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Method error is the component of measurement error resulting from the imperfection of the measurement method.
The error of the method E is an error resulting from the replacement of the exact solution algorithm with an approximate one. Therefore, the calculation method must be chosen so that its error at the last calculation step does not exceed a given value.
The error of the method does not exceed one and a half divisions. Since the number of teeth of the dividing wheel of the machine is not a multiple of the number of grooves in the sensor disk, at the moment the signal is given, the worm of the dividing gear of the machine is in different angular positions. This makes it possible to determine the total accuracy of the dividing gear, and, if necessary, also highlight the error of the wheel and worm. To do this, use the methods of harmonic analysis. If the table sensor has 40 slots, then the amplitudes and phases of 19 harmonics can be calculated, by which the chain links that are sources of errors are found out, or a correction device can be configured.
The error of the method, of course, is not taken into account, since in both cases the measurement method is the same.
The error of the method arises due to the insufficient development of the theory of those phenomena that underlie the measurement, and those relationships that are used to evaluate the measured quantity.
The error of the method E is the error resulting from the replacement of the exact solution algorithm with an approximate one. Therefore, the calculation method must be chosen so that its error at the last calculation step does not exceed a given value.
The error of the method is estimated at 1% of the measured humidity. Calibration dependences make it possible to estimate the range of measured humidity values from 0 to 20%; at high humidity, the presence of a condensate film significantly overestimates the measurement results. The method is inapplicable in low velocity flows due to significant errors introduced by a sufficiently thick film on the walls of the sensor chamber. The reasonable range of operating flow rates of wet steam is M0 3 - g - I. The disadvantages of the method include the complexity of the equipment and probes, as well as the need to adjust the zero of the device over time.
The error of the method for other combinations of boundary conditions will be within the limits presented in Table 7.2. In this case, the correspondence is always observed: if the load is a piecewise continuous function, then the results of the method are greater than the reference ones, if the load is concentrated, then it is less. Obviously, this is due to the fact that one expansion term describes a piecewise continuous load with excess, and a lumped one - with a disadvantage.
The error of the method is 5 µg of nitrogen.
The error of the method is otherwise called the theoretical error.
The error of the method is determined by the accuracy of measuring the distance from the body surface to the proximal surface of the liver, which was measured by ultrasound.
INTRODUCTION
Any measurements, no matter how carefully they are performed, are accompanied by errors (errors), i.e., deviations of the measured values from their true value. This is explained by the fact that in the process of measurement conditions are constantly changing: the state of the environment, the measuring device and the object being measured, as well as the attention of the performer. Therefore, when measuring a quantity, its approximate value is always obtained, the accuracy of which must be estimated. Another problem also arises: to choose an instrument, conditions and technique in order to perform measurements with a given accuracy. The theory of errors helps to solve these problems, which studies the laws of distribution of errors, establishes evaluation criteria and tolerances for measurement accuracy, methods for determining the most probable value of the quantity being determined, and rules for predicting the expected accuracy.
12.1. MEASUREMENTS AND THEIR CLASSIFICATION
Measurement is the process of comparing a measured value with another known value, taken as a unit of measurement.
All quantities with which we are dealing are divided into measured and calculated. measured the value is called its approximate value, found by comparison with a homogeneous unit of measure. So, sequentially laying the survey tape in a given direction and counting the number of layings, they find the approximate value of the length of the section.
Computed a quantity is its value determined from other measured quantities that are functionally related to it. For example, the area of a rectangular area is the product of its measured length and width.
To detect misses (gross errors) and improve the accuracy of the results, the same value is measured several times. By accuracy, such measurements are divided into equal and unequal. Equivalent
- homogeneous multiple measurement results of the same quantity, performed by the same instrument (or different instruments of the same accuracy class), in the same way and in the same number of steps, under identical conditions. unequal
- measurements made in case of non-compliance with the conditions of equal accuracy.
In the mathematical processing of measurement results, the number of measured values is of great importance. For example, to get the value of each angle of a triangle, it is enough to measure only two of them - this will be necessary
number of values. In the general case, to solve any topographic-geodesic problem, it is necessary to measure a certain minimum number of quantities that ensures the solution of the problem. They are called the number of required quantities
or measurements. But in order to judge the quality of the measurements, check their correctness and improve the accuracy of the result, the third angle of the triangle is also measured - excess
. The number of redundant values
(k
) is the difference between the number of all measured quantities ( P
) and the number of required quantities ( t
):
In topographic and geodetic practice, redundant measured values are indispensable. They make it possible to detect errors (errors) in measurements and calculations and increase the accuracy of the determined values.
By physical performance
measurements can be direct, indirect and remote.
Direct
measurements are the simplest and historically the first types of measurements, for example, measuring the lengths of lines with a survey tape or tape measure.
Indirect
measurements are based on the use of certain mathematical relationships between the sought and directly measured quantities. For example, the area of a rectangle on the ground is determined by measuring the lengths of its sides.
remote
measurements are based on the use of a number of physical processes and phenomena and, as a rule, are associated with the use of modern technical means: light range finders, electronic total stations, phototheodolites, etc.
Measuring instruments used in topographic and geodetic production can be divided into three main classes :
- high-precision (precision);
- accurate;
- technical.
12.2. MEASUREMENT ERRORS
With repeated measurement of the same value, each time slightly different results are obtained, both in absolute value and in signs, no matter how experienced the performer has and no matter what high-precision instruments he uses.
Errors are distinguished: gross, systematic and random.
Appearance rough
errors ( misses
) is associated with serious errors in the production of measurement work. These errors are easily identified and eliminated as a result of measurement control.
Systematic errors
are included in each measurement result according to a strictly defined law. They are due to the influence of the design of measuring instruments, errors in the calibration of their scales, wear, etc. ( instrumental errors)
or arise due to underestimation of the measurement conditions and the patterns of their changes, the approximation of some formulas, etc. ( methodological errors).
Systematic errors are divided into permanent
(invariant in sign and magnitude) and variables
(changing their value from one dimension to another according to a certain law).
Such errors are predetermined and can be reduced to the required minimum by introducing appropriate corrections.
For example, the influence of the curvature of the Earth on the accuracy of determining vertical distances, the influence of air temperature and atmospheric pressure can be taken into account in advance when determining the lengths of lines with light range finders or electronic total stations, the influence of atmospheric refraction can be taken into account in advance, etc.
If gross errors are not allowed and systematic errors are eliminated, then the quality of measurements will be determined only random errors. These errors are unavoidable, but their behavior is subject to the laws of large numbers. They can be analyzed, controlled and reduced to the necessary minimum.
To reduce the influence of random errors on the measurement results, they resort to repeated measurements, to improve working conditions, choose more advanced instruments, measurement methods and carry out their careful production.
Comparing the series of random errors of equally accurate measurements, it can be found that they have the following properties:
a) for a given type and measurement conditions, random errors cannot exceed a certain limit in absolute value;
b) errors that are small in absolute value appear more often than large ones;
c) positive errors appear as often as negative ones equal in absolute value;
d) the arithmetic mean of random errors of the same value tends to zero with an unlimited increase in the number of measurements.
The distribution of errors corresponding to the specified properties is called normal (Fig. 12.1).
Rice. 12.1. Curve of normal distribution of Gaussian random errors
The difference between the measurement result of some quantity ( l) and its true meaning ( X) called absolute (true) error .
Δ = l - XThe true (absolutely accurate) value of the measured quantity cannot be obtained, even using the highest accuracy instruments and the most advanced measurement technique. Only in some cases can the theoretical value of the quantity be known. The accumulation of errors leads to the formation of discrepancies between the measurement results and their actual values.
The difference between the sum of practically measured (or calculated) values and its theoretical value is called inviscid.
For example, the theoretical sum of the angles in a flat triangle is 180º, and the sum of the measured angles turned out to be 180º02"; then the error of the sum of the measured angles will be +0º02". This error will be the angular discrepancy of the triangle.
Absolute error is not a complete indicator of the accuracy of the work performed. For example, if some line whose actual length is 1000 m, measured with a survey tape with an error of 0.5 m, and a segment of length 200 m- with an error of 0.2 m, then, despite the fact that the absolute error of the first measurement is greater than the second, the first measurement was nevertheless performed with an accuracy twice as high. Therefore, the concept is introduced relative errors:
The ratio of the absolute error of the measured valueΔ to the measured valuelcalled relative error.
Relative errors are always expressed as a fraction with a numerator equal to one (aliquot fraction). So, in the above example, the relative error of the first measurement is
and the second
12.3 MATHEMATICAL PROCESSING OF THE RESULTS OF EQUAL-ACCURACY MEASUREMENTS OF A SINGLE VALUE
Let some quantity with true value X measured equally n
times and the results are: l 1
, l 2
, l 3
, …
li
(i = 1, 2, 3, … n), which is often referred to as a series of measurements. It is required to find the most reliable value of the measured quantity, which is called most likely
,
and evaluate the accuracy of the result.
In the theory of errors, the most probable value for a series of equally accurate measurement results is average
,
i.e.
(12.1)
In the absence of systematic errors, the arithmetic mean with an unlimited increase in the number of measurements tends to the true value of the measured value.
To enhance the influence of larger errors on the result of estimating the accuracy of a series of measurements, one uses root mean square error
(UPC). If the true value of the measured quantity is known, and the systematic error is negligible, then the root mean square error ( m
) of a single result of equally accurate measurements is determined by the Gauss formula:
m =
(12.2)
,
Where Δ i is true error.
In geodetic practice, the true value of the measured quantity in most cases is not known in advance. Then the root-mean-square error of a single measurement result is calculated from the most probable errors ( δ ) individual measurement results ( l i ); according to the Bessel formula:
m
= 
(12.3)
Where are the most likely errors ( δ i ) are defined as the deviation of the measurement results from the arithmetic mean
δ i =l i - µ
Often, next to the most probable value of a quantity, its root-mean-square error is also written ( m), e.g. 70°05" ± 1". This means that the exact value of the angle can be more or less than the specified value by 1 ". However, this minute cannot be added to the angle or subtracted from it. It characterizes only the accuracy of obtaining results under given measurement conditions.
An analysis of the Gaussian normal distribution curve shows that with a sufficiently large number of measurements of the same value, the random measurement error can be:
- greater than rms m in 32 cases out of 100;
- greater than twice the root mean square 2m in 5 cases out of 100;
- more than three times the root mean square 3m in 3 cases out of 1000.
It is unlikely that the random measurement error is greater than three times the root mean square, so tripled root mean square error is considered limiting:
Δ prev. = 3m
The marginal error is such a value of random error, the occurrence of which under the given measurement conditions is unlikely.
The root mean square error is also taken as the limiting error, equal to
Δprev = 2.5m ,
With an error probability of about 1%.
RMS error of the sum of the measured values
The square of the mean square error of the algebraic sum of the argument is equal to the sum of the squares of the mean square errors of the terms
m S 2 = m 1 2+m 2 2+m 3 2 + ..... + m n 2
In the particular case when m 1 = m 2 = m 3 = m n= m to determine the root mean square error of the arithmetic mean, use the formula
m S =
The root mean square error of the algebraic sum of equal measurements is several times greater than the root mean square error of one term.
Example.
If 9 angles are measured with a 30-second theodolite, then the root mean square error of the angle measurements will be
m coal = 30 " = ±1.5"
RMS error of the arithmetic mean
(accuracy of determining the arithmetic mean)
RMS error of the arithmetic mean (mµ )times less than the root mean square of one measurement.
This property of the root mean square error of the arithmetic mean allows you to improve the accuracy of measurements by increasing the number of measurements .
For example, it is required to determine the value of the angle with an accuracy of ± 15 seconds in the presence of a 30-second theodolite.
If you measure the angle 4 times ( n) and determine the arithmetic mean, then the root mean square error of the arithmetic mean ( mµ ) will be ± 15 seconds.
The root mean square error of the arithmetic mean ( m µ ) shows to what extent the influence of random errors is reduced during repeated measurements.
Example
A 5-fold measurement of the length of one line was made.
Based on the measurement results, calculate: the most probable value of its length L(average); probable errors (deviations from the arithmetic mean); root mean square error of one measurement m; accuracy of determining the arithmetic mean mµ, and the most probable value of the line length, taking into account the root-mean-square error of the arithmetic mean ( L).
Processing distance measurements (example)
Table 12.1.
Measurement number |
measurement result, |
Most likely errors di, cm |
The square of the most probable error, cm 2 |
Characteristic |
|---|---|---|---|---|
m=±=±19cm |
||||
Σ di = 0 |
[Σ di]2 = 1446 |
L= (980.65 ±0.08) m |
12.4. WEIGHTS OF THE RESULTS OF UNEQUAL MEASUREMENTS
With unequal measurements, when the results of each measurement cannot be considered equally reliable, it is no longer possible to get by with the definition of a simple arithmetic mean. In such cases, the merit (or reliability) of each measurement result is taken into account.
The dignity of the measurement results is expressed by a certain number called the weight of this measurement.
. Obviously, the arithmetic average will carry more weight than a single measurement, and measurements made with a more advanced and accurate instrument will have a greater degree of confidence than the same measurements made with a less accurate instrument.
Since the measurement conditions determine a different value of the root-mean-square error, it is customary to take the latter as basics of estimating weight values,
measurements. In this case, the weights of the measurement results are taken inversely proportional to the squares of their corresponding root-mean-square errors
.
So, if denoted by R And R measurement weights having root-mean-square errors, respectively m And µ
, then we can write the proportionality relation:
For example, if µ the root mean square error of the arithmetic mean, and m- respectively, one dimension, then, as follows from
can be written:
i.e. the weight of the arithmetic mean in n times the weight of a single measurement.
Similarly, it can be found that the weight of an angle measurement made with a 15-second theodolite is four times the weight of an angle measurement made with a 30-second instrument.
In practical calculations, the weight of any one quantity is usually taken as a unit, and under this condition, the weights of the remaining measurements are calculated. So, in the last example, if we take the weight of the result of an angular measurement with a 30-second theodolite as R= 1, then the weight value of the measurement result with a 15-second theodolite will be R = 4.
12.5. REQUIREMENTS FOR FORMATTING THE RESULTS OF FIELD MEASUREMENTS AND THEIR PROCESSING
All materials of geodetic measurements consist of field documentation, as well as documentation of computational and graphic works. Many years of experience in the production of geodetic measurements and their processing allowed us to develop the rules for maintaining this documentation.
Registration of field documents
Field documents include materials for checking geodetic instruments, measurement logs and special forms, outlines, picket logs. All field documentation is considered valid only in the original. It is compiled in a single copy and, in case of loss, can be restored only by repeated measurements, which is practically not always possible.
The rules for keeping field logs are as follows.
1. Field journals should be filled out carefully, all numbers and letters should be written clearly and legibly.
2. Correction of numbers and their erasure, as well as writing numbers by numbers are not allowed.
3. Erroneous records of readings are crossed out with one line and “erroneous” or “misprint” is indicated on the right, and the correct results are inscribed on top.
4. All entries in the journals are made with a simple pencil of medium hardness, ink or a ballpoint pen; the use of chemical or colored pencils for this is not recommended.
5. When performing each type of geodetic survey, records of the measurement results are made in the appropriate journals of the established form. Before the start of work, the pages of the magazines are numbered and their number is certified by the head of the work.
6. In the process of field work, pages with rejected measurement results are crossed out diagonally with one line, the reason for the rejection and the number of the page containing the results of repeated measurements are indicated.
7. In each journal, on the title page, fill in information about the geodetic instrument (brand, number, standard error of measurement), record the date and time of observations, weather conditions (weather, visibility, etc.), names of performers, provide the necessary schemes, formulas and notes.
8. The journal must be filled in in such a way that another performer who is not involved in field work can accurately perform the subsequent processing of the measurement results. When filling out field journals, the following entry forms should be followed:
a) the numbers in the columns are written so that all the digits of the corresponding digits are located one below the other without offset.
b) all results of measurements performed with the same accuracy are recorded with the same number of decimal places.
Example
356.24 and 205.60 m - correct,
356.24 and 205.6 m - wrong;
c) the values of minutes and seconds in angular measurements and calculations are always written in two-digit numbers.
Example
127°07"05 "
, not 127º7"5 "
;
d) in the numerical values of the measurement results, write down such a number of digits that allows you to get the reading device of the corresponding measuring instrument. For example, if the length of the line is measured with a tape measure with millimeter divisions and the reading is carried out with an accuracy of 1 mm, then the reading should be recorded as 27.400 m, not 27.4 m. Or if the goniometer only allows reading whole minutes, then the reading will be written as 47º00 " , not 47º or 47º00"00".
12.5.1. The concept of the rules of geodetic calculations
The processing of the measurement results is started after checking all field materials. At the same time, one should adhere to the rules and techniques developed by practice, the observance of which facilitates the work of the calculator and allows him to rationally use computer technology and auxiliary means.
1. Before processing the results of geodetic measurements, a detailed computational scheme should be developed, which indicates the sequence of actions that allows obtaining the desired result in the simplest and fastest way.
2. Taking into account the amount of computational work, choose the most optimal means and methods of calculations that require the least cost while ensuring the required accuracy.
3. The accuracy of the calculation results cannot be higher than the measurement accuracy. Therefore, sufficient, but not excessive, accuracy of computational operations should be specified in advance.
4. When calculating, one should not use drafts, since rewriting digital material takes a lot of time and is often accompanied by errors.
5. To record the results of calculations, it is recommended to use special schemes, forms and statements that determine the procedure for calculations and provide intermediate and general control.
6. Without control, the calculation cannot be considered complete. Control can be performed using a different move (method) for solving the problem or by performing repeated calculations by another performer (in "two hands").
7. Calculations always end with the determination of errors and their mandatory comparison with the tolerances provided for by the relevant instructions.
8. Special requirements for computational work are imposed on the accuracy and clarity of recording numbers in computational forms, since carelessness in entries leads to errors.
As in field journals, when writing columns of numbers in computational schemes, digits of the same digits should be placed one under the other. In this case, the fractional part of the number is separated by a comma; it is desirable to write multi-digit numbers at intervals, for example: 2 560 129.13. Calculation records should be kept only in ink, in roman type; erroneous results are carefully crossed out and the corrected values are written on top.
When processing measurement materials, one should know with what accuracy the results of calculations should be obtained in order not to operate with an excessive number of characters; if the final result of the calculation is obtained with more digits than necessary, then the numbers are rounded off.
12.5.2. Rounding numbers
Round up to n signs - means to keep in it the first n significant digits.
The significant digits of a number are all of its digits from the first non-zero digit on the left to the last digit written on the right. In this case, zeros on the right are not considered significant figures if they replace unknown figures or are put in place of other figures when rounding a given number.
For example, the number 0.027 has two significant digits, and the number 139.030 has six significant digits.
When rounding numbers, the following rules should be followed.
1. If the first of the discarded digits (counting from left to right) is less than 5, then the last remaining digit is retained unchanged.
For example, the number 145.873, after rounding to five significant digits, would be 145.87.
2. If the first of the discarded digits is greater than 5, then the last remaining digit is increased by one.
For example, the number 73.5672, after rounding it to four significant digits, will be 73.57.
3. If the last digit of the rounded number is the number 5 and it must be discarded, then the digit preceding it in the number is increased by one only if it is odd (even number rule).
For example, the numbers 45.175 and 81.325, after rounding to 0.01, will be 45.18 and 81.32, respectively.
12.5.3. Graphic works
The value of graphic materials (plans, maps and profiles), which are the final result of geodetic surveys, is largely determined not only by the accuracy of field measurements and the correctness of their computational processing, but also by the quality of graphic execution. Graphic work should be carried out using carefully checked drawing tools: rulers, triangles, geodesic protractors, measuring compasses, sharpened pencils (T and TM), etc. The organization of the workplace has a great influence on the quality and productivity of drawing work. Drawing work should be carried out on sheets of high-quality drawing paper, fixed on a flat table or on a special drawing board. The drawn pencil original of the graphic document, after careful checking and correction, is drawn up in ink in accordance with the established conventional signs.
Questions and tasks for self-control
- What does the expression "measure something" mean?
- How are measurements classified?
- How are measuring devices classified?
- How are measurement results classified by accuracy?
- What measurements are called equal?
- What do the concepts mean: necessary And excess number of measurements?
- How are measurement errors classified?
- What causes systematic errors?
- What are the properties of random errors?
- What is called absolute (true) error?
- What is referred to as relative error?
- What is called the arithmetic mean in the theory of errors?
- What is called the mean square error in the theory of errors?
- What is the marginal mean square error?
- How is the root mean square error of the algebraic sum of equally accurate measurements and the root mean square error of one term related?
- What is the relationship between the root mean square error of the arithmetic mean and the root mean square error of one measurement?
- What does the root mean square error of the arithmetic mean show?
- What parameter is taken as the basis for estimating the weight values?
- What is the relationship between the weight of the arithmetic mean and the weight of a single measurement?
- What are the rules adopted in geodesy for keeping field logs?
- List the basic rules of geodetic calculations.
- Round to 0.01 the numbers 31.185 and 46.575.
- List the basic rules for performing graphic work.
Terms measurement error And measurement error are used as synonyms.) It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. In this case, the average value obtained by statistical processing of the results of a series of measurements is taken as the true value. This value obtained is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, along with the result obtained, the measurement error is indicated. For example, the entry T=2.8±0.1 c. means that the true value of the quantity T lies in the interval from 2.7 s. before 2.9 s. some specified probability (see confidence interval, confidence probability, standard error).
In 2006, a new document was adopted at the international level, dictating the conditions for carrying out measurements and establishing new rules for comparing state standards. The concept of "error" became obsolete, instead of it the concept of "measurement uncertainty" was introduced.
Definition of error
Depending on the characteristics of the measured quantity, various methods are used to determine the measurement error.
- The Kornfeld method consists in choosing a confidence interval ranging from the minimum to the maximum measurement result, and an error as half the difference between the maximum and minimum measurement result:
- Root mean square error:
- The mean square error of the arithmetic mean:
Error classification
According to the form of presentation
- Absolute error - Δ X is an estimate of the absolute measurement error. The value of this error depends on the method of its calculation, which, in turn, is determined by the distribution of the random variable X meas . In this case, the equality:
Δ X = | X true − X meas | ,
Where X true is the true value, and X meas - the measured value, should be performed with some probability close to 1. If the random variable X meas distributed according to the normal law, then, usually, its standard deviation is taken as an absolute error. Absolute error is measured in the same units as the value itself.
- Relative error- the ratio of the absolute error to the value that is taken as true:
Relative error is a dimensionless quantity, or is measured in percent.
- Reduced error- relative error, expressed as the ratio of the absolute error of the measuring instrument to the conditionally accepted value of the quantity, constant over the entire measurement range or in part of the range. Calculated according to the formula
Where X n- normalizing value, which depends on the type of measuring instrument scale and is determined by its graduation:
If the scale of the device is one-sided, i.e. the lower measurement limit is zero, then X n is determined equal to the upper limit of measurements;
- if the scale of the device is two-sided, then the normalizing value is equal to the width of the measuring range of the device.
The given error is a dimensionless value (it can be measured as a percentage).
Due to the occurrence
- Instrumental / Instrumental Errors- errors that are determined by the errors of the measuring instruments used and are caused by the imperfection of the operating principle, the inaccuracy of the scale graduation, and the lack of visibility of the device.
- Methodological errors- errors due to the imperfection of the method, as well as simplifications underlying the methodology.
- Subjective / operator / personal errors- errors due to the degree of attentiveness, concentration, preparedness and other qualities of the operator.
In engineering, devices are used to measure only with a certain predetermined accuracy - the main error allowed by the normal under normal operating conditions for this device.
If the device is operated under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by the deviation of the ambient temperature from normal, installation, due to the deviation of the position of the device from the normal operating position, etc. 20°C is taken as normal ambient temperature, and 01.325 kPa as normal atmospheric pressure.
A generalized characteristic of measuring instruments is an accuracy class determined by the limit values of the permissible basic and additional errors, as well as other parameters that affect the accuracy of measuring instruments; the value of the parameters is established by the standards for certain types of measuring instruments. The accuracy class of measuring instruments characterizes their accuracy properties, but is not a direct indicator of the accuracy of measurements performed using these instruments, since the accuracy also depends on the measurement method and the conditions for their implementation. Measuring instruments, the limits of the permissible basic error of which are given in the form of reduced basic (relative) errors, are assigned accuracy classes selected from a number of the following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0 ;5.0;6.0)*10n, where n = 1; 0; -1; -2 etc.
According to the nature of the manifestation
- random error- error, changing (in magnitude and in sign) from measurement to measurement. Random errors can be associated with the imperfection of devices (friction in mechanical devices, etc.), shaking in urban conditions, with the imperfection of the object of measurement (for example, when measuring the diameter of a thin wire, which may not have a completely round cross section as a result of the imperfection of the manufacturing process ), with the features of the measured quantity itself (for example, when measuring the number of elementary particles passing per minute through a Geiger counter).
- Systematic error- an error that changes over time according to a certain law (a special case is a constant error that does not change over time). Systematic errors can be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
- Progressive (drift) error is an unpredictable error that changes slowly over time. It is a non-stationary random process.
- Gross error (miss)- an error resulting from an oversight of the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the division number on the scale of the device, if there was a short circuit in the electrical circuit).
According to the method of measurement
- Accuracy of direct measurements
- Uncertainty of indirect measurements- error of the calculated (not measured directly) value:
If F = F(x 1 ,x 2 ...x n) , Where x i- directly measured independent quantities with an error Δ x i, Then:
see also
- Measurement of physical quantities
- System for automated data collection from meters over the air
Literature
- Nazarov N. G. Metrology. Basic concepts and mathematical models. M.: Higher school, 2002. 348 p.
- Laboratory classes in physics. Textbook / Goldin L. L., Igoshin F. F., Kozel S. M. and others; ed. Goldina L. L. - M .: Science. Main edition of physical and mathematical literature, 1983. - 704 p.
Wikimedia Foundation. 2010 .
ERRORS OF MEASUREMENTS OF PHYSICAL QUANTITIES AND
MEASUREMENT RESULTS PROCESSING
by measurement called finding the values of physical quantities empirically with the help of special technical means. Measurements are either direct or indirect. At direct measurement, the desired value of a physical quantity is found directly with the help of measuring instruments (for example, measuring the dimensions of bodies using a caliper). Indirect called a measurement in which the desired value of a physical quantity is found on the basis of a known functional relationship between the measured quantity and the quantities subjected to direct measurements. For example, when determining the volume V of a cylinder, its diameter D and height H are measured, and then according to the formula p D 2 /4 calculate its volume.
Due to the inaccuracy of measuring instruments and the difficulty of taking into account all side effects in measurements, measurement errors inevitably arise. error or mistake measurement refers to the deviation of the measurement result from the true value of the measured physical quantity. The measurement error is usually unknown, as is the true value of the measured quantity. Therefore, the task of elementary processing of measurement results is to establish the interval within which the true value of the measured physical quantity is located with a given probability.
Classification of measurement errors
Errors are divided into three types:
1) gross or misses,
2) systematic,
3) random.
gross errors- these are erroneous measurements resulting from careless reading on the device, illegible recording of readings. For example, writing a result of 26.5 instead of 2.65; reading on a scale of 18 instead of 13, etc. If a gross error is detected, the result of this measurement should be immediately discarded, and the measurement itself should be repeated.
Systematic errors- errors that remain constant during repeated measurements or change according to a certain law. These errors may be due to the wrong choice of measurement method, imperfection or malfunction of the instruments (for example, measurements using an instrument that has a zero offset). In order to eliminate systematic errors as much as possible, one should always carefully analyze the measurement method, compare instruments with standards. In the future, we will assume that all systematic errors have been eliminated, except for those caused by inaccuracies in the manufacture of devices and reading errors. We will call this error hardware.
Random errors - These are errors, the cause of which cannot be taken into account in advance. Random errors depend on the imperfection of our sense organs, on the continuous action of changing external conditions (changes in temperature, pressure, humidity, air vibration, etc.). Random errors are unavoidable, they are inevitably present in all measurements, but they can be estimated using the methods of probability theory.
Processing the results of direct measurements
Let, as a result of direct measurements of a physical quantity, a series of its values be obtained:
x 1 , x 2 , ... x n .
Knowing this series of numbers, you need to indicate the value closest to the true value of the measured value, and find the value of the random error. This problem is solved on the basis of probability theory, a detailed presentation of which is beyond the scope of our course.
The most probable value of the measured physical quantity (close to the true value) is the arithmetic mean
. (1)
Here x i is the result of the i-th measurement; n is the number of measurements. Random measurement error can be estimated by the absolute error D x, which is calculated by the formula
,
(2)
where t(a ,n) - Student's coefficient, depending on the number of measurements n and the confidence level a . Confidence value a set by the experimenter.
Probability random event is the ratio of the number of cases favorable for this event to the total number of equally likely cases. The probability of a sure event is 1, and an impossible one is 0.
The value of the Student's coefficient corresponding to a given confidence level a and a certain number of measurements n, find according to the table. 1.
Table 1
|
Number measurements n |
Confidence probability a |
|||
|
0,95 |
0,98 |
|||
|
1,38 |
12,7 |
31,8 |
||
|
1,06 |
||||
|
0,98 |
||||
|
0,94 |
||||
|
0,92 |
||||
|
0,90 |
||||
|
0,90 |
||||
|
0,90 |
||||
|
0,88 |
||||
|
0,84 |
||||
From Table. 1 it can be seen that the value of the Student's coefficient and the random measurement error are the smaller, the larger n and the smaller a . Practically choose a =0.95. However, a simple increase in the number of measurements cannot reduce the total error to zero, since any measuring device gives an error.
Let us explain the meaning of the terms absolute error D x and confidence level a using the number line. Let the average value of the measured quantity
Fig.1
If the measurements of the same quantity are repeated by the same instruments under the same conditions, then the true value of the measured quantity x est will fall into the same confidence interval, but the hit will not be reliable, but with a probability a.
Calculating the magnitude of the absolute error D x by formula (2), the true value x of the measured physical quantity can be written as x=
To assess the accuracy of measuring a physical quantity, calculate relative error which is usually expressed as a percentage
. (3)
Thus, when processing the results of direct measurements, it is necessary to do the following:
1. Take measurements n times.
2. Calculate the arithmetic mean using formula (1).
3. Set a confidence level a (usually take a = 0.95).
4. According to Table 1, find the Student's coefficient corresponding to the given confidence level a and the number of dimensions n.
5. Calculate the absolute error using formula (2) and compare it with the instrumental one. For further calculations, take the one that is larger.
6. Using formula (3), calculate the relative error e.
7. Write down the final result
x=
Processing the results of indirect measurements
Let the desired physical quantity y be associated with other quantities x 1 , x 2 , ... x k by some functional dependence
Y=f(x 1 , x 2 , ... x k) (4)
Among the values x 1 , x 2 , ... x k there are values obtained from direct measurements and tabular data. It is required to determine the absolute D y and relative e errors in the value of y.
In most cases, it is easier to calculate the relative error first, and then the absolute error. From the theory of probability, the relative error of indirect measurement
.
(5)
Here 
, where is the partial derivative of the function with respect to the variable x i, in the calculation of which all values, except for x i , are considered constant; D x i is the absolute error of x i . If x i is obtained as a result of direct measurements, then its average value
Let's write the final result:
y=
Here
Usually, both random and systematic (instrumental) errors are present in real measurements. If the calculated random error of direct measurements is equal to zero or less than the hardware error by two or more times, then when calculating the error of indirect measurements, the hardware error should be taken into account. If these errors differ by less than two times, then the absolute error is calculated by the formula
.
Consider an example. Let it be necessary to calculate the volume of the cylinder:
. (6)
Here D is the diameter of the cylinder, H is its height, measured with a vernier caliper with a division value of 0.1 mm. As a result of repeated measurements, we find the average values
,
(7)
where D D and D H are absolute errors of direct measurements of diameter and height. Their values are calculated by formula (2): D D=0.01 mm; D H=0.13 mm. Let's compare the calculated errors with the hardware one, equal to the division value of the caliper. D D<0.1, поэтому в формуле (7) подставим вместо D D is not 0.01 mm, but 0.1 mm.
p value must be chosen such that the relative error Dp/p in formula (7) could be neglected. From analysis of measured values and calculated absolute errors D D and D H, it can be seen that the height measurement error makes the greatest contribution to the relative volume measurement error. Calculating the relative height error gives e H =0.01. Therefore, the value p you need to take 3.14. In this case Dp / p » 0.001 (Dp =3.142-3.14=0.002).
One significant figure is left in the absolute error.
Notes.
1. If the measurements are made once or the results of multiple measurements are the same, then the absolute measurement error should be taken as the instrumental error, which for most of the instruments used is equal to the division value of the instrument (for more details on the instrumental error, see the “Measuring instruments” section).
2. If tabular or experimental data are given without specifying the error, then the absolute error of such numbers is taken equal to half the order of the last significant digit.
Actions with approximate numbers
The issue of different calculation accuracy is very important, since overestimation of the calculation accuracy leads to a large amount of unnecessary work. Students often calculate the value they are looking for to five or more significant digits. It should be understood that this precision is excessive. It makes no sense to carry out calculations beyond the limit of accuracy, which is provided by the accuracy of determining directly measured quantities. After processing the measurements, they often do not calculate the errors of individual results and judge the error of the approximate value of the quantity, indicating the number of correct significant digits in this number.
Significant figures An approximate number is called all digits except zero, as well as zero in two cases:
1) when it stands between significant figures (for example, in the number 1071 - four significant figures);
2) when it stands at the end of the number and when it is known that the unit of the corresponding digit is not available in the given number. Example. There are three significant figures in the number 5.20, and this means that when measuring we took into account not only units, but also tenths and hundredths, and in the number 5.2 - only two significant figures, which means that we took into account only integers and tenths.
Approximate calculations should be made in compliance with the following rules.
1. When adding and subtracting as a result, retain as many decimal places as there are in the number with the least number of decimal places. For example: 0.8934+3.24+1.188=5.3214» 5.32. The amount should be rounded to hundredths, i.e. take equal to 5.32.
2. When multiplying and dividing as a result, as many significant digits are retained as the approximate number with the fewest significant digits has. For example, you need to multiply 8.632´ 2.8´ 3.53. Instead, expressions should be evaluated
8.6 ´ 2.8 ´ 3.5 » 81.
When calculating intermediate results, they save one digit more than the rules recommend (the so-called spare digit). In the final result, the spare digit is discarded. To refine the value of the last significant digit of the result, you need to calculate the digit behind it. If it turns out to be less than five, it should simply be discarded, and if five or more than five, then, having discarded it, the previous figure should be increased by one. Usually, one significant digit is left in the absolute error, and the measured value is rounded up to the digit in which the significant digit of the absolute error is located.
3. The result of calculating the values of the functions x n , , lg( x) some approximate number x must contain as many significant digits as there are in the number x. For example: 
.
Plotting
The results obtained during the performance of laboratory work are often important and must be presented in a graphical relationship. In order to build a graph, it is necessary, on the basis of the measurements made, to compile a table in which each value of one of the quantities corresponds to a certain value of the other.
Graphs are made on graph paper. When constructing a graph, the values of the independent variable should be plotted on the abscissa, and the values of the function on the ordinate. Near each axis, you need to write the designation of the displayed value and indicate in what units it is measured (Fig. 2).
Fig.2
For the correct construction of the graph, the choice of scale is important: the curve occupies the entire sheet, and the dimensions of the graph in length and height are approximately the same. The scale should be simple. The easiest way is if the unit of the measured value (0.1; 10; 100, etc.) corresponds to 1, 2 or 5 cm. It should be borne in mind that the intersection of the coordinate axes does not have to coincide with the zero values of the values being plotted (Fig. 2).
Each experimental value obtained is plotted on the graph in a fairly noticeable way: a dot, a cross, etc.
Errors are indicated for the measured values in the form of segments with a length of a confidence interval, in the center of which the experimental points are located. Since the indication of errors clutters up the graph, this is done only when information about the errors is really needed: when constructing a curve from experimental points, when determining errors using a graph, when comparing experimental data with a theoretical curve (Figure 2). Often it is enough to specify the error for one or more points.
It is necessary to draw a smooth curve through the experimental points. Often, the experimental points are connected by a simple broken line. Thus, as it were, it is indicated that the quantities depend on each other in some jumpy way. And this is incredible. The curve must be smooth and may pass not through the marked points, but close to them so that these points are on both sides of the curve at the same distance from it. If any point strongly falls out of the graph, then this measurement should be repeated. Therefore, it is desirable to build a graph directly during the experiment. The graph can then serve to control and improve observations.
MEASURING INSTRUMENTS AND ACCOUNTING FOR THEIR ERRORS
Measuring instruments are used for direct measurements of physical quantities. Any measuring instruments do not give the true value of the measured value. This is due, firstly, to the fact that it is impossible to accurately read the measured value on the scale of the instrument, and secondly, to the inaccuracy in the manufacture of measuring instruments. To take into account the first factor, the reading error Δx o is introduced, for the second - the allowable errorΔ x d. The sum of these errors forms the instrumental or absolute error of the deviceΔ x:
.
The permissible error is normalized by state standards and indicated in the passport or description of the device.
The reading error is usually taken equal to half the division of the instrument, but for some instruments (stopwatch, aneroid barometer) - equal to the division of the instrument (since the position of the arrow of these instruments changes in jumps by one division) and even several divisions of the scale, if the conditions of the experiment do not allow confidently count up to one division (for example, with a thick pointer or poor lighting). Thus, the counting error is set by the experimenter himself, actually reflecting the conditions of a particular experiment.
If the allowable error is much less than the reading error, then it can be ignored. Usually, the absolute error of the instrument is taken equal to the scale division of the instrument.
Measuring rulers usually have millimeter divisions. For measurement, it is recommended to use steel or drawing rulers with a bevel. The permissible error of such rulers is 0.1 mm and it can be ignored, since it is much less than the reading error equal to ± 0.5 mm. Permissible error of wooden and plastic rulers± 1 mm.
The permissible measurement error of a micrometer depends on the upper limit of measurement and can be ± (3-4) µm (for micrometers with measuring range 0-25 mm). Half of the division value is taken as the reading error. Thus, the absolute error of the micrometer can be taken equal to the division value, i.e. 0.01 mm.
When weighing, the permissible error of technical scales depends on the load and amounts to 50 mg for a load of 20 to 200 g, and 25 mg for a load of less than 20 g.
The error of digital instruments is determined by the accuracy class.
1. Measurement methods.
2. Measurement errors.
3. Choice of method and measuring instruments.
4. Choice of measurements.
1. Measurement methods . The measurement of a physical quantity can be carried out by various methods (methods), the choice of which in each individual case depends on the nature of the measured quantity, on the measurement conditions, on the device and principle of operation, as well as the required accuracy.
According to the method of obtaining the numerical value of the measured quantity, measurement methods are divided into 3 types:
2. Indirect
3. Cumulative
They differ in the nature of the use of measures.
The most important methods of direct measurements constantly encountered in practice include the following:
1. Method of direct evaluation.
2. Method of comparison, consisting of four varieties:
a) null method;
b) differential method;
c) substitution method;
d) matching method.
The essence of the method of direct evaluation It consists in the fact that the value of the measured quantity is judged by the indication of one or more direct conversion instruments, pre-calibrated in units of the measured quantity or in units of other quantities on which the measured quantity depends. It is one of the most common in technical practice (due to its simplicity), and a typical example of it is the measurement of electrical quantities with pointer instruments. The accuracy of this method is usually limited by the accuracy of measuring instruments. A distinctive feature of this method is that the measure does not directly participate in the measurement process.
The essence of the comparison method is that when using these methods, the measured value in the measurement process is compared with the value reproduced by the measure.
Thus, a distinctive feature of comparison methods is the direct participation of the measure in the measurement process. They differ in the nature of the use of measures.
A) Zero Method- this is a method in which the resulting effect of the impact of the measured quantity and exemplary measure on the comparator (zero indicator) is brought to zero. Examples of the use of zero methods in electrical engineering are bridge and compensation circuits. Zero methods are much more difficult than methods of direct estimation, require much more time, but their accuracy is incomparably higher (0.02% and higher).
Zero methods are used mainly in the verification of instruments used for direct evaluation.
B) Differential method - this is a method in which the difference between the measured value and the exemplary measure, or the difference in the effects produced by them, is directly estimated by measuring instruments.
Aiz-A=a
Аiz is the measured value; A - instrument reading; a is the error.
Knowing A and measuring a, one can find Aiz. The accuracy of this method is the higher, the smaller the measured difference and with the greater accuracy it is measured (if the difference between Aiz and A is 1% and measured with an accuracy of 1%, then the measurement accuracy will be already 0.01%).
Differential methods are used for precise laboratory measurements (calibration of exemplary resistances, calibration of measuring transformers, etc.).
C) Substitution method . This method consists in the fact that during the measurement process, the measured value Aiz is replaced in the measuring installation by the known value A, and by measuring the value A, the measuring installation is brought to its previous state, that is, the same instrument readings are achieved as when the value Aiz is used. Under such conditions Aiz =.
D) Match method . This method consists in the fact that the difference between the desired value and the exemplary measure is measured using the coincidence of scale marks or periodic signals. The essence of this method can be explained by the example of determining the size of an inch.
1inch= 127/5=254/10=25.4mm
Measurement errors.
When carrying out measurements, due to a number of reasons, the numerical value of the measured quantity obtained as a result of the experiment is only more or less approximate.
The deviation of the measurement results from the true value of the measured quantity is called Measurement error.
True (true) value A variable quantity is called its value, free from measurement errors.
Actual value - this is the value obtained as a result of a measurement with an allowable error (error).
Measurement errors can be classified according to a number of criteria:
1. By way of numerical expression measurement errors are divided into:
A) Absolute and b ) are relative.
Absolute error The difference between the measured and the actual value of the measured quantity is called.
∆ A=Aiz-Aq
For the actual values of the measured quantity, the readings of the exemplary instrument are taken.
Absolute error is measured in units of the measured quantity.
The reciprocal value of the absolute error is called the correction.
σ \u003d -ΔA
Relative error The ratio of the absolute error to the actual value of the measured quantity is called.
β = ΔA/AD\u003d Aiz - Hell / Hell; or β = ΔA/Ad 100%.
2. By the nature of the change Measurement errors are divided into:
A) systematic;
B) random;
C) gross errors (blunders).
Systematic Errors are called, subject to a certain law or remaining in
The measurement process is constant. These include errors due to the inaccuracy of the implementation of the measure, the incorrect calibration of the measuring device, the influence of the ambient temperature on the measures and measuring devices.
There are the following types of systematic errors:
1. Instrumental.
2. Device installation errors.
3. Personal errors (subjective).
4. Method errors (or theoretical).
Depending on the change in time, systematic errors are divided into: a) permanent; b) progressive; c) periodic.
To take into account and eliminate systematic errors, it is necessary to have, possibly, complete data on the presence of certain types of errors and on the causes of their occurrence.
Systematic errors can be eliminated or significantly reduced by eliminating sources of errors or introducing corrections that are stopped on the basis of a preliminary study of errors, by checking the measures and instruments used in the measurement, by introducing correction formulas and curves expressing the dependence of instrument readings on external conditions.
Random Errors are called, the change of which does not obey any regularity. They are found during repeated measurements of the desired value, when repeated measurements are carried out equally carefully and, it would seem, under the same conditions.
Random errors cannot be excluded empirically, but their influence on the measurement result can be theoretically taken into account by applying the methods of probability theory and mathematical statistics when processing the measurement results.
Gross mistakes - these are errors that are significantly larger than expected under given conditions. An example of gross errors can be incorrect readings of the readings of measuring instruments. Gross measurement errors are revealed during repeated measurements and should be discarded as credible.
General methods for improving the accuracy of measuring instruments.
In an effort to create more accurate measuring instruments, measuring technics has developed a number of general methods for achieving accuracy, which can be divided into four groups:
1. Stabilization of the most important parameters of measuring instruments by technological means, i.e. by using the most stable parts, materials and appropriate manufacturing techniques.
2. Method of passive protection against rapidly changing influencing quantities, i.e. reduction of random errors of measuring instruments by applying filtering, damping, thermal insulation, etc.
3. Methods of active protection against slowly changing influencing quantities by stabilizing these quantities.
4. Methods for correcting systematic and progressive errors and static processing of random errors.
Increasing the accuracy of measurements is usually associated with the complexity of the equipment and the increase in time.
(large repetition) measurements. And this is not always justified. It is also obvious that the special accuracy of measuring quantities that have little effect on the numerical value of the overall final result is inexpedient.
So, for example, when measuring the values x1, x2 and x3 to determine the value y=x12*x2β*x3γ, it is hardly advisable to achieve special accuracy in measuring x1 if the exponent α = 1, β = 2, γ = 3.
The required accuracy must correspond to the tasks and conditions of measurement.
The choice of method and means of measurement.
When choosing a measurement method, one should be guided by the required accuracy of the measurement results.
According to the accuracy of the results obtained, they can be divided into three groups:
1. The result of the measurement should have the highest possible accuracy with the existing level of measuring technology.
Such measurements are called Accurate (precision). For example, measurements of physical constants, reference measurements, some special measurements related to the most accurate operation of individual devices.
2. Measurements, the error of the result of which should not exceed a certain specified value.
Such measurements are called Superficial control. They are carried out in calibration control and measurement laboratories with such measuring instruments and according to such a procedure as to guarantee the error of the result, not exceeding some predetermined value.
3. Measurements in which the error of the result is determined by the characteristics of the measuring devices.
Such measurements are called Technical.
These include laboratory measurements carried out during various types of processing and research, and research, and production, and acceptance, and operational measurements carried out to ensure the necessary mode of operation of various objects and devices.
Instruments for measurements are selected according to a number of indicators: type of current, frequency, range of the measured value, accuracy, input parameters, degree of influence of external factors.
1. The type of current of the circuit under study determines the principle of operation and the system of the measuring device chosen for it.(U, I, R at direct current - ME, R-ED, accurate measurement of I, U, P, cosγ to a voltmeter - cf. D., measurements of average, effective values of current and voltage in audio and high frequency transmitted current circuits are used - rectifier, thermoelectric, electronic and electrostatic devices.Instantaneous values of variables are measured - oscelographs).
2. The rated frequency or frequency range of the measuring instrument or measure must correspond to the current frequency of the circuit under test.
The more the frequency of the circuit under study differs from the nominal frequency of the device or measure, the greater the measurement error.
3. The nominal limits of the device or measure should not exceed the upper limit of the measured value by more than 25%.
The more they differ, the less accurate the measurement results. For a given accuracy class, the relative error of the device or measure is allowed, the larger the smaller the measured value.
4. The accuracy classes of the selected measuring instrument or measure must be such that the allowable basic errors are 3 times smaller than the allowable errors of measurement data, since the maximum measurement error possible under these conditions cannot exceed
Three times the root-mean-square error of a series of measurements.
5. Depending on the switching circuit of the measuring device, its input resistance should possibly be greater or lesser.
The more accurate the measurements, the greater should be the input resistances of the measuring instruments connected in parallel, and the smaller they should be for the instruments connected in series in the circuit under study.
6. When choosing the right measuring device, the specific measurement conditions and technical characteristics of the device should be taken into account.
Types of measurements.
The measurement process can be carried out in different ways depending on the type of the measured quantity and measurement methods.
By way of getting results distinguish between the following types of measurements:
1. Direct measurements.
2. indirect measurements.
3. Cumulative measurements.
To direct measurements Measurements, the result of which is obtained directly from the experimental measurement data, are included.
Direct measurement can be conditionally expressed by the formula Y=X, where
Y is the desired value of the measured value;
X is a value directly obtained from experimental data.
This type of measurement includes measurements of various physical quantities using instruments calibrated in established units (current - with an ammeter, temperature - with a thermometer). This type of measurement also includes measurements in which the desired value of a quantity is determined by directly comparing it with a measure.
indirect Such a measurement is called in which the desired value of a quantity is found on the basis of a known relationship between this quantity and the quantities subjected to direct measurements. In indirect measurements, the numerical value of the measured quantity is determined by calculation according to the formula.
Y = F (X1 , X2 , … , xn),
where y is the desired value of the measured quantity;
x1, x2, …, xn are measured values (R = U/I, P = U*I – in DC circuits).
Aggregate Such measurements are called for which the sought values of quantities are determined by solving a system of equations relating the values of the sought quantities with directly measured quantities, i.e., by solving a system of equations.
An example of this type of measurement is the determination of the temperature coefficients of resistance:
Rt = R20
Here Rt and t are measured by direct measurement, and α, β and R 20 are the desired values.
By changing the thermal regime of the coil and measuring Rt at a number of given temperatures t1; t2 and t3, we obtain a system of equations, the joint solution of which allows us to determine the numerical values of the unknown quantities.
