How to calculate the mean absolute error. Calculation of errors of direct measurements

The exact natural sciences are based on measurements. 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, which put one value into correspondence with another associated with it, accessible to quantitative assessment with the help of 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. Therefore, much attention is paid to processing 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 imperfection. For example, friction between various parts in the pointer block of an ammeter leads to the fact that a change in current by some small, but finite, amount 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 disabilities our eye.

· 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 the true value of the 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 we are interested in. 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 value of a certain quantity that is not known in advance 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- this is the ratio of the absolute error of the device to the maximum value of the measured value, which can be determined using this device (this is the systematic relative error of this 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 with a very large number of 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, the mathematical description of 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 arithmetic mean 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 root mean square absolute error:

.

4). Calculated absolute random error. If not large number 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 value falls within the confidence interval.

Confidence interval is a numerical 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, than larger number 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. Numerical values ​​of Student's coefficients at this reliability for different numbers 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 the exact value of the error is 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 measuring instrument?

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 fall 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.

Due to the errors inherent in the measuring instrument, the chosen method and measurement procedure, differences external conditions, in which the measurement is performed, for established and other reasons, the result of almost every measurement is burdened with error. This error is calculated or estimated and assigned to the result obtained.

Measurement result error(in short - measurement error) - the deviation of the measurement result from the true value of the measured value.

The true value of the quantity remains unknown due to the presence of errors. It is used in solving theoretical problems of metrology. In practice, the actual value of the quantity is used, which replaces the true value.

The measurement error (Δx) is found by the formula:

x = x meas. - x valid (1.3)

where x meas. - the value of the quantity obtained on the basis of measurements; x valid — the value of the quantity taken as real.

For single measurements, the actual value is often taken to be the value obtained using a standard measuring instrument; for multiple measurements, the arithmetic mean of the values ​​of individual measurements included in a given series.

Measurement errors can be classified according to the following criteria:

By the nature of the manifestations - systematic and random;

According to the method of expression - absolute and relative;

According to the conditions of change in the measured value - static and dynamic;

According to the method of processing a number of measurements - arithmetic averages and root mean squares;

According to the completeness of coverage of the measurement task - partial and complete;

In relation to a unit of physical quantity - errors in reproducing the unit, storing the unit and transmitting the size of the unit.

Systematic measurement error(in short - systematic error) - a component of the error of a measurement result that remains constant for a given series of measurements or changes naturally with repeated measurements of the same physical quantity.

According to the nature of their manifestation, systematic errors are divided into permanent, progressive and periodic. Constant systematic errors(in short - constant errors) - errors, long time retaining their value (for example, throughout the entire series of measurements). This is the most common type of error.

Progressive systematic errors(in short - progressive errors) - continuously increasing or decreasing errors (for example, errors from wear of measuring tips that come into contact with the part during the grinding process when monitoring it with an active control device).

Periodic systematic error(briefly - periodic error) - an error, the value of which is a function of time or a function of the movement of the pointer of a measuring device (for example, the presence of eccentricity in goniometer devices with a circular scale causes a systematic error that varies according to a periodic law).

Based on the reasons for the appearance of systematic errors, a distinction is made between instrumental errors, method errors, subjective errors and errors due to deviations of external measurement conditions from those established by the methods.

Instrumental measurement error(briefly - instrumental error) is a consequence of a number of reasons: wear of device parts, excessive friction in the device mechanism, inaccurate marking of strokes on the scale, discrepancy between the actual and nominal values ​​of the measure, etc.

Measurement method error(in short - method error) may arise due to the imperfection of the measurement method or its simplifications established by the measurement methodology. For example, such an error may be due to insufficient performance of the measuring instruments used when measuring the parameters of fast processes or unaccounted for impurities when determining the density of a substance based on the results of measuring its mass and volume.

Subjective measurement error(in short - subjective error) is due to the individual errors of the operator. This error is sometimes called personal difference. It is caused, for example, by a delay or advance in the operator's acceptance of a signal.

Error due to deviation(in one direction) the external measurement conditions from those established by the measurement technique leads to the emergence of a systematic component of the measurement error.

Systematic errors distort the measurement result, so they must be eliminated as far as possible by introducing corrections or adjusting the device to bring systematic errors to an acceptable minimum.

Unexcluded systematic error(in short - non-excluded error) is the error of the measurement result, due to the error in calculation and introduction of a correction for the action of a systematic error, or a small systematic error, the correction for which is not introduced due to its smallness.

Sometimes this type of error is called non-excluded residuals of systematic error(in short - non-excluded balances). For example, when measuring the length of a line meter in wavelengths of reference radiation, several non-excluded systematic errors were identified (i): due to inaccurate temperature measurement - 1; due to inaccurate determination of the refractive index of air - 2, due to inaccurate wavelength - 3.

Usually the sum of non-excluded systematic errors is taken into account (their boundaries are set). When the number of terms is N ≤ 3, the limits of non-excluded systematic errors are calculated using the formula

When the number of terms is N ≥ 4, the formula is used for calculations

(1.5)

where k is the coefficient of dependence of non-excluded systematic errors on the selected confidence probability P when they are uniformly distributed. At P = 0.99, k = 1.4, at P = 0.95, k = 1.1.

Random measurement error(in short - random error) - a component of the error of a measurement result that changes randomly (in sign and value) in a series of measurements of the same size of a physical quantity. Reasons for random errors: rounding errors when taking readings, variation in readings, changes in measurement conditions random etc.

Random errors cause scattering of measurement results in a series.

The theory of errors is based on two principles, confirmed by practice:

1. With a large number of measurements, random errors of the same numerical value, But different sign, occur equally often;

2. Large (in absolute value) errors are less common than small ones.

From the first position follows an important conclusion for practice: as the number of measurements increases, the random error of the result obtained from a series of measurements decreases, since the sum of the errors of individual measurements of a given series tends to zero, i.e.

(1.6)

For example, as a result of measurements, a number of values ​​were obtained electrical resistance(corrected for systematic errors): R 1 = 15.5 Ohm, R 2 = 15.6 Ohm, R 3 = 15.4 Ohm, R 4 = 15.6 Ohm and R 5 = 15.4 Ohm . Hence R = 15.5 Ohm. Deviations from R (R 1 = 0.0; R 2 = +0.1 Ohm, R 3 = -0.1 Ohm, R 4 = +0.1 Ohm and R 5 = -0.1 Ohm) are random errors of individual measurements in this series. It is easy to verify that the sum R i = 0.0. This indicates that the errors in individual measurements of this series were calculated correctly.

Despite the fact that as the number of measurements increases, the sum of random errors tends to zero (in in this example it accidentally turned out to be equal to zero), the random error of the measurement result must be assessed. In the theory of random variables, the characteristic of scattering of values ​​is random variable serves as dispersion o2. "|/o2 = a is called the mean square deviation of the population or standard deviation.

It is more convenient than dispersion, since its dimension coincides with the dimension of the measured quantity (for example, the value of the quantity is obtained in volts, the standard deviation will also be in volts). Since in measurement practice we deal with the term “error,” the derivative term “mean square error” should be used to characterize a number of measurements. A characteristic of a series of measurements can be the arithmetic mean error or the range of measurement results.

Range of measurement results (span for short) — algebraic difference the largest and smallest results of individual measurements forming a series (or sample) of n measurements:

R n = X max - X min (1.7)

where R n is the range; X max and X min - the greatest and smallest value values ​​in a given series of measurements.

For example, out of five measurements of the hole diameter d, the values ​​R 5 = 25.56 mm and R 1 = 25.51 mm turned out to be its maximum and minimum values. In this case, R n = d 5 - d 1 = 25.56 mm - 25.51 mm = 0.05 mm. This means that the remaining errors in this series are less than 0.05 mm.

Arithmetic mean error of an individual measurement in a series(briefly - arithmetic mean error) - a generalized characteristic of the scattering (due to random reasons) of individual measurement results (of the same quantity) included in a series of n equal-precision independent measurements, calculated by the formula

(1.8)

where X i is the result of the i-th measurement included in the series; x is the arithmetic mean of n values: |Х і - X| — absolute value of the error of the i-th measurement; r is the arithmetic mean error.

The true value of the average arithmetic error p is determined from the relation

p = lim r, (1.9)

With the number of measurements n > 30 between the arithmetic mean (r) and the root mean square (s) there are correlations between errors

s = 1.25 r; r and= 0.80 s. (1.10)

The advantage of the arithmetic mean error is the simplicity of its calculation. But still, the mean square error is more often determined.

Mean square error individual measurement in a series (in short - mean square error) - a generalized characteristic of the scattering (due to random reasons) of individual measurement results (of the same value) included in a series of n equal-precision independent measurements, calculated by the formula

(1.11)

Mean square error for general sample o, which is the statistical limit of S, can be calculated at /i-mx > using the formula:

Σ = lim S (1.12)

In reality, the number of measurements is always limited, so it is not σ , and its approximate value (or estimate), which is s. The more p, the closer s is to its limit σ .

At normal law distribution, the probability that the error of an individual measurement in a series will not exceed the calculated mean square error is small: 0.68. Therefore, in 32 cases out of 100 or 3 cases out of 10, the actual error may be greater than the calculated one.

Figure 1.2 Decrease in the value of the random error of the result of multiple measurements with an increase in the number of measurements in a series

In a series of measurements, there is a relationship between the root mean square error of an individual measurement s and the root mean square error of the arithmetic mean S x:

which is often called the “U n rule”. From this rule it follows that the measurement error due to random causes can be reduced by n times if n measurements of the same size of any quantity are performed, and the arithmetic mean is taken as the final result (Fig. 1.2).

Performing at least 5 measurements in a series makes it possible to reduce the influence of random errors by more than 2 times. With 10 measurements, the influence of random error is reduced by 3 times. A further increase in the number of measurements is not always economically feasible and, as a rule, is carried out only for critical measurements that require high accuracy.

The root mean square error of a single measurement from a number of homogeneous double measurements S α is calculated by the formula

(1.14)

where x" i and x"" i are the i-th results of measurements of the same size quantity in the forward and reverse directions with one measuring instrument.

In case of unequal measurements, the root mean square error of the arithmetic average in the series is determined by the formula

(1.15)

where p i is the weight of the i-th measurement in a series of unequal measurements.

The root mean square error of the result of indirect measurements of the value Y, which is a function of Y = F (X 1, X 2, X n), is calculated using the formula

(1.16)

where S 1, S 2, S n are the root mean square errors of the measurement results of the quantities X 1, X 2, X n.

If, for greater reliability in obtaining a satisfactory result, several series of measurements are carried out, the root mean square error of an individual measurement from m series (S m) is found by the formula

(1.17)

Where n is the number of measurements in the series; N— total number measurements in all series; m is the number of series.

With a limited number of measurements, it is often necessary to know the root mean square error. To determine the error S, calculated by formula (2.7), and the error S m, calculated by formula (2.12), you can use the following expressions

(1.18)

(1.19)

where S and S m are the mean square errors of S and S m , respectively.

For example, when processing the results of a number of measurements of length x, we obtained

= 86 mm 2 at n = 10,

= 3.1 mm

= 0.7 mm or S = ±0.7 mm

The value S = ±0.7 mm means that due to the calculation error, s is in the range from 2.4 to 3.8 mm, therefore tenths of a millimeter are unreliable here. In the case considered, we must write: S = ±3 mm.

To have greater confidence in assessing the error of a measurement result, calculate the confidence error or confidence limits of the error. Under the normal distribution law, the confidence limits of the error are calculated as ±t-s or ±t-s x, where s and s x are the mean square errors, respectively, of an individual measurement in the series and the arithmetic mean; t is a number depending on the confidence probability P and the number of measurements n.

An important concept is the reliability of the measurement result (α), i.e. the probability that the desired value of the measured quantity will fall within a given confidence interval.

For example, when processing parts on machine tools in a stable technological mode, the distribution of errors obeys the normal law. Let's assume that the part length tolerance is set to 2a. In this case, the confidence interval in which the desired value of the length of the part a is located will be (a - a, a + a).

If 2a = ±3s, then the reliability of the result is a = 0.68, i.e. in 32 cases out of 100 one should expect the part size to exceed tolerance 2a. When assessing the quality of a part according to a tolerance of 2a = ±3s, the reliability of the result will be 0.997. In this case, we can expect only three parts out of 1000 to exceed the established tolerance. However, an increase in reliability is possible only by reducing the error in the length of the part. Thus, to increase reliability from a = 0.68 to a = 0.997, the error in the length of the part must be reduced by three times.

IN lately The term “measurement reliability” has become widespread. In some cases, it is unreasonably used instead of the term “measurement accuracy.” For example, in some sources you can find the expression “establishing the unity and reliability of measurements in the country.” Whereas it would be more correct to say “establishing the unity and required accuracy of measurements.” We consider reliability as a qualitative characteristic that reflects the proximity to zero of random errors. It can be quantitatively determined through the unreliability of measurements.

Unreliability of measurements(in short - unreliability) - an assessment of the discrepancy between the results in a series of measurements due to the influence of the total influence of random errors (determined by statistical and non-statistical methods), characterized by the range of values ​​in which the true value of the measured value is located.

In accordance with the recommendations of the International Bureau of Weights and Measures, unreliability is expressed in the form of a total mean square measurement error - Su, including the mean square error S (determined by statistical methods) and the mean square error u (determined by non-statistical methods), i.e.

(1.20)

Maximum measurement error(briefly - maximum error) - the maximum measurement error (plus, minus), the probability of which does not exceed the value P, while the difference 1 - P is insignificant.

For example, with a normal distribution law, the probability of a random error equal to ±3s is 0.997, and the difference 1-P = 0.003 is insignificant. Therefore, in many cases, the confidence error of ±3s is taken as the maximum, i.e. pr = ±3s. If necessary, pr may have other relationships with s at a sufficiently large P (2s, 2.5s, 4s, etc.).

Due to the fact that in the GSI standards, instead of the term “mean square error,” the term “mean square deviation” is used, in further discussions we will adhere to this very term.

Absolute measurement error(in short - absolute error) - measurement error expressed in units of the measured value. Thus, the error X in measuring the length of a part X, expressed in micrometers, represents an absolute error.

The terms “absolute error” and “absolute value of error” should not be confused, which is understood as the value of the error without taking into account the sign. So, if the absolute measurement error is ±2 μV, then the absolute value of the error will be 0.2 μV.

Relative measurement error(in short - relative error) - measurement error, expressed in fractions of the value of the measured value or as a percentage. The relative error δ is found from the relations:

(1.21)

For example, there is a real value of the part length x = 10.00 mm and an absolute value of the error x = 0.01 mm. The relative error will be

Static error— error of the measurement result due to the conditions of static measurement.

Dynamic error— error of the measurement result due to the conditions of dynamic measurement.

Unit reproduction error— error in the result of measurements performed when reproducing a unit of physical quantity. Thus, the error in reproducing a unit using a state standard is indicated in the form of its components: the non-excluded systematic error, characterized by its boundary; random error characterized by standard deviation s and instability over the year ν.

Unit size transmission error— error in the result of measurements performed when transmitting the size of a unit. The error in transmitting the unit size includes non-excluded systematic errors and random errors of the method and means of transmitting the unit size (for example, a comparator).

In this topic I will write something like a short cheat sheet on errors. Again, this text is in no way official and reference to it is unacceptable. I would be grateful for the correction of any errors or inaccuracies that may be in this text.

What is error?

Recording the result of an experiment of the form () means that if we conduct a lot of identical experiments, then in 70% the results obtained will lie in the interval, and in 30% they will not.

Or, which is the same thing, if we repeat the experiment, then new result will fall within the confidence interval with a probability equal to the confidence probability.

How to round the error and the result?

The error is rounded to the first significant digit, if it is not one. If one - then up to two. At the same time significant figure any digit of the result except leading zeros is called.

Round to or or but under no circumstances or , since there are 2 significant figures - 2 and 0 after the two.

Round up to or

Round up to or or

We round the result so that the last significant figure the result corresponded to the last significant digit of the error.

Examples correct entry:

mm

Um, let's keep the error here to 2 significant figures because the first significant figure in the error is one.

mm

Examples incorrect entry:

Mm. Here extra sign as a result. mm will be correct.

mm. Here extra sign both in error and as a result. mm will be correct.

In my work I use the value given to me simply as a number. For example, a mass of weights. What is its margin of error?

If the error is not explicitly indicated, you can take one in the last digit. That is, if m = 1.35 g is written, then the error should be taken as 0.01 g.

There is a function of several quantities. Each of these quantities has its own error. To find the error of the function you need to do the following:

The symbol means the partial derivative of f with respect to x. Read more about partial derivatives.

Suppose you measured the same quantity x several (n) times. We received a set of values. . You need to calculate the scatter error, calculate the instrument error and add them together.

Point by point.

1. We calculate the spread error

If all the values ​​coincide, you have no spread. Otherwise, there is a scatter error that needs to be calculated. To begin with, the root mean square error of the average is calculated:

Here means the average over all.
The scatter error is obtained by multiplying the root mean square error of the mean by the Student coefficient, which depends on the confidence probability you choose and the number of measurements n:

We take Student's coefficients from the table below. The confidence probability is generated arbitrarily, the number of measurements n we also know.

2. We consider the instrument error of the average

If the errors of different points are different, then according to the formula

Naturally, everyone’s confidence probability should be the same.

3. Add the average with the spread

Errors always add up as the root of squares:

In this case, you need to make sure that the confidence probabilities with which were calculated and coincide.

How to determine the instrument error of the average from a graph? Well, that is, using the method of paired points or the method least squares, we will find the error in the spread of the average resistance. How to find the instrument error of the average resistance?

Both the least squares method and the paired point method can give a strict answer to this question. For the MLS forum in Svetozarov there is ("Basics...", a section about the least squares method), and for paired points the first thing that comes to mind (in the forehead, as they say) is to calculate the instrumental error of each slope. Well, further on all points...

If you don’t want to suffer, then in the lab books there is a simple way to assessments instrument error of the angular coefficient, namely from the following MNC (for example, before work 1 in the lab book "Electrical measuring instruments...." last page of Methodological recommendations).

Where is the maximum deviation along the Y axis of a point with an error from the drawn straight line, and the denominator is the width of the area of ​​our graph along the Y axis. Likewise for the X axis.

The accuracy class is written on the resistance magazine: 0.05/4*10^-6? How to find the instrument error from this?

This means that the maximum relative error of the device (in percent) has the form:
, Where
- highest value store resistance, a is the nominal value of the included resistance.
It is easy to see that the second term is important when we are working at very low resistances.

More details can always be found in the device passport. The passport can be found on the Internet by typing the brand of the device into Google.

Literature about errors

Much more information on this subject can be found in the book recommended for freshmen:
V.V. Svetozarov "Elementary processing of measurement results"

As additional (for freshmen additional) literature we can recommend:
V.V. Svetozarov "Fundamentals of statistical processing of measurement results"

And those who want to finally understand everything should definitely look here:
J. Taylor. "Introduction to Error Theory"

Thank you for finding and posting these wonderful books on your site.

1. Introduction

The work of chemists, physicists and representatives of other natural science professions often involves performing quantitative measurements of various quantities. In this case, the question arises of analyzing the reliability of the obtained values, processing the results of direct measurements and assessing the errors of calculations that use the values ​​of directly measured characteristics (the latter process is also called processing of results indirect measurements). For a number of objective reasons, the knowledge of graduates of the Faculty of Chemistry of Moscow State University about calculating errors is not always sufficient for correct processing of the received data. One of these reasons is the absence in the faculty curriculum of a course on statistical processing of measurement results.

TO at this moment the issue of calculating errors has, of course, been studied exhaustively. Exists large number methodological developments, textbooks, etc., in which you can find information about calculating errors. Unfortunately, most of these works are overloaded with additional and not always necessary information. In particular, most of the work of student workshops does not require such actions as comparing samples, assessing convergence, etc. Therefore, it seems appropriate to create a brief development that outlines the algorithms for the most frequently used calculations, which is what this development is devoted to.

2. Notation adopted in this work

The measured value, - the average value of the measured value, - the absolute error of the average value of the measured value, - the relative error of the average value of the measured value.

3. Calculation of errors of direct measurements

So, let's assume that they were carried out n measurements of the same quantity under the same conditions. In this case, you can calculate the average value of this value in the measurements taken:

(1)

How to calculate the error? According to the following formula:

(2)

This formula uses the Student coefficient. Its values ​​at different confidence probabilities and values ​​are given in.

3.1. An example of calculating the errors of direct measurements:

Task.

The length of the metal bar was measured. 10 measurements were made and the following values ​​were obtained: 10 mm, 11 mm, 12 mm, 13 mm, 10 mm, 10 mm, 11 mm, 10 mm, 10 mm, 11 mm. It is required to find the average value of the measured value (length of the bar) and its error.

Solution.

Using formula (1) we find:

mm

Now, using formula (2), we find the absolute error of the average value with confidence probability and the number of degrees of freedom (we use the value = 2.262, taken from):

Let's write down the result:

10.8±0.7 0.95 mm

4. Calculation of errors of indirect measurements

Let us assume that during the experiment the quantities are measured and then c Using the obtained values, the value is calculated using the formula . In this case, the errors of directly measured quantities are calculated as described in paragraph 3.

The calculation of the average value of a quantity is carried out according to the dependence using the average values ​​of the arguments.

The error value is calculated using the following formula:

,(3)

where is the number of arguments, is the partial derivative of the function with respect to the arguments, is the absolute error of the average value of the argument.

The absolute error, as in the case of direct measurements, is calculated using the formula.

4.1. An example of calculating the errors of direct measurements:

Task.

5 direct measurements of and were carried out. The following values ​​were obtained for the value: 50, 51, 52, 50, 47; the following values ​​were obtained for the quantity: 500, 510, 476, 354, 520. It is required to calculate the value of the quantity determined by the formula and find the error of the obtained value.

In our age, man has invented and uses a huge variety of all kinds of measuring instruments. But no matter how perfect the technology for their manufacture is, they all have a greater or lesser error. This parameter, as a rule, is indicated on the instrument itself, and to assess the accuracy of the value being determined, you need to be able to understand what the numbers indicated on the marking mean. In addition, relative and absolute errors inevitably arise during complex mathematical calculations. It is widely used in statistics, industry (quality control) and in a number of other areas. How this value is calculated and how to interpret its value - this is exactly what will be discussed in this article.

Absolute error

Let us denote by x the approximate value of a quantity, obtained, for example, through a single measurement, and by x 0 its exact value. Now let's calculate the magnitude of the difference between these two numbers. The absolute error is exactly the value that we got as a result of this simple operation. In the language of formulas, this definition can be written in this form: Δ x = | x - x 0 |.

Relative error

Absolute deviation has one important drawback - it does not allow assessing the degree of importance of the error. For example, we buy 5 kg of potatoes at the market, and an unscrupulous seller, when measuring the weight, made a mistake of 50 grams in his favor. That is, the absolute error was 50 grams. For us, such an oversight will be a mere trifle and we will not even pay attention to it. Can you imagine what will happen if a similar error occurs while preparing the medicine? Here everything will be much more serious. And when loading a freight car, deviations are likely to occur much larger than this value. Therefore, the absolute error itself is not very informative. In addition to it, very often they additionally calculate the relative deviation, equal to the ratio of the absolute error to exact value numbers. This is written by the following formula: δ = Δ x / x 0 .

Error Properties

Suppose we have two independent quantities: x and y. We need to calculate the deviation of the approximate value of their sum. In this case, we can calculate the absolute error as the sum of the pre-calculated absolute deviations of each of them. In some measurements, it may happen that errors in the determination of x and y values ​​cancel each other out. Or it may happen that as a result of addition, the deviations become maximally intensified. Therefore, when total absolute error is calculated, the worst-case scenario must be considered. The same is true for the difference between errors of several quantities. This property is characteristic only of absolute error, and cannot be applied to relative deviation, since this will inevitably lead to an incorrect result. Let's look at this situation using the following example.

Suppose measurements inside the cylinder showed that the inner radius (R 1) is 97 mm, and the outer radius (R 2) is 100 mm. It is necessary to determine the thickness of its wall. First, let's find the difference: h = R 2 - R 1 = 3 mm. If the problem does not indicate what the absolute error is, then it is taken as half the scale division of the measuring device. Thus, Δ(R 2) = Δ(R 1) = 0.5 mm. The total absolute error is: Δ(h) = Δ(R 2) + Δ(R 1) = 1 mm. Now let’s calculate the relative deviation of all values:

δ(R 1) = 0.5/100 = 0.005,

δ(R 1) = 0.5/97 ≈ 0.0052,

δ(h) = Δ(h)/h = 1/3 ≈ 0.3333>> δ(R 1).

As you can see, the error in measuring both radii does not exceed 5.2%, and the error in calculating their difference - the thickness of the cylinder wall - was as much as 33.(3)%!

The following property states: the relative deviation of the product of several numbers is approximately equal to the sum of the relative deviations of the individual factors:

δ(xy) ≈ δ(x) + δ(y).

Moreover this rule is true regardless of the number of values ​​being evaluated. The third and final property of relative error is that the relative estimate kth numbers degree approximately in | k | times the relative error of the original number.