Measurement error assessment is absolute and relative. Absolute and relative error

Errors in measurements of physical quantities

1.Introduction(measurement and measurement error)

2.Random and systematic errors

3.Absolute and relative errors

4. Errors of measuring instruments

5. Accuracy class of electrical measuring instruments

6.Reading error

7.Total absolute error of direct measurements

8.Recording the final result of direct measurement

9. Errors of indirect measurements

10.Example

1. Introduction(measurement and measurement error)

Physics as a science was born more than 300 years ago, when Galileo essentially created the scientific study of physical phenomena: physical laws are established and tested experimentally by accumulating and comparing experimental data, represented by a set of numbers, laws are formulated in the language of mathematics, i.e. using formulas that connect numerical values of physical quantities by functional dependence. Therefore, physics is an experimental science, physics is a quantitative science.

Let's get acquainted with some characteristic features of any measurements.

Measurement is finding the numerical value of a physical quantity experimentally using measuring instruments (ruler, voltmeter, watch, etc.).

Measurements can be direct or indirect.

Direct measurement is finding the numerical value of a physical quantity directly by means of measurement. For example, length - with a ruler, atmospheric pressure - with a barometer.

Indirect measurement is finding the numerical value of a physical quantity using a formula that connects the desired quantity with other quantities determined by direct measurements. For example, the resistance of a conductor is determined by the formula R=U/I, where U and I are measured by electrical measuring instruments.

Let's look at an example of measurement.

Measure the length of the bar with a ruler (division value is 1 mm). We can only say that the length of the bar is between 22 and 23 mm. The width of the interval of “unknown” is 1 mm, that is, equal to the division price. Replacing the ruler with a more sensitive device, such as a caliper, will reduce this interval, which will lead to increased measurement accuracy. In our example, the measurement accuracy does not exceed 1mm.

Therefore, measurements can never be made absolutely accurately. The result of any measurement is approximate. Uncertainty in measurement is characterized by error - the deviation of the measured value of a physical quantity from its true value.

Let us list some of the reasons leading to errors.

1. Limited manufacturing accuracy of measuring instruments.

2. Impact on measurement external conditions(temperature change, voltage fluctuation...).

3. Actions of the experimenter (delay in starting the stopwatch, different eye positions...).

4. The approximate nature of the laws used to find measured quantities.

The listed causes of errors cannot be eliminated, although they can be minimized. To establish the reliability of conclusions obtained as a result of scientific research, there are methods for assessing these errors.

2. Random and systematic errors

Errors arising during measurements are divided into systematic and random.

Systematic errors are errors corresponding to the deviation of the measured value from the true value of a physical quantity, always in one direction (increase or decrease). With repeated measurements, the error remains the same.

Reasons for systematic errors:

1) non-compliance of measuring instruments with the standard;

2) incorrect installation of measuring instruments (tilt, imbalance);

3) discrepancy between the initial indicators of the instruments and zero and ignoring the corrections that arise in connection with this;

4) discrepancy between the measured object and the assumption about its properties (presence of voids, etc.).

Random errors are errors that change their numerical value in an unpredictable way. Such errors are caused by a large number of uncontrollable reasons that affect the measurement process (irregularities on the surface of the object, wind blowing, power surges, etc.). The influence of random errors can be reduced by repeating the experiment many times.

3. Absolute and relative errors

To quantify the quality of measurements, the concepts of absolute and relative measurement errors are introduced.

As already mentioned, any measurement gives only an approximate value of a physical quantity, but you can specify an interval that contains its true value:

A pr - D A< А ист < А пр + D А

Value D A is called the absolute error in measuring the quantity A. The absolute error is expressed in units of the quantity being measured. The absolute error is equal to the modulus of the maximum possible deviation of the value of a physical quantity from the measured value. And pr is the value of a physical quantity obtained experimentally; if the measurement was carried out repeatedly, then the arithmetic mean of these measurements.

But to assess the quality of measurement it is necessary to determine the relative error e. e = D A/A pr or e= (D A/A pr)*100%.

If a relative error of more than 10% is obtained during a measurement, then they say that only an estimate of the measured value has been made. In physics workshop laboratories, it is recommended to carry out measurements with a relative error of up to 10%. In scientific laboratories, some precise measurements (for example, determining the wavelength of light) are performed with an accuracy of millionths of a percent.

4. Errors of measuring instruments

These errors are also called instrumental or instrumental. They are determined by the design of the measuring device, the accuracy of its manufacture and calibration. Usually they are content with the permissible instrumental errors reported by the manufacturer in the passport for this device. These permissible errors are regulated by GOSTs. This also applies to standards. Usually the absolute instrumental error is denoted D and A.

If there is no information about the permissible error (for example, with a ruler), then half the division value can be taken as this error.

When weighing, the absolute instrumental error consists of the instrumental errors of the scales and weights. The table shows the most common permissible errors

measuring instruments encountered in school experiments.

Measuring	Measurement limit	Value of division	Allowable error
student ruler
demonstration ruler
measuring tape
beaker
weights 10,20, 50 mg
weights 100,200 mg
weights 500 mg







calipers
micrometer
dynamometer
training scales
Stopwatch			1s in 30 min
aneroid barometer	720-780 mm Hg.	1 mmHg	3 mmHg
laboratory thermometer	0-100 degrees C
school ammeter
school voltmeter

5. Accuracy class of electrical measuring instruments

Switch electric measuring instruments By acceptable values errors are divided into accuracy classes, which are indicated on instrument scales by numbers 0.1; 0.2; 0.5; 1.0; 1.5; 2.5; 4.0. Accuracy class g pr The device shows what percentage the absolute error is from the entire scale of the device.

g pr = (D and A/A max)*100% .

For example, the absolute instrumental error of a class 2.5 device is 2.5% of its scale.

If the accuracy class of the device and its scale are known, then the absolute instrumental measurement error can be determined

D and A = (g pr * A max)/100.

To increase the accuracy of measurements with a pointer electrical measuring instrument, it is necessary to select a device with such a scale that during the measurement process it is located in the second half of the instrument scale.

6. Reading error

The reading error results from insufficiently accurate readings of the measuring instruments.

In most cases, the absolute reading error is taken equal to half the division value. Exceptions are made when measuring with a clock (the hands move jerkily).

The absolute error of reading is usually denoted D oA

7. Total absolute error of direct measurements

When performing direct measurements of physical quantity A, the following errors must be assessed: D and A, D oA and D сА (random). Of course, other sources of errors associated with incorrect installation instruments, misalignment of the initial position of the instrument needle with 0, etc. must be excluded.

The total absolute error of direct measurement must include all three types of errors.

If the random error is small compared to the smallest value that can be measured by a given measuring instrument (compared to the division value), then it can be neglected and then one measurement is sufficient to determine the value of a physical quantity. Otherwise, probability theory recommends finding the measurement result as the arithmetic mean value of the results of the entire series of multiple measurements, and calculating the error of the result using the method of mathematical statistics. Knowledge of these methods goes beyond the school curriculum.

8. Recording the final result of direct measurement

The final result of measuring the physical quantity A should be written in this form;

A=A pr + D A, e= (D A/A pr)*100%.

And pr is the value of a physical quantity obtained experimentally; if the measurement was carried out repeatedly, then the arithmetic mean of these measurements. D A is the total absolute error of direct measurement.

Absolute error is usually expressed in one significant figure.

Example: L=(7.9 + 0.1) mm, e=13%.

9. Errors of indirect measurements

When processing the results of indirect measurements of a physical quantity that is functionally related to physical quantities A, B and C, which are measured directly, the relative error of the indirect measurement is first determined e=D X/X pr, using the formulas given in the table (without evidence).

The absolute error is determined by the formula D X=X pr *e,

where e expressed as a decimal fraction rather than a percentage.

The final result is recorded in the same way as in the case of direct measurements.

Function type	Formula
X=A+B+C
X=A-B
X=ABC
X=A n
X=A/B

Example: Let's calculate the error in measuring the friction coefficient using a dynamometer. The experiment consists of pulling a block evenly over a horizontal surface and measuring the applied force: it is equal to the sliding friction force.

Using a dynamometer, weigh the block with weights: 1.8 N. F tr =0.6 N

μ = 0.33. The instrumental error of the dynamometer (we find it from the table) is Δ and = 0.05 N, Reading error (half the division value)

Δ o =0.05 N. The absolute error in measuring weight and friction force is 0.1 N.

Relative measurement error (5th line in the table)

, therefore the absolute error of indirect measurement μ is 0.22*0.33=0.074

Based on precise natural sciences measurements lie. When measuring, the values of quantities are expressed in the form of numbers that indicate how many times the measured quantity is greater or less than another quantity, the value of which is taken as a unit. The numerical values of various quantities obtained as a result of measurements may depend on each other. The relationship between such quantities is expressed in the form of formulas that show how the numerical values of some quantities can be found from the numerical values of others.

Errors inevitably occur during measurements. It is necessary to master the methods used in processing the results obtained from measurements. This will allow you to learn how to obtain results that are closest to the truth from a set of measurements, notice inconsistencies and errors in a timely manner, intelligently organize the measurements themselves and correctly assess the accuracy of the obtained values.

If the measurement consists of comparing a given quantity with another, homogeneous quantity taken as a unit, then the measurement in this case is called direct.

Direct (direct) measurements- these are measurements in which we obtain the numerical value of the measured quantity either by direct comparison with a measure (standard), or with the help of instruments calibrated in units of the measured quantity.

However, such a comparison is not always made directly. In most cases, it is not the quantity that interests us that is measured, but other quantities associated with it by certain relationships and patterns. In this case, to measure the required quantity, it is necessary to first measure several other quantities, the value of which determines the value of the desired quantity by calculation. This measurement is called indirect.

Indirect measurements consist of direct measurements of one or more quantities associated with the quantity being determined by a quantitative dependence, and calculations of the quantity being determined from these data.

Measurements always involve measuring instruments that match one value with another associated with it, available quantification using our senses. For example, the current strength is matched by the angle of deflection of the arrow on a graduated scale. In this case, two main conditions of the measurement process must be met: unambiguity and reproducibility of the result. these two conditions are always only approximately satisfied. That's why The measurement process contains, along with finding the desired value, an assessment of the measurement inaccuracy.

A modern engineer must be able to evaluate the error of measurement results taking into account the required reliability. That's why great attention is devoted to the processing of measurement results. Familiarity with the basic methods of calculating errors is one of the main tasks of the laboratory workshop.

Why do errors occur?

There are many reasons for measurement errors to occur. Let's list some of them.

· processes occurring during the interaction of the device with the measurement object inevitably change the measured value. For example, measuring the dimensions of a part using a caliper leads to compression of the part, that is, to a change in its dimensions. Sometimes the influence of the device on the measured value can be made relatively small, but sometimes it is comparable or even exceeds the measured value itself.

· Any device has limited capabilities for unambiguously determining the measured value due to its design imperfections. For example, friction between various parts in the pointer block of an ammeter leads to the fact that a change in current by a certain small, but finite, value will not cause a change in the angle of deflection of the pointer.

· Always participates in all processes of interaction between the device and the measurement object. external environment, the parameters of which can change and, often, in unpredictable ways. This limits the reproducibility of the measurement conditions, and therefore the measurement result.

· When taking instrument readings visually, there may be ambiguity in reading the instrument readings due to the limited capabilities of our eye meter.

· Most quantities are determined indirectly based on our knowledge of the relationship of the desired quantity with other quantities directly measured by instruments. Obviously, the error of indirect measurement depends on the errors of all direct measurements. In addition, the limitations of our knowledge about the measured object, the simplification of the mathematical description of the relationships between quantities, and ignoring the influence of those quantities whose influence is considered insignificant during the measurement process contribute to errors in indirect measurement.

Error classification

Error value measurements of a certain quantity are usually characterized by:

1. Absolute error - the difference between the experimentally found (measured) and the true value of a certain quantity

. (1)

The absolute error shows how much we are mistaken when measuring a certain value of X.

2. Relative error equal to the ratio absolute error to true meaning measured quantity X

The relative error shows by what fraction of the true value of X we are mistaken.

Quality the results of measurements of some quantity are characterized by a relative error. The value can be expressed as a percentage.

From formulas (1) and (2) it follows that in order to find the absolute and relative measurement errors, 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 unknown value of a certain quantity and to find, if not its true value, then at least a value that differs quite slightly from it. Therefore, formulas (1) and (2), which determine the magnitude of errors, are not suitable in practice. In practical measurements, errors are not calculated, but rather estimated. The assessments take into account the experimental conditions, the accuracy of the methodology, the quality of the instruments and a number of other factors. Our task: to learn how to construct an experimental methodology and correctly use the data obtained from experience in order to find values of measured quantities that are sufficiently close to the true values, and to reasonably evaluate measurement errors.

Speaking about measurement errors, we should first of all mention gross errors (misses) arising due to the experimenter’s oversight or equipment malfunction. Serious mistakes should be avoided. If it is determined that they have occurred, the corresponding measurements must be discarded.

Experimental errors not associated with gross errors are divided into random and systematic.

Withrandom errors. Repeating the same measurements many times, you can notice that quite often their results are not exactly equal to each other, but “dance” around some average (Fig. 1). Errors that change magnitude and sign from experiment to experiment are called random. Random errors are involuntarily introduced by the experimenter due to the imperfection of the sense organs, random external factors etc. If the error of each individual measurement is fundamentally unpredictable, then they randomly change the value of the measured quantity. These errors can only be assessed using statistical processing of multiple measurements of the desired quantity.

Systematic errors may be associated with instrument errors (incorrect scale, unevenly stretching spring, uneven micrometer screw pitch, unequal balance arms, etc.) and with the experiment itself. They retain their magnitude (and sign!) during the experiment. As a result of systematic errors, the experimental results scattered due to random errors do not fluctuate around the true value, but around a certain biased value (Fig. 2). the error of each measurement of the desired quantity can be predicted in advance, knowing the characteristics of the device.

Calculation of errors of direct measurements

Systematic errors. Systematic errors naturally change the values of the measured quantity. The errors introduced into measurements by instruments are most easily assessed if they are related to design features the devices themselves. These errors are indicated in the passports for the devices. The errors of some devices can be assessed without referring to the data sheet. For many electrical measuring instruments, their accuracy class is indicated directly on the scale.

Instrument accuracy class is the ratio of the absolute error of the device to maximum value measured quantity, which can be determined using a given device (this is the systematic relative error of a given device, expressed as a percentage of the scale rating).

Then the absolute error of such a device is determined by the relation:

For electrical measuring instruments, 8 accuracy classes have been introduced: 0.05; 0.1; 0.5; 1.0; 1.5; 2.0; 2.5; 4.

The closer the measured value is to the nominal value, the more accurate the measurement result will be. The maximum accuracy (i.e., the smallest relative error) that a given device can provide is equal to the accuracy class. This circumstance must be taken into account when using multiscale instruments. The scale must be selected in such a way that the measured value, while remaining within the scale, is as close as possible to the nominal value.

If the accuracy class for the device is not specified, then the following rules must be followed:

· The absolute error of instruments with a vernier is equal to the accuracy of the vernier.

· The absolute error of instruments with a fixed arrow pitch is equal to the division value.

· The absolute error of digital devices is equal to one minimum digit.

· For all other instruments, the absolute error is assumed to be equal to half the division value.

Random errors. These errors are statistical in nature and are described by probability theory. It has been established that at very large quantities measurements, the probability of obtaining one or another result in each individual measurement can be determined using the Gaussian normal distribution. With a small number of measurements mathematical description the probability of obtaining one or another measurement result is called the Student distribution (you can read more about this in the manual “Measurement errors of physical quantities”).

How to evaluate the true value of the measured quantity?

Suppose that when measuring a certain value we received N results: . The arithmetic mean of a series of measurements is closer to the true value of the measured quantity than most individual measurements. To obtain the result of measuring a certain value, the following algorithm is used.

1). Calculated average series of N direct measurements:

2). Calculated absolute random error of each measurement is the difference between the arithmetic mean of a series of N direct measurements and this measurement:

3). Calculated mean square absolute error:

4). Calculated absolute random error. With a small number of measurements, the absolute random error can be calculated through the mean square error and a certain coefficient called the Student coefficient:

The Student coefficient depends on the number of measurements N and the reliability coefficient (Table 1 shows the dependence of the Student coefficient on the number of measurements at a fixed value of the reliability coefficient).

Reliability factor is the probability with which the true value of the measured quantity falls within confidence interval.

Confidence interval - This numeric interval, into which the true value of the measured quantity falls with a certain probability.

Thus, the Student coefficient is the number by which the mean square error must be multiplied in order to ensure the specified reliability of the result for a given number of measurements.

The greater the reliability required for a given number of measurements, the greater the Student coefficient. On the other hand, the greater the number of measurements, the lower the Student coefficient for a given reliability. In the laboratory work of our workshop, we will assume that the reliability is given and equal to 0.9. Numeric values Student's coefficients for this reliability for different numbers of measurements are given in Table 1.

Table 1

Number of measurements N

Student's coefficient

5). Calculated total absolute error. In any measurement, there are both random and systematic errors. Calculating the total (total) absolute measurement error is not an easy task, since these errors are of different natures.

For engineering measurements, it makes sense to sum up the systematic and random absolute errors

For simplicity of calculations, it is customary to estimate the total absolute error as the sum of the absolute random and absolute systematic (instrumental) errors, if the errors are of the same order of magnitude, and to neglect one of the errors if it is more than an order of magnitude (10 times) less than the other.

6). The error and the result are rounded. Since the measurement result is presented as an interval of values, the value of which is determined by the total absolute error, correct rounding of the result and error is important.

Rounding begins with absolute error!!! The number of significant figures that is left in the error value, generally speaking, depends on the reliability coefficient and the number of measurements. However, even for very precise measurements (for example, astronomical), in which exact value errors are important, do not leave more than two significant figures. A larger number of numbers does not make sense, since the definition of error itself has its own error. Our practice has a relatively small reliability coefficient and a small number of measurements. Therefore, when rounding (with excess), the total absolute error is left to one significant figure.

The digit of the significant digit of the absolute error determines the digit of the first doubtful digit in the result value. Consequently, the value of the result itself must be rounded (with correction) to that significant digit whose digit coincides with the digit of the significant digit of the error. The formulated rule should also be applied in cases where some of the numbers are zeros.

If the result obtained when measuring body weight is , then it is necessary to write zeros at the end of the number 0.900. The recording would mean that nothing was known about the next significant figures, while the measurements showed that they were zero.

7). Calculated relative error.

When rounding the relative error, it is enough to leave two significant figures.

R the result of a series of measurements of a certain physical quantity is presented in the form of an interval of values, indicating the probability of the true value falling into this interval, that is, the result must be written in the form:

Here is the total absolute error, rounded to the first significant digit, and is the average value of the measured value, rounded taking into account the already rounded error. When recording a measurement result, you must indicate the unit of measurement of the value.

Let's look at a few examples:

1. Suppose that when measuring the length of a segment, we obtained the following result: cm and cm. How to correctly write down the result of measuring the length of a segment? First, we round off the absolute error with excess, leaving one significant digit, see. Significant digit of the error in the hundredths place. Then, with the correction, we round the average value to the nearest hundredth, i.e., to the significant digit whose digit coincides with the digit of the significant digit of the error see Calculate the relative error

cm; ; .

2. Let us assume that when calculating the conductor resistance we obtained the following result: And . First, we round the absolute error, leaving one significant figure. Then we round the average to the nearest integer. Calculate the relative error

We write the measurement result as follows:

; ; .

3. Suppose that when calculating the mass of the load we received the following result: kg and kg. First, we round the absolute error, leaving one significant figure kg. Then we round the average to the nearest tens kg. Calculate the relative error

Questions and tasks on the theory of errors

1. What does it mean to measure a physical quantity? Give examples.

2. Why do measurement errors occur?

3. What is absolute error?

4. What is relative error?

5. What error characterizes the quality of measurement? Give examples.

6. What is a confidence interval?

7. Define the concept of “systematic error”.

8. What are the causes of systematic errors?

9. What is the accuracy class of a measuring device?

10. How are the absolute errors of various physical instruments determined?

11. What errors are called random and how do they arise?

12. Describe the procedure for calculating the mean square error.

13. Describe the procedure for calculating the absolute random error of direct measurements.

14. What is a “reliability factor”?

15. On what parameters and how does the Student coefficient depend?

16. How is the total absolute error of direct measurements calculated?

17. Write formulas for determining the relative and absolute errors of indirect measurements.

18. Formulate the rules for rounding the result with an error.

19. Find the relative error in measuring the length of the wall using a tape measure with a division value of 0.5 cm. The measured value was 4.66 m.

20. When measuring the length of sides A and B of the rectangle, absolute errors ΔA and ΔB were made, respectively. Write a formula to calculate the absolute error ΔS obtained when determining the area from the results of these measurements.

21. The measurement of the cube edge length L had an error ΔL. Write a formula to determine the relative error of the volume of a cube based on the results of these measurements.

22. A body moved uniformly accelerated from a state of rest. To calculate the acceleration, we measured the path S traveled by the body and the time of its movement t. The absolute errors of these direct measurements were ΔS and Δt, respectively. Derive a formula to calculate the relative acceleration error from these data.

23. When calculating the power of the heating device according to measurement data, the values Pav = 2361.7893735 W and ΔР = 35.4822 W were obtained. Record the result as a confidence interval, rounding as necessary.

24. When calculating the resistance value based on measurement data, the following values were obtained: Rav = 123.7893735 Ohm, ΔR = 0.348 Ohm. Record the result as a confidence interval, rounding as necessary.

25. When calculating the friction coefficient based on measurement data, the values μav = 0.7823735 and Δμ = 0.03348 were obtained. Record the result as a confidence interval, rounding as necessary.

26. A current of 16.6 A was determined using a device with an accuracy class of 1.5 and a scale rating of 50 A. Find the absolute instrumental and relative errors of this measurement.

27. In a series of 5 measurements of the period of oscillation of the pendulum, the following values were obtained: 2.12 s, 2.10 s, 2.11 s, 2.14 s, 2.13 s. Find the absolute random error in determining the period from these data.

28. The experiment of dropping a load from a certain height was repeated 6 times. In this case, the following values of the load falling time were obtained: 38.0 s, 37.6 s, 37.9 s, 37.4 s, 37.5 s, 37.7 s. Find the relative error in determining the time of fall.

The division value is a measured value that causes the pointer to deviate by one division. The division value is determined as the ratio of the upper limit of measurement of the device to the number of scale divisions.

Let some random variable a measured n times under the same conditions. The measurement results gave a set n different numbers

Absolute error- dimensional value. Among n Absolute error values are necessarily both positive and negative.

For the most probable value of the quantity A usually taken average value of measurement results

The greater the number of measurements, the closer the average value is to the true value.

Absolute errori

Relative errori-th measurement is called quantity

Relative error is a dimensionless quantity. Usually the relative error is expressed as a percentage, for this e i multiply by 100%. The magnitude of the relative error characterizes the accuracy of the measurement.

Average absolute error is defined like this:

We emphasize the need to sum the absolute values (modules) of the quantities D and i. Otherwise, the result will be identically zero.

Average relative error is called the quantity

For a large number of measurements.

Relative error can be considered as the error value per unit of the measured value.

The accuracy of measurements is judged by comparing the errors of the measurement results. Therefore, measurement errors are expressed in such a form that to assess the accuracy it is enough to compare only the errors of the results, without comparing the sizes of the objects being measured or knowing these sizes very approximately. It is known from practice that the absolute error in measuring an angle does not depend on the value of the angle, and the absolute error in measuring length depends on the value of the length. The greater the length, the this method and measurement conditions, the absolute error will be greater. Consequently, the absolute error of the result can be used to judge the accuracy of the angle measurement, but the accuracy of the length measurement cannot be judged. Expressing the error in relative form makes it possible to compare the accuracy of angular and linear measurements in known cases.

Basic concepts of probability theory. Random error.

Random error called 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.

Random errors are an inevitable consequence of any measurements and are caused by:

a) inaccuracy of readings on the scale of instruments and instruments;

b) non-identity of conditions for repeated measurements;

c) random changes in external conditions (temperature, pressure, force field etc.) that cannot be controlled;

d) all other influences on measurements, the causes of which are unknown to us. The magnitude of random error can be minimized by repeating the experiment many times and corresponding mathematical processing of the results obtained.

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.

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 of 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 exclude random errors in individual measurements, mathematical theory random phenomena allow 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.

Due to the fact that the occurrence of random errors is inevitable and unavoidable, the main task of any measurement process is to reduce errors to a minimum.

The theory of errors is based on two main assumptions, confirmed by experience:

1. With a large number of measurements, random errors are of the same magnitude, but different sign, that is, errors in the direction of increasing and decreasing the result occur quite often.

2. Errors that are large in absolute value are less common than small ones, thus, the probability of an error occurring decreases as its magnitude increases.

The behavior of random variables is described by statistical patterns, which are the subject of probability theory. Statistical definition probabilities w i events i is the attitude

Where n- total number of experiments, n i- the number of experiments in which the event i happened. In this case, the total number of experiments should be very large ( n®¥). With a large number of measurements, random errors obey a normal distribution (Gaussian distribution), the main features of which are the following:

1. The greater the deviation of the measured value from the true value, the less likely it is for such a result.

2. Deviations in both directions from the true value are equally probable.

From the above assumptions it follows that in order to reduce the influence of random errors it is necessary to measure this value several times. Suppose we are measuring some quantity x. Let it be produced n measurements: x 1 , x 2 , ... x n- using the same method and with the same care. It can be expected that the number dn obtained results, which lie in some fairly narrow interval from x before x + dx, must be proportional:

The size of the interval taken dx;

Total number of measurements n.

Probability dw(x) that some value x lies in the range from x before x + dx, is defined as follows :

(with the number of measurements n ®¥).

Function f(X) is called the distribution function or probability density.

As a postulate of the error theory, it is accepted that the results of direct measurements and their random errors, when there are a large number of them, obey the law of normal distribution.

The continuous distribution function found by Gauss random variablex has the following form:

, where mis - distribution parameters .

The parameter m of the normal distribution is equal to the mean value b xñ a random variable, which, for an arbitrary known distribution function, is determined by the integral

Thus, the value m is the most probable value of the measured quantity x, i.e. her best estimate.

The parameter s 2 of the normal distribution is equal to the variance D of the random variable, which in general case is determined by the following integral

The square root of the variance is called the standard deviation of the random variable.

The average deviation (error) of the random variable ásñ is determined using the distribution function as follows

The average measurement error ásñ, calculated from the Gaussian distribution function, is related to the value of the standard deviation s as follows:

< s > = 0.8s.

The parameters s and m are related to each other as follows:

This expression allows you to find the average standard deviation s if there is a normal distribution curve.

The graph of the Gaussian function is presented in the figures. Function f(x) is symmetrical about the ordinate drawn at the point x = m; passes through a maximum at the point x = m and has an inflection at points m ±s. Thus, variance characterizes the width of the distribution function, or shows how widely the values of a random variable are scattered relative to its true value. The more accurate the measurements, the closer to the true value the results of individual measurements, i.e. the value s is less. Figure A shows the function f(x) for three values of s .

Area of a figure enclosed by a curve f(x) and vertical lines drawn from points x 1 and x 2 (Fig. B) , numerically equal to the probability of the measurement result falling into the interval D x = x 1 - x 2, which is called the confidence probability. Area under the entire curve f(x) is equal to the probability of a random variable falling into the interval from 0 to ¥, i.e.

since the probability of a reliable event is equal to one.

Using the normal distribution, error theory poses and solves two main problems. The first is an assessment of the accuracy of the measurements taken. The second is an assessment of the accuracy of the arithmetic mean value of the measurement results.5. Confidence interval. Student's coefficient.

Probability theory allows us to determine the size of the interval in which, with a known probability w the results of individual measurements are found. This probability is called confidence probability, and the corresponding interval (<x>±D x)w called confidence interval. The confidence probability is also equal to the relative proportion of results that fall within the confidence interval.

If the number of measurements n is sufficiently large, then the confidence probability expresses the proportion of the total number n those measurements in which the measured value was within the confidence interval. Each confidence probability w corresponds to its confidence interval. w 2 80%. The wider the confidence interval, the greater the likelihood of getting a result within that interval. In probability theory, a quantitative relationship is established between the value of the confidence interval, confidence probability and the number of measurements.

If we choose as a confidence interval the interval corresponding to the average error, that is, D a =áD Añ, then for a sufficiently large number of measurements it corresponds to the confidence probability w 60%. As the number of measurements decreases, the confidence probability corresponding to such a confidence interval (á Añ ± áD Añ), decreases.

Thus, to estimate the confidence interval of a random variable, one can use the value of the average error áD Añ .

To characterize the magnitude of the random error, it is necessary to specify two numbers, namely, the value of the confidence interval and the value of the confidence probability . Indicating only the magnitude of the error without the corresponding confidence probability is largely meaningless.

If the average measurement error ásñ is known, the confidence interval written as (<x> ± ásñ) w, determined with confidence probability w= 0,57.

If the standard deviation s is known distribution of measurement results, the specified interval has the form (<x>± t w s) w, Where t w- coefficient depending on the confidence probability value and calculated using the Gaussian distribution.

Most commonly used quantities D x are given in table 1.

Basic qualitative characteristics of any instrumentation sensor is the measurement error of the controlled parameter. The measurement error of a device is the amount of discrepancy between what the instrumentation sensor showed (measured) and what actually exists. The measurement error for each specific type of sensor is indicated in the accompanying documentation (passport, operating instructions, verification procedure), which is supplied with this sensor.

According to the form of presentation, errors are divided into absolute, relative And given errors.

Absolute error is the difference between the value of Xiz measured by the sensor and the actual value of Xd of this value.

The actual value Xd of the measured quantity is the experimentally found value of the measured quantity that is as close as possible to its true value. Speaking in simple language The actual value of Xd is the value measured by a reference device, or generated by a calibrator or setter of a high accuracy class. The absolute error is expressed in the same units as the measured value (for example, m3/h, mA, MPa, etc.). Since the measured value may be either greater or less than its actual value, the measurement error can be either with a plus sign (the device readings are overestimated) or with a minus sign (the device underestimates).

Relative error is the ratio of the absolute measurement error Δ to the actual value Xd of the measured quantity.

The relative error is expressed as a percentage, or is a dimensionless quantity, and can also take on both positive and negative values.

Reduced error is the ratio of the absolute measurement error Δ to the normalizing value Xn, constant over the entire measurement range or part of it.

The normalizing value Xn depends on the type of instrumentation sensor scale:

If the sensor scale is one-sided and the lower measurement limit is zero (for example, the sensor scale is from 0 to 150 m3/h), then Xn is taken equal to the upper measurement limit (in our case, Xn = 150 m3/h).
If the sensor scale is one-sided, but the lower measurement limit is not zero (for example, the sensor scale is from 30 to 150 m3/h), then Xn is taken equal to the difference between the upper and lower measurement limits (in our case, Xn = 150-30 = 120 m3/h ).
If the sensor scale is two-sided (for example, from -50 to +150 ˚С), then Xn is equal to the width of the sensor measurement range (in our case, Xn = 50+150 = 200 ˚С).

The given error is expressed as a percentage, or is a dimensionless quantity, and can also take both positive and negative values.

Quite often, the description of a particular sensor indicates not only the measurement range, for example, from 0 to 50 mg/m3, but also the reading range, for example, from 0 to 100 mg/m3. The given error in this case is normalized to the end of the measurement range, that is, to 50 mg/m3, and in the reading range from 50 to 100 mg/m3 the measurement error of the sensor is not determined at all - in fact, the sensor can show anything and have any measurement error. The measuring range of the sensor can be divided into several measuring subranges, for each of which its own error can be determined, both in magnitude and in the form of presentation. In this case, when checking such sensors, each sub-range can use its own standard measuring instruments, the list of which is indicated in the verification procedure for this device.

For some devices, the passports indicate the accuracy class instead of the measurement error. Such instruments include mechanical pressure gauges, indicating bimetallic thermometers, thermostats, flow indicators, dial ammeters and voltmeters for panel mounting and so on. An accuracy class is a generalized characteristic of measuring instruments, determined by the limits of permissible basic and additional errors, as well as a number of other properties that affect the accuracy of measurements made with their help. Moreover, the accuracy class is not a direct characteristic of the accuracy of measurements performed by this device; it only indicates the possible instrumental component of the measurement error. The accuracy class of the device is applied to its scale or body in accordance with GOST 8.401-80.

When assigning an accuracy class to a device, it is selected from the series 1·10 n; 1.5 10 n; (1.6·10 n); 2·10n; 2.5 10 n; (3·10 n); 4·10n; 5·10n; 6·10n; (where n =1, 0, -1, -2, etc.). The values of accuracy classes indicated in brackets are not established for newly developed measuring instruments.

The measurement error of sensors is determined, for example, during their periodic verification and calibration. Using various setpoints and calibrators with high accuracy generate certain values of one or another physical quantity and compare the readings of the sensor being verified with the readings of a standard measuring instrument to which the same value of the physical quantity is supplied. Moreover, the measurement error of the sensor is controlled both during the forward stroke (increase in the measured physical quantity from the minimum to the maximum of the scale) and during the reverse stroke (decreasing the measured value from the maximum to the minimum of the scale). This is due to the fact that due to the elastic properties of the sensor’s sensitive element (pressure sensor membrane), different flow rates chemical reactions(electrochemical sensor), thermal inertia, etc. sensor readings will vary depending on how the force acting on the sensor changes. physical quantity: decreases or increases.

Quite often, in accordance with the verification methodology, the readings of the sensor during verification should be performed not according to its display or scale, but according to the value of the output signal, for example, according to the value of the output current of the current output 4...20 mA.

For the pressure sensor being verified with a measurement scale from 0 to 250 mbar, the main relative measurement error over the entire measurement range is 5%. The sensor has a current output of 4...20 mA. The calibrator applied a pressure of 125 mbar to the sensor, while its output signal is 12.62 mA. It is necessary to determine whether the sensor readings are within acceptable limits.

First, it is necessary to calculate what the output current of the sensor Iout.t should be at a pressure Рт = 125 mbar.

Iout.t = Ish.out.min + ((Ish.out.max – Ish.out.min)/(Rsh.max – Rsh.min))*Рт

where Iout.t is the output current of the sensor at a given pressure of 125 mbar, mA.

Ish.out.min – minimum output current of the sensor, mA. For a sensor with an output of 4…20 mA, Ish.out.min = 4 mA, for a sensor with an output of 0…5 or 0…20 mA, Ish.out.min = 0.

Ish.out.max - maximum output current of the sensor, mA. For a sensor with an output of 0...20 or 4...20 mA, Ish.out.max = 20 mA, for a sensor with an output of 0...5 mA, Ish.out.max = 5 mA.

Рш.max – maximum of the pressure sensor scale, mbar. Psh.max = 250 mbar.

Rsh.min – minimum scale of the pressure sensor, mbar. Rsh.min = 0 mbar.

Рт – pressure supplied from the calibrator to the sensor, mbar. RT = 125 mbar.

Substituting the known values we get:

Iout.t = 4 + ((20-4)/(250-0))*125 = 12 mA

That is, with a pressure of 125 mbar applied to the sensor, its current output should be 12 mA. We consider the limits within which it can vary calculated value output current, taking into account that the main relative measurement error is ± 5%.

ΔIout.t =12 ± (12*5%)/100% = (12 ± 0.6) mA

That is, with a pressure of 125 mbar applied to the sensor at its current output, the output signal should be in the range from 11.40 to 12.60 mA. According to the conditions of the problem, we have an output signal of 12.62 mA, which means that our sensor did not meet the measurement error specified by the manufacturer and requires adjustment.

The main relative measurement error of our sensor is:

δ = ((12.62 – 12.00)/12.00)*100% = 5.17%

Verification and calibration of instrumentation devices must be carried out when normal conditions environment By atmospheric pressure, humidity and temperature and at the rated supply voltage of the sensor, since higher or low temperature and supply voltage may lead to additional measurement errors. The verification conditions are specified in the verification procedure. Devices whose measurement error does not fall within the limits established by the verification method are either re-adjusted and adjusted, after which they are re-verified, or, if the adjustment does not bring results, for example, due to aging or excessive deformation of the sensor, they are repaired. If repair is impossible, the devices are rejected and taken out of service.

If, nevertheless, the devices were able to be repaired, then they are no longer subject to periodic, but to primary verification with the implementation of all the points set out in the verification procedure for this type of verification. In some cases, the device is specially subjected to minor repairs () since according to the verification method, performing primary verification turns out to be much easier and cheaper than periodic verification, due to differences in the set of standard measuring instruments that are used for periodic and primary verification.

To consolidate and test the knowledge gained, I recommend doing this.

Subject " ” is studied fluently in 9th grade. And students, as a rule, do not fully develop the skills to calculate it.

But with practical application relative error of the number , as well as with absolute error, we encounter at every step.

During repair work measured (in centimeters) the thickness m carpeting and width n threshold. We got the following results:

m≈0.8 (with an accuracy of 0.1);

n≈100.0 (accurate to 0.1).

Note that the absolute error of each measurement data is no more than 0.1.

However, 0.1 is a solid part of the number 0.8. As fornumber 100 it represents insignificant his. This shows that the quality of the second dimension is much higher than the first.

To assess the quality of measurement it is used relative error of the approximate number.

Definition.

Relative error of the approximate number (values) is the ratio of the absolute error to the absolute value of the approximate value.

They agreed to express the relative error as a percentage.

Example 1.

Consider the fraction 14.7 and round it to whole numbers. We will also find relative error of the approximate number:

14,7≈15.

To calculate the relative error, in addition to the approximate value, as a rule, you also need to know the absolute error. The absolute error is not always known. Therefore calculate impossible. And in this case, it is enough to indicate an estimate of the relative error.

Let's remember the example that was given at the beginning of the article. The thickness measurements were indicated there. m carpet and width n threshold.

Based on the results of measurements m≈0.8 with an accuracy of 0.1. We can say that the absolute measurement error is no more than 0.1. This means that the result of dividing the absolute error by the approximate value (and this is the relative error) is less than or equal to 0.1/0.8 = 0.125 = 12.5%.

Thus, the relative approximation error is ≤ 12.5%.

In a similar way, we calculate the relative error in approximating the width of the sill; it is no more than 0.1/100 = 0.001 = 0.1%.

They say that in the first case the measurement was carried out with a relative accuracy of up to 12.5%, and in the second - with a relative accuracy of up to 0.1%.

Summarize.

Absolute error approximate number - this is the differencebetween the exact number x and its approximate value a.

If the difference modulus | x – a| less than some D a, then the value D a called absolute error approximate number a.

Relative error of the approximate number is the ratio of the absolute error D a to the modulus of a number a, that isD a / |a| =d a .

Example 2.

Let's consider the known approximate value of the number π≈3.14.

Considering its value with an accuracy of one hundred thousandths, you can indicate its error as 0.00159... (it will help to remember the digits of the number π )

The absolute error of the number π is equal to: | 3,14 – 3,14159 | = 0,00159 ≈0,0016.

The relative error of the number π is equal to: 0.0016/3.14 = 0.00051 = 0.051%.

Example 3.

Try to calculate it yourself relative error of the approximate number √2. There are several ways to remember the digits of a number “ Square root from 2″.