instrument for measuring length. More complex devices are represented by a thermometer - so that a hygrometer - to determine humidity, an ammeter - to measure the level of force with which an electric current propagates.

How accurate will the measurements be?

Take a ruler and a simple pencil. Our task is to measure the length of this stationery.

First you need to determine what is the division value indicated on the scale of the measuring device. On the two divisions, which are the nearest strokes of the scale, numbers are written, for example, "1" and "2".

It is necessary to calculate how many divisions are enclosed in the interval of these numbers. If you count correctly, you get "10". Subtract from the number that is greater, the number that will be less, and divide by the number that makes up the divisions between the digits:

(2-1)/10 = 0.1 (cm)

So we determine that the price that determines the division of stationery is the number 0.1 cm or 1 mm. It is clearly shown how the price indicator for division is determined using any measuring device.

By measuring a pencil with a length that is slightly less than 10 cm, we will use the knowledge gained. If there were no small divisions on the ruler, the conclusion would follow that the object has a length of 10 cm. This approximate value is called the measurement error. It indicates the level of inaccuracy that can be tolerated in the measurement.

Determining the parameters of the length of a pencil with more high level accuracy, a larger division value achieves a greater measuring accuracy, which provides a smaller error.

In this case, absolutely accurate measurements cannot be made. And the indicators should not exceed the size of the division price.

It has been established that the dimensions of the measurement error are ½ of the price, which is indicated on the divisions of the instrument used to determine the dimensions.

After measuring the pencil at 9.7 cm, we determine the indicators of its error. This is a gap of 9.65 - 9.85 cm.

The formula that measures such an error is the calculation:

A = a ± D (a)

A - in the form of a quantity for measuring processes;

a - the value of the measurement result;

D - designation absolute error.

When subtracting or adding values ​​with an error, the result will be equal to the sum of the error indicators, which is each individual value.

Introduction to the concept

If we consider depending on the way it is expressed, we can distinguish the following varieties:

• Absolute.
• Relative.
• Given.

The absolute measurement error is indicated by the capital letter "Delta". This concept is defined as the difference between the measured and actual values ​​of that physical quantity which is measured.

The expression of the absolute measurement error is the units of the quantity that needs to be measured.

When measuring mass, it will be expressed, for example, in kilograms. This is not a measurement accuracy standard.

How to calculate the error of direct measurements?

There are ways to represent and calculate them. To do this, it is important to be able to determine the physical quantity with the required accuracy, to know what the absolute measurement error is, that no one will ever be able to find it. You can only calculate its boundary value.

Even if this term is conditionally used, it indicates precisely the boundary data. Absolute and relative measurement errors are indicated by the same letters, the difference is in their spelling.

When measuring length, the absolute error will be measured in those units in which the length is calculated. And the relative error is calculated without dimensions, since it is the ratio of the absolute error to the measurement result. This value is often expressed as a percentage or fractions.

The absolute and relative measurement errors have several different ways of calculating, depending on what physical quantities.

The concept of direct measurement

The absolute and relative error of direct measurements depend on the accuracy class of the device and the ability to determine the weighing error.

Before talking about how the error is calculated, it is necessary to clarify the definitions. A direct measurement is a measurement in which the result is directly read from the instrument scale.

When we use a thermometer, ruler, voltmeter or ammeter, we always carry out direct measurements, since we use a device with a scale directly.

There are two factors that affect performance:

• Instrument error.
• The error of the reference system.

The absolute error limit for direct measurements will be equal to the sum of the error that the device shows and the error that occurs during the reading process.

D = D (pr.) + D (absent)

Medical thermometer example

Accuracy values ​​are indicated on the instrument itself. An error of 0.1 degrees Celsius is registered on a medical thermometer. The reading error is half the division value.

D = C/2

If the division value is 0.1 degrees, then for a medical thermometer, calculations can be made:

D \u003d 0.1 o C + 0.1 o C / 2 \u003d 0.15 o C

On the back side of the scale of another thermometer there is a technical specification and it is indicated that for the correct measurements it is necessary to immerse the thermometer with the entire back part. The measurement accuracy is not specified. The only remaining error is the counting error.

If the division value of the scale of this thermometer is 2 o C, then you can measure the temperature with an accuracy of 1 o C. These are the limits of the permissible absolute measurement error and the calculation of the absolute measurement error.

A special system for calculating accuracy is used in electrical measuring instruments.

Accuracy of electrical measuring instruments

To specify the accuracy of such devices, a value called the accuracy class is used. For its designation, the letter "Gamma" is used. To accurately determine the absolute and relative measurement errors, you need to know the accuracy class of the device, which is indicated on the scale.

Take, for example, an ammeter. Its scale indicates the accuracy class, which shows the number 0.5. It is suitable for measurements on direct and alternating current, refers to the devices of the electromagnetic system.

This is a fairly accurate device. If you compare it with a school voltmeter, you can see that it has an accuracy class of 4. This value must be known for further calculations.

Application of knowledge

Thus, D c \u003d c (max) X γ / 100

This formula will be used for concrete examples. Let's use a voltmeter and find the error in measuring the voltage that the battery gives.

Let's connect the battery directly to the voltmeter, having previously checked whether the arrow is at zero. When the device was connected, the arrow deviated by 4.2 divisions. This state can be described as follows:

1. It can be seen that the maximum value of U for this item is 6.
2. Accuracy class -(γ) = 4.
3. U(o) = 4.2 V.
4. C=0.2 V

Using these formula data, the absolute and relative measurement errors are calculated as follows:

D U \u003d DU (ex.) + C / 2

D U (pr.) \u003d U (max) X γ / 100

D U (pr.) \u003d 6 V X 4/100 \u003d 0.24 V

This is the error of the device.

The calculation of the absolute measurement error in this case will be performed as follows:

D U = 0.24 V + 0.1 V = 0.34 V

Using the considered formula, you can easily find out how to calculate the absolute measurement error.

There is a rule for rounding errors. It allows you to find the average between the absolute error limit and the relative one.

Learning to determine the weighing error

This is one example of direct measurements. In a special place is weighing. After all, lever scales do not have a scale. Let's learn how to determine the error of such a process. The accuracy of mass measurement is affected by the accuracy of the weights and the perfection of the scales themselves.

We use a balance scale with a set of weights that must be placed exactly on the right side of the scale. Take a ruler for weighing.

Before starting the experiment, you need to balance the scales. We put the ruler on the left bowl.

The mass will be equal to the sum of the installed weights. Let us determine the measurement error of this quantity.

D m = D m (weights) + D m (weights)

The mass measurement error consists of two terms associated with scales and weights. To find out each of these values, at the factories for the production of scales and weights, products are supplied with special documents that allow you to calculate the accuracy.

Application of tables

Let's use a standard table. The error of the scale depends on how much mass is put on the scale. The larger it is, the larger the error, respectively.

Even if you put a very light body, there will be an error. This is due to the process of friction occurring in the axles.

The second table refers to a set of weights. It indicates that each of them has its own mass error. The 10-gram has an error of 1 mg, as well as the 20-gram. We calculate the sum of the errors of each of these weights, taken from the table.

It is convenient to write the mass and the mass error in two lines, which are located one under the other. The smaller the weight, the more accurate the measurement.

Results

In the course of the considered material, it was established that it is impossible to determine the absolute error. You can only set its boundary indicators. For this, the formulas described above in the calculations are used. This material proposed for study at school for students in grades 8-9. Based on the knowledge gained, it is possible to solve problems for determining the absolute and relative errors.

Absolute and relative error are used to evaluate the inaccuracy in the calculations made with high complexity. They are also used in various measurements and for rounding off calculation results. Consider how to determine the absolute and relative error.

Absolute error

The absolute error of the number name the difference between this number and its exact value.
Consider an example : 374 students study at the school. If this number is rounded up to 400, then the absolute measurement error is 400-374=26.

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

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

Δa=A-a. How to find the absolute error by the formula, we discussed above.

In practice, the absolute error is not enough to accurately evaluate the measurement. It is rarely possible to know exactly the value of the measured quantity in order to calculate the absolute error. If you measure a book 20 cm long and allow an error of 1 cm, you can read the measurement 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, the determination of the relative measurement error is more important.

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

Relative error

Relative error called the ratio of the absolute error of a number to the number itself. To calculate the relative error in the student example, 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 the 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 10% and 0.1%, respectively. For a segment with a length of 10 cm, the error of 1 cm is very large, this is an error of 10%. And for a ten-meter segment, 1 cm does not matter, only 0.1%.

There are systematic and random errors. The systematic error 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 their absolute errors;
• when dividing and multiplying numbers, it is required to add relative errors;
• when exponentiated, 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 the correct and doubtful numbers.

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

In physics and other sciences, it is very often necessary to measure 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 devices.

Measuring device called a device by which a measured quantity is compared 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 scales 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 dependence. For example, determining the circumference of a circle based on the results of measuring the diameter or determining the volume of a body based on the results of measuring its linear dimensions.

Due to the imperfection of measuring instruments, our sense organs, the influence of external influences on the measuring equipment and the object of measurement, 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 quantity, but only an approximate one. If, for example, body weight is determined with an accuracy of 0.1 mg, then this means that the found weight differs from the true body weight by less than 0.1 mg.

Accuracy of measurements - a characteristic of the quality of measurements, reflecting the proximity of the measurement results to the true value of the measured quantity.

The smaller the measurement errors, the greater the measurement accuracy. The measurement accuracy depends on the instruments used in the measurements and on the general measurement methods. It is absolutely useless to try to go beyond this limit of accuracy when making measurements under given conditions. It is possible to minimize the impact of causes 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 made not once, but several times under the same experimental conditions.

As a result of the i-th measurement (i is the measurement number) of the value "X", an approximate number X i is obtained, which differs from the true value Xist by some value ∆X i = |X i - X|, which is a mistake 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 the X value obtained during the measurement (that is, the measured value); ∆X is the absolute error in determining the value of X.

Absolute error (error) of measurement ∆X is the absolute value of the difference between the true value of the measured quantity Xist and the measurement result X i: ∆X = |X ist - X i |.

Relative error (error) measurement δ (characterizing the measurement accuracy) is numerically equal to the ratio of the absolute measurement error ∆X to the true value of the measured value X sist (often expressed as a percentage): δ \u003d (∆X / X sist) 100% .

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

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

In any measuring device, one or another systematic error is inherent, 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 of 0.01 g more than indicated on it, then the found value of the body weight 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, which we can only say that they do not exceed a certain value.

Random mistakes called errors that change their magnitude and sign in an unpredictable way from experience to experience. The appearance of random errors is due to the action of many diverse and uncontrollable causes.

For example, when weighing with a balance, these reasons can be air vibrations, settled dust particles, different friction in the left and right suspension of the cups, etc. Random errors manifest themselves in the fact that, having measured the same X value under the same experimental conditions, we different values: X1, X2, X3,…, X i ,…, X n , where X i is the result of the i-th measurement. It is not possible to establish any regularity between the results, therefore the result of the i -th measurement of X is considered random variable. Random errors may have a certain effect 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.

Misses and blunders– 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 reading of the device “212”, a completely different number is written - “221”). Measurements containing misses and gross errors should be discarded.

Measurements can be made in terms of their accuracy by technical and laboratory methods.

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

With laboratory measurement methods, it is required to more accurately indicate the value of the measured quantity than its single measurement by the technical method allows. 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 (accounting for random errors).

From the possibility of carrying out measurements by two methods, the existence of two methods for assessing the accuracy of measurements follows: technical and laboratory.

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.

Since it is impossible to find out with absolute accuracy the true value of any quantity, it is also impossible to indicate the magnitude of the deviation of the measured value from the true one. (This deviation is usually called the measurement error. In a number of sources, for example, in the Great Soviet Encyclopedia, the terms measurement error and measurement error are used as synonyms, but according to RMG 29-99 the term measurement error not recommended as less successful). It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. In practice, instead of the true value, we use actual value x d, that is, the value of a physical quantity obtained experimentally and so close to the true value that it can be used instead of it in the set measurement task. Such a value is usually calculated as the average value obtained by statistical processing of the results of a series of measurements. This value obtained is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, along with the result obtained, the measurement error is indicated. For example, the entry T=2.8±0.1 c. means that the true value of the quantity T lies in the interval from 2.7 s before 2.9 s with some specified probability

In 2004, at the international level was adopted new document, dictating the conditions for carrying out measurements and establishing new rules for comparing state standards. The concept of "error" became obsolete, the concept of "measurement uncertainty" was introduced instead, however, GOST R 50.2.038-2004 allows the use of the term error for documents used in Russia.

There are the following types of errors:

The absolute error

Relative error

the reduced error;

The main error

Additional error

· systematic error;

Random error

Instrumental error

· methodical error;

· personal error;

· static error;

dynamic error.

Measurement errors are classified according to the following criteria.

· According to the method of mathematical expression, the errors are divided into absolute errors and relative errors.

· According to the interaction of changes in time and the input value, the errors are divided into static errors and dynamic errors.

By the nature of the occurrence of errors are divided into systematic errors and random errors.

· According to the nature of the dependence of the error on the influencing values, the errors are divided into basic and additional.

· According to the nature of the dependence of the error on the input value, the errors are divided into additive and multiplicative.

Absolute error is the value calculated as the difference between the value of the quantity obtained during the measurement process and the real (actual) value of the given quantity. The absolute error is calculated using the following formula:

AQ n =Q n /Q 0 , where AQ n is the absolute error; Qn- the value of a certain quantity obtained in the process of measurement; Q0- the value of the same quantity, taken as the base of comparison (real value).

Absolute error of measure is the value calculated as the difference between the number, which is the nominal value of the measure, and the real (actual) value of the quantity reproduced by the measure.

Relative error is a number that reflects the degree of accuracy of the measurement. The relative error is calculated using the following formula:

Where ∆Q is the absolute error; Q0 is the real (actual) value of the measured quantity. Relative error is expressed as a percentage.

Reduced error is the value calculated as the ratio of the absolute error value to the normalizing value.

The normalizing value is defined as follows:

For measuring instruments for which a nominal value is approved, this nominal value is taken as a normalizing value;

· for measuring instruments, in which the zero value is located on the edge of the measurement scale or outside the scale, the normalizing value is taken equal to the final value from the measurement range. The exception is measuring instruments with a significantly uneven measurement scale;

· for measuring instruments, in which the zero mark is located within the measurement range, the normalizing value is taken equal to the sum of the final numerical values ​​of the measurement range;

For measuring instruments (measuring instruments) with an uneven scale, the normalizing value is taken equal to the entire length of the measurement scale or the length of that part of it that corresponds to the measurement range. The absolute error is then expressed in units of length.

Measurement error includes instrumental error, methodological error and reading error. Moreover, the reading error arises due to the inaccuracy in determining the division fractions of the measurement scale.

Instrumental error- this is the error arising due to the errors made in the manufacturing process of the functional parts of the error measuring instruments.

Methodological error is an error due to the following reasons:

· inaccuracy in building a model of the physical process on which the measuring instrument is based;

Incorrect use of measuring instruments.

Subjective error- this is an error arising due to the low degree of qualification of the operator of the measuring instrument, as well as due to the error of the human visual organs, i.e. the human factor is the cause of the subjective error.

Errors in the interaction of changes in time and the input value are divided into static and dynamic errors.

Static error- this is the error that occurs in the process of measuring a constant (not changing in time) value.

Dynamic error- this is an error, the numerical value of which is calculated as the difference between the error that occurs when measuring a non-constant (variable in time) quantity, and a static error (the error in the value of the measured quantity at a certain point in time).

According to the nature of the dependence of the error on the influencing quantities, the errors are divided into basic and additional.

Basic error is the error obtained under normal operating conditions of the measuring instrument (at normal values ​​of the influencing quantities).

Additional error- this is the error that occurs when the values ​​of the influencing quantities do not correspond to their normal values, or if the influencing quantity goes beyond the boundaries of the area of ​​normal values.

Normal conditions are the conditions under which all values ​​of the influencing quantities are normal or do not go beyond the boundaries of the range of normal values.

Working conditions- these are conditions in which the change in the influencing quantities has a wider range (the values ​​of the influencing ones do not go beyond the boundaries of the working range of values).

Working range of values ​​of the influencing quantity is the range of values ​​in which the values ​​of the additional error are normalized.

According to the nature of the dependence of the error on the input value, the errors are divided into additive and multiplicative.

Additive error- this is the error that occurs due to the summation of numerical values ​​and does not depend on the value of the measured quantity, taken modulo (absolute).

Multiplicative error- this is an error that changes along with a change in the values ​​​​of the quantity being measured.

It should be noted that the value of the absolute additive error is not related to the value of the measured quantity and the sensitivity of the measuring instrument. Absolute additive errors are unchanged over the entire measurement range.

The value of the absolute additive error determines the minimum value of the quantity that can be measured by the measuring instrument.

The values ​​of multiplicative errors change in proportion to changes in the values ​​of the measured quantity. The values ​​of multiplicative errors are also proportional to the sensitivity of the measuring instrument. The multiplicative error arises due to the influence of influencing quantities on the parametric characteristics of the instrument elements.

Errors that may occur during the measurement process are classified according to the nature of their occurrence. Allocate:

systematic errors;

random errors.

Gross errors and misses may also appear in the measurement process.

Systematic error- this is component the entire error of the measurement result, which does not change or changes naturally with repeated measurements of the same value. Usually, systematic error is tried to be eliminated. possible ways(for example, by using measurement methods that reduce the likelihood of its occurrence), but if a systematic error cannot be excluded, then it is calculated before the start of measurements and appropriate corrections are made to the measurement result. In the process of normalizing the systematic error, the boundaries of its admissible values ​​are determined. The systematic error determines the correctness of measurements of measuring instruments (metrological property). Systematic errors in some cases can be determined experimentally. The measurement result can then be refined by introducing a correction.

Methods for eliminating systematic errors are divided into four types:

elimination of the causes and sources of errors before the start of measurements;

· Elimination of errors in the process of already begun measurement by methods of substitution, compensation of errors in sign, oppositions, symmetrical observations;

Correction of measurement results by making an amendment (elimination of errors by calculations);

Determining the limits of systematic error in case it cannot be eliminated.

Elimination of the causes and sources of errors before the start of measurements. This method is the best option, since its use simplifies the further course of measurements (there is no need to eliminate errors in the process of an already started measurement or make corrections to the result).

To eliminate systematic errors in the process of an already started measurement, apply various ways

Amendment Method is based on knowledge of the systematic error and the current patterns of its change. When using this method, the measurement result obtained with systematic errors is subject to corrections equal in magnitude to these errors, but opposite in sign.

substitution method consists in the fact that the measured value is replaced by a measure placed in the same conditions in which the object of measurement was located. The substitution method is used when measuring the following electrical parameters: resistance, capacitance and inductance.

Sign error compensation method consists in the fact that the measurements are performed twice in such a way that the error, unknown in magnitude, is included in the measurement results with the opposite sign.

Contrasting method similar to sign-based compensation. This method consists in the fact that measurements are performed twice in such a way that the source of error in the first measurement has the opposite effect on the result of the second measurement.

random error- this is a component of the error of the measurement result, which changes randomly, irregularly during repeated measurements of the same value. The occurrence of a random error cannot be foreseen and predicted. Random error cannot be completely eliminated; it always distorts the final measurement results to some extent. But you can make the measurement result more accurate by taking repeated measurements. The cause of a random error can be, for example, a random change in external factors affecting the measurement process. A random error during multiple measurements with a sufficiently high degree of accuracy leads to scattering of the results.

Misses and blunders are errors that are much larger than the systematic and random errors expected under the given measurement conditions. Slips and gross errors may appear due to gross errors in the measurement process, a technical malfunction of the measuring instrument, and unexpected changes in external conditions.

Measurement error

Measurement error- assessment of the deviation of the value of the measured value of the 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 conditionally accepted value of the quantity, which is constant over the entire measurement range or in part of the range. Calculated according to the formula

where X n- normalizing value, which depends on the type of measuring instrument scale and is determined by its graduation:

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

The given error is a dimensionless value (it 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 the imperfection of the operating principle, the inaccuracy of the scale graduation, and the 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 due to the degree of attentiveness, concentration, preparedness and other qualities of the operator.

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

If the device is operated under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by temperature deviation environment from normal, installation, due to the deviation of the position of the device from the normal operating position, etc. 20°C is taken as normal ambient temperature, and 01.325 kPa as normal atmospheric pressure.

A generalized characteristic of measuring instruments is an accuracy class determined by the limit values ​​of the permissible basic and additional errors, as well as other parameters that affect the accuracy of measuring instruments; the value of the parameters is established by the standards for certain types of measuring instruments. The accuracy class of measuring instruments characterizes their accuracy 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 given in the form of reduced basic (relative) errors, are assigned accuracy classes selected from a number of the following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0 ;5.0;6.0)*10n, where n = 1; 0; -one; -2 etc.

According to the nature of the manifestation

• random error- error, changing (in magnitude and in sign) from measurement to measurement. Random errors can be associated with the imperfection of devices (friction in mechanical devices, etc.), shaking in urban conditions, with the imperfection of the object of measurement (for example, when measuring the diameter of a thin wire, which may not have a completely round cross section as a result of the imperfection of the manufacturing process ), with the features 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 can be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
• Progressive (drift) error is an unpredictable error that changes slowly over time. It is a non-stationary random process.
• Gross error (miss)- an error resulting from an oversight of the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the division number on the scale of the device, if there was a short circuit in the electrical circuit).

According to the method of measurement

• Accuracy of direct measurements
• Uncertainty of indirect measurements- error of the calculated (not measured directly) value:

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

see also

• Measurement of physical quantities
• System for automated data collection from meters over the air

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. and others; ed. Goldina L. L. - M .: Science. Main edition of physical and mathematical literature, 1983. - 704 p.

Wikimedia Foundation. 2010 .

time measurement error- laiko matavimo paklaida statusas T sritis automatika atitikmenys: engl. time measuring error vok. Zeitmeßfehler, m rus. time measurement error, fpranc. erreur de mesure de temps, f … Automatikos terminų žodynas

systematic error (measurement)- introduce a systematic error - Topics oil and gas industry Synonyms introduce a systematic error EN bias ...

STANDARD ERRORS OF MEASUREMENT- Evaluation of the extent to which it can be expected that a certain set of measurements obtained in a given situation (for example, in a test or in one of several parallel forms of the test) will deviate from the true values. Designated as a (M) ...

overlay error- Caused by the superposition of short response output pulses when the time interval between input current pulses is less than the duration of a single response output pulse. Overlay errors can be ... ... Technical Translator's Handbook

error- 01.02.47 error (digital data) (1-4): The result of collecting, storing, processing and transmitting data, in which the bit or bits take inappropriate values, or there are not enough bits in the data stream. 4) Terminological ... ... Dictionary-reference book of terms of normative and technical documentation

There is no movement, said the bearded sage. The other was silent and began to walk before him. He could not have objected more strongly; All praised the convoluted answer. But, gentlemen, this funny case Another example brings me to mind: After all, every day ... Wikipedia

ERROR OPTIONS- The size of the variance, which cannot be explained by controllable factors. The error of variance is offset by sampling errors, measurement errors, experimental errors, etc… Dictionary in psychology