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. Expressed in the language of formulas, this definition can be written in the following 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. 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 this, the relative deviation is often additionally calculated, equal to the ratio of the absolute error to the exact value of the number. 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 it 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 valid regardless of the number of values being assessed. The third and final property of the relative error is that the relative estimate of the kth power is approximately | k | times the relative error of the original number.
Often in life we have to deal with various approximate quantities. Approximate calculations are always calculations with some error.
The absolute error of an approximate value is the magnitude of the difference between the exact value and the approximate value.
That is, you need to subtract the approximate value from the exact value and take the resulting number modulo. Thus, the absolute error is always positive.
Let's show what this might look like in practice. For example, we have a graph of a certain value, let it be a parabola: y=x^2.
From the graph we can determine the approximate value at some points. For example, at x=1.5 the value of y is approximately equal to 2.2 (y≈2.2).
Using the formula y=x^2 we can find the exact value at the point x=1.5 y= 2.25.
Now let's calculate the absolute error of our measurements. |2.25-2.2|=|0.05| = 0.05.
The absolute error is 0.05. In such cases, they also say the value is calculated with an accuracy of 0.05.
It often happens that the exact value cannot always be found, and therefore the absolute error cannot always be found.
For example, if we calculate the distance between two points using a ruler, or the value of the angle between two straight lines using a protractor, then we will get approximate values. But the exact value is impossible to calculate. In this case, we can specify a number such that the absolute error value cannot be greater.
In the example with a ruler, this will be 0.1 cm, since the division value on the ruler is 1 millimeter. In the example for the protractor, 1 degree because the protractor scale is graduated at every degree. Thus, the absolute error values in the first case are 0.1, and in the second case 1.
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 the 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. Wherein 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 digit of the result corresponds 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.
The points.
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 paired point method or the least squares method, 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 least squares forum in Svetozarov there is ("Basics...", section on 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 angular coefficient. 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
- the highest value of the magazine resistance, and - 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.
The main qualitative characteristic of any instrumentation sensor is the measurement error of the controlled parameter. The measurement error of a device is the amount of discrepancy between what the instrumentation sensor showed (measured) and what actually exists. The measurement error for each specific type of sensor is indicated in the accompanying documentation (passport, operating instructions, verification procedure), which is supplied with this sensor.
According to the form of presentation, errors are divided into absolute, relative And given errors.
Absolute error is the difference between the value of Xiz measured by the sensor and the actual value of Xd of this value.
The actual value Xd of the measured quantity is the experimentally found value of the measured quantity that is as close as possible to its true value. In simple terms, the actual value of Xd is the value measured by a reference device, or generated by a calibrator or setter of a high accuracy class. The absolute error is expressed in the same units as the measured value (for example, m3/h, mA, MPa, etc.). Since the measured value may be either greater or less than its actual value, the measurement error can be either with a plus sign (the device readings are overestimated) or with a minus sign (the device underestimates).
Relative error is the ratio of the absolute measurement error Δ to the actual value Xd of the measured quantity.
The relative error is expressed as a percentage, or is a dimensionless quantity, and can also take on both positive and negative values.
Reduced error is the ratio of the absolute measurement error Δ to the normalizing value Xn, constant over the entire measurement range or part of it.
The normalizing value Xn depends on the type of instrumentation sensor scale:
The given error is expressed as a percentage, or is a dimensionless quantity, and can also take both positive and negative values.
Quite often, the description of a particular sensor indicates not only the measurement range, for example, from 0 to 50 mg/m3, but also the reading range, for example, from 0 to 100 mg/m3. The given error in this case is normalized to the end of the measurement range, that is, to 50 mg/m3, and in the reading range from 50 to 100 mg/m3 the measurement error of the sensor is not determined at all - in fact, the sensor can show anything and have any measurement error. The measuring range of the sensor can be divided into several measuring subranges, for each of which its own error can be determined, both in magnitude and in the form of presentation. In this case, when checking such sensors, each sub-range can use its own standard measuring instruments, the list of which is indicated in the verification procedure for this device.
For some devices, the passports indicate the accuracy class instead of the measurement error. Such instruments include mechanical pressure gauges, indicating bimetallic thermometers, thermostats, flow indicators, pointer ammeters and voltmeters for panel mounting, etc. An accuracy class is a generalized characteristic of measuring instruments, determined by the limits of permissible basic and additional errors, as well as a number of other properties that affect the accuracy of measurements made with their help. Moreover, the accuracy class is not a direct characteristic of the accuracy of measurements performed by this device; it only indicates the possible instrumental component of the measurement error. The accuracy class of the device is applied to its scale or body in accordance with GOST 8.401-80.
When assigning an accuracy class to a device, it is selected from the series 1·10 n; 1.5 10 n; (1.6·10 n); 2·10n; 2.5 10 n; (3·10 n); 4·10n; 5·10n; 6·10n; (where n =1, 0, -1, -2, etc.). The values of accuracy classes indicated in brackets are not established for newly developed measuring instruments.
The measurement error of sensors is determined, for example, during their periodic verification and calibration. With the help of various setters and calibrators, certain values of one or another physical quantity are generated with high accuracy and the readings of the sensor being verified are compared with the readings of a standard measuring instrument to which the same value of the physical quantity is supplied. Moreover, the measurement error of the sensor is controlled both during the forward stroke (increase in the measured physical quantity from the minimum to the maximum of the scale) and during the reverse stroke (decreasing the measured value from the maximum to the minimum of the scale). This is due to the fact that due to the elastic properties of the sensor’s sensitive element (pressure sensor membrane), different rates of chemical reactions (electrochemical sensor), thermal inertia, etc. The sensor readings will be different depending on how the physical quantity affecting the sensor changes: decreases or increases.
Quite often, in accordance with the verification methodology, the readings of the sensor during verification should be performed not according to its display or scale, but according to the value of the output signal, for example, according to the value of the output current of the current output 4...20 mA.
For the pressure sensor being verified with a measurement scale from 0 to 250 mbar, the main relative measurement error over the entire measurement range is 5%. The sensor has a current output of 4...20 mA. The calibrator applied a pressure of 125 mbar to the sensor, while its output signal is 12.62 mA. It is necessary to determine whether the sensor readings are within acceptable limits.
First, it is necessary to calculate what the output current of the sensor Iout.t should be at a pressure Рт = 125 mbar.
Iout.t = Ish.out.min + ((Ish.out.max – Ish.out.min)/(Rsh.max – Rsh.min))*Рт
where Iout.t is the output current of the sensor at a given pressure of 125 mbar, mA.
Ish.out.min – minimum output current of the sensor, mA. For a sensor with an output of 4…20 mA, Ish.out.min = 4 mA, for a sensor with an output of 0…5 or 0…20 mA, Ish.out.min = 0.
Ish.out.max - maximum output current of the sensor, mA. For a sensor with an output of 0...20 or 4...20 mA, Ish.out.max = 20 mA, for a sensor with an output of 0...5 mA, Ish.out.max = 5 mA.
Рш.max – maximum of the pressure sensor scale, mbar. Psh.max = 250 mbar.
Rsh.min – minimum scale of the pressure sensor, mbar. Rsh.min = 0 mbar.
Рт – pressure supplied from the calibrator to the sensor, mbar. RT = 125 mbar.
Substituting the known values we get:
Iout.t = 4 + ((20-4)/(250-0))*125 = 12 mA
That is, with a pressure of 125 mbar applied to the sensor, its current output should be 12 mA. We consider the limits within which the calculated value of the output current can change, taking into account that the main relative measurement error is ± 5%.
ΔIout.t =12 ± (12*5%)/100% = (12 ± 0.6) mA
That is, with a pressure of 125 mbar applied to the sensor at its current output, the output signal should be in the range from 11.40 to 12.60 mA. According to the conditions of the problem, we have an output signal of 12.62 mA, which means that our sensor did not meet the measurement error specified by the manufacturer and requires adjustment.
The main relative measurement error of our sensor is:
δ = ((12.62 – 12.00)/12.00)*100% = 5.17%
Verification and calibration of instrumentation devices must be carried out under normal environmental conditions of atmospheric pressure, humidity and temperature and at the rated supply voltage of the sensor, since higher or lower temperatures and supply voltage may lead to additional measurement errors. The verification conditions are specified in the verification procedure. Devices whose measurement error does not fall within the limits established by the verification method are either re-adjusted and adjusted, after which they are re-verified, or, if the adjustment does not bring results, for example, due to aging or excessive deformation of the sensor, they are repaired. If repair is impossible, the devices are rejected and taken out of service.
If, nevertheless, the devices were able to be repaired, then they are no longer subject to periodic, but to primary verification with the implementation of all the points set out in the verification procedure for this type of verification. In some cases, the device is specially subjected to minor repairs () since according to the verification method, performing primary verification turns out to be much easier and cheaper than periodic verification, due to differences in the set of standard measuring instruments that are used for periodic and primary verification.
To consolidate and test the knowledge gained, I recommend doing this.
In the practical implementation of the measurement process, regardless of the accuracy of the measuring instruments, the correctness of the methodology and thoroughness
When performing measurements, the measurement results differ from the true value of the measured value, i.e. measurement errors are inevitable. When assessing the error, the real value is taken instead of the true value; therefore, only an approximate estimate of the measurement error can be given. Assessment of the reliability of the measurement result, i.e. determining measurement error is one of the main tasks of metrology.
Error is the deviation of a measurement result from the true value of the measured value. Errors can be roughly divided into errors of measuring instruments and errors of measurement results.
Errors of measuring instruments were discussed in Chapter 3.
Measurement result error is a number indicating the possible limits of uncertainty in the value of the measured quantity.
Below we will give a classification and consider the errors of the measurement results.
By numerical expression method differentiate absolute and relative errors.
Depending on the source of occurrence there are errors instrumental, methodological, counting and installations.
According to the patterns of manifestation measurement errors are divided by systematic, progressive, random and gross.
Let us consider these measurement errors in more detail.
Absolute error D is the difference between the measured X and the true X and the values of the measured quantity. The absolute error is expressed in units of the measured value: D = X - Chi.
Since the true value of the measured quantity cannot be determined, in practice the actual value of the measured quantity Xd is used instead. The actual value is found experimentally, by using fairly accurate methods and measuring instruments. It differs little from the true value and can be used instead to solve the problem. During verification, the readings of standard measuring instruments are usually taken as the actual value. Thus, in practice, the absolute error is found using the formula D » X - Xd. Relative error d is the ratio of the absolute measurement error to the true (actual) value of the measured quantity (it is usually expressed as a percentage): .
Instrumental(instrumental or instrumental) errors are those that belong to a given measuring instrument, can be determined during its tests and are entered in its passport.
These errors are due to design and technological shortcomings of measuring instruments, as well as the result of their wear, aging or malfunction. Instrumental errors, caused by the errors of the measuring instruments used, were discussed in Chapter 3.
However, in addition to instrumental errors, during measurements there are also errors that cannot be attributed to a given device, cannot be indicated in its passport and are called methodical, those. associated not with the device itself, but with the method of its use.
Methodological errors may arise due to imperfect development of the theory of phenomena underlying the measurement method, inaccuracy of the relationships used to find an estimate of the measured value, as well as due to the discrepancy between the measured value and its model.
Let's consider examples illustrating the methodological measurement error.
The object of study is an alternating voltage source, the amplitude value of which Um need to be measured. Based on a preliminary study of the research object, a sinusoidal voltage generator was adopted as its model. Using a voltmeter designed for measuring the effective values of alternating voltages, and knowing the relationship between the effective and amplitude values of the sinusoidal voltage, we obtain the measurement result in the form Um =
×
Uv, Where UV- voltmeter reading. A more thorough study of the object could reveal that the shape of the measured voltage differs from sinusoidal and a more correct relationship between the value of the measured quantity and the voltmeter reading Um =k×
UV, Where k¹
.
Thus, the imperfection of the adopted model of the research object leads to a methodological measurement error DU =
×
UV-k×
Uv.
This error can be reduced either by calculating the value k based on an analysis of the shape of the measured voltage curve, or by replacing the measuring instrument by taking a voltmeter designed for measuring the amplitude values of alternating voltages.
A very common reason for the occurrence of methodological errors is the fact that, when organizing measurements, we are forced to measure (or consciously measure) not the value that should be measured, but some other value that is close, but not equal to it.
An example of such a methodological error is the error in measuring voltage with a voltmeter with a finite resistance (Fig. 4.1). 
Due to the voltmeter shunting the section of the circuit on which the voltage is measured, it turns out to be less than it was before connecting the voltmeter. Indeed, the voltage that the voltmeter will show is determined by the expression U = I×Rv. Considering that the current in the circuit I =E/(Ri +Rv), That 
< .
Therefore, for the same voltmeter, connected alternately to different sections of the circuit under study, this error is different: in low-resistance sections it is negligible, but in high-resistance sections it can be very large. This error could be eliminated if the voltmeter were constantly connected to this section of the circuit for the entire time the device is operating (as on a power plant switchboard), but this is unprofitable for many reasons.
There are often cases when it is generally difficult to indicate a measurement method that excludes methodological error. Let, for example, the temperature of the hot ingots coming from the furnace to the rolling mill be measured. The question is, where to place the temperature sensor (for example, a thermocouple): under the blank, on the side or above the blank? Wherever we place it, we will not measure the internal temperature of the body of the blank, i.e. we will have a significant methodological error, since we are measuring not what is needed, but what is simpler (it is not possible to drill a channel in each blank to place a thermocouple in its center).
Thus, the main distinctive feature of methodological errors is the fact that they cannot be indicated in the instrument’s passport, but must be assessed by the experimenter himself when organizing the chosen measurement technique, therefore he must clearly distinguish between the actual measurable they are the size of subject to measurement.
Reading error occurs due to insufficiently accurate readings. It is due to the subjective characteristics of the observer (for example, interpolation error, i.e. inaccurate reading of division fractions on the instrument scale) and the type of reading device (for example, parallax error). There are no reading errors when using digital measuring instruments, which is one of the reasons for the prospects of the latter.
Installation error caused by deviation of measurement conditions from normal, i.e. conditions under which calibration and verification of measuring instruments were carried out. This includes, for example, errors from incorrect installation of the device in space or its pointer to the zero mark, from changes in temperature, supply voltage and other influencing quantities.
The types of errors considered are equally suitable for characterizing the accuracy of both individual measurement results and measuring instruments.
Systematic measurement error Dc is a component of the measurement error that remains constant or changes naturally with repeated measurements of the same quantity.
The causes of systematic errors can usually be established during the preparation and conduct of measurements. These reasons are very diverse: imperfection of the measuring instruments and methods used, incorrect installation of the measuring instrument, the influence of external factors (influencing quantities) on the parameters of the measuring instruments and on the measurement object itself, shortcomings of the measurement method (methodological errors), individual characteristics of the operator (subjective errors) and etc. According to the nature of their manifestation, systematic errors are divided into constant and variable. Constants include, for example, errors caused by inaccurate adjustment of the measure value, incorrect calibration of the instrument scale, incorrect installation of the instrument relative to the direction of magnetic fields, etc. Variable systematic errors are caused by the influence of influencing quantities on the measurement process and can arise, for example, when changing the voltage of the device's power supply, external magnetic fields, the frequency of the measured alternating voltage, etc. The main feature of systematic errors is that their dependence on the influencing quantities is subject to a certain law. This law can be studied, and the measurement result can be clarified by introducing amendments if the numerical values of these errors are determined. Another way to reduce the influence of systematic errors is to use measurement methods that make it possible to eliminate the influence of systematic errors without determining their values (for example, the substitution method).
The result of measurements is the closer to the true value of the measured value, the smaller the remaining non-excluded systematic errors. The presence of excluded systematic errors determines the accuracy of measurements, quality reflecting the closeness to zero of systematic errors. The measurement result will be as correct as it is undistorted by systematic errors, and the smaller these errors, the more correct it is.
Progressive(or drift) are unpredictable errors that change slowly over time. These errors, as a rule, are caused by the aging processes of certain parts of the equipment (discharge of power supplies, aging of resistors, capacitors, deformation of mechanical parts, shrinkage of paper tape in recorders, etc.). A feature of progressive errors is that they can be corrected by introducing an amendment only at a given point in time, and then increase again unpredictably. Therefore, unlike systematic errors, which can be corrected by a correction found once for the entire service life of the device, progressive errors require continuous repetition of the correction and the more often, the smaller their residual value should be. Another feature of progressive errors is that their change over time is a non-stationary random process and therefore, within the framework of a well-developed theory of stationary random processes, they can be described only with reservations.
Random measurement error— component of the measurement error that changes randomly during repeated measurements of the same quantity. The value and sign of random errors cannot be determined; they cannot be directly taken into account due to their chaotic changes caused by the simultaneous influence of various factors independent of each other on the measurement result. Random errors are detected during repeated measurements of the same quantity (individual measurements in this case are called observations) using the same measuring instruments under the same conditions by the same observer, i.e. for equal-precision (equidispersed) measurements. The influence of random errors on the measurement result is taken into account by the methods of mathematical statistics and probability theory.
Gross measurement errors - random measurement errors that significantly exceed the errors expected under given conditions.
Gross errors (misses) are usually caused by incorrect readings from the instrument, an error in recording observations, the presence of a strongly influencing quantity, malfunction of measuring instruments and other reasons. As a rule, measurement results containing gross errors are not taken into account, so gross errors have little effect on the accuracy of the measurement. It is not always easy to detect a mistake, especially with a single measurement; It is often difficult to distinguish a gross error from a large random error. If gross errors occur frequently, we will question all measurement results. Therefore, gross errors affect the validity of measurements.
In conclusion of the described division of errors of instruments and measurement results into random, progressive and systematic components, it is necessary to pay attention to the fact that such division is a very simplified method of their analysis. Therefore, one should always remember that in reality, these error components appear together and form a single non-stationary random process. The error of the measurement result can be represented in the form of the sum of random and systematic errors Dс: D = Dс +. The measurement errors include a random component, so it should be considered a random variable.
Consideration of the nature of the manifestation of measurement errors shows us that the only correct way to evaluate errors is provided by probability theory and mathematical statistics.
Laws of distribution of random errors. Random errors are detected when a number of measurements of the same quantity are carried out. The results of measurements, as a rule, do not coincide with each other, since due to the total influence of many different factors that cannot be taken into account, each new measurement also gives a new random value of the measured value. If measurements are carried out correctly, there are a sufficient number of them and systematic errors and errors are excluded, it can be argued that the true value of the measured quantity does not go beyond the values obtained from these measurements. It remains unknown until the theoretically probable value of the random error is determined.
Let the quantity A be measured P times and observed the values a1, a2, a3,...,a i,...,an. The random absolute error of a single measurement is determined by the difference
Di = ai - A. (4.1)
Graphically, the results of individual measurements are presented in Fig. 4.2.
With a sufficiently large number P the same errors, if they have a number of discrete values, are repeated and therefore it is possible to establish the relative frequency (frequency) of their occurrence, i.e. ratio of the number of identical data received mi to the total number of measurements taken P. When continuing to measure the value A this frequency will not change, so it can be considered the probability of an error occurring in these measurements: p(Ai) =
mi /
n.
The statistical dependence of the probability of occurrence of random errors on their value is called the law of error distribution or law of probability distribution. This law determines the nature of the appearance of various results of individual measurements. There are two types of descriptions of distribution laws: integral And differential.
Integral law, or probability distribution functionF( D )
random error Di Vi-th experience, call a function whose value for each D is the probability of the event R(D), which consists in the fact that the random error Di takes values less than a certain value D, i.e. function F( D ) = P[ Di <
D ].
When D changes from -¥ to +¥, this function takes values from 0 to 1 and is non-decreasing. It exists for all random variables, both discrete and continuous (Figure 4.3 a).
If F(D) symmetrical about a point A, the corresponding probability is 0.5, then the distribution of observation results will be symmetrical relative to the true value A. In this case it is advisable F(D) shift along the x-axis by the value DA, i.e. eliminate systematic error (DA =Dс) and obtain the distribution function of the random component of the error D=(Fig. 4.3 b). Error probability distribution function D differs from the probability distribution function of the random component of the error only by a shift along the x-axis by the value of the systematic component of the error Dc.
Differential law probability distributions for random error with continuous and differentiable distribution function F(D) call the function 
. This dependence exists probability distribution density. The probability density distribution graph can have different shapes depending on the law of error distribution. For F(D), shown in Fig. 4.3 b, distribution curve f(D) has a shape close to the shape of a bell (Fig. 4.3 c).
The probability of random errors is determined by the area bounded by the curve f(D) or part of it and the abscissa axis (Fig. 4.3 c). Depending on the considered error interval 
.
Meaning f(D)dD there is an element of probability equal to the area of the rectangle with the base dD and abscissa D1,D2, called quantiles. Because F(+¥)=
1, then the equality is true 
,
those. area under the curve f(D) according to the normalization rule, it is equal to one and reflects the probability of all possible events.
In the practice of electrical measurements, one of the most common laws of distribution of random errors is normal law(Gauss).
The mathematical expression of the normal law has the form 
,
Where f(D)- probability density of random error D = ai-A; s - standard deviation. The standard deviation can be expressed in terms of random deviations of the observation results Di (see formula (4.1)):
.
The nature of the curves described by this equation for two values of s is shown in Fig. 4.4. From these curves it is clear that the smaller s, the more often small random errors occur, i.e. the more accurate the measurements are. In measurement practice, there are other distribution laws that can be established on the basis of statistical processing 
experimental data. Some of the most common distribution laws are given in GOST 8.011-84 “Indicators of measurement accuracy and forms of presentation of measurement results.”
The main characteristics of distribution laws are expected value And dispersion.
Expectation of a random variable- this is its value around which the results of individual observations are grouped. Mathematical expectation of a discrete random variable M[X] is defined as the sum of the products of all possible values of a random variable by the probability of these values 
.
For continuous random variables one has to resort to integration, for which it is necessary to know the dependence of the probability density on X, i.e. f(x), Where x=D. Then 
.
This expression means that the mathematical expectation is equal to the sum of an infinitely large number of products of all possible values of the random variable X to infinitesimal areas f(x)dx, Where f(x) — ordinates for each X, a dx - elementary segments of the abscissa axis.
If a normal distribution of random errors is observed, then the mathematical expectation of the random error is zero (Fig. 4.4). If we consider the normal distribution of results, then the mathematical expectation will correspond to the true value of the measured value, which we denote by A.
The systematic error is the deviation of the mathematical expectation of the observation results from the true value A measured quantity: Dc = M[X]-A, and random error is the difference between the result of a single observation and the mathematical expectation: 
.
The dispersion of a number of observations characterizes the degree of dispersion (scatter) of the results of individual observations around the mathematical expectation:
D[X] =Dx=M[(ai-mx)2].
The smaller the dispersion, the smaller the scatter of individual results, the more accurate the measurements are. However, dispersion is expressed in units squared of the measured value. Therefore, the standard deviation (MSD), equal to the square root of the variance, is most often used to characterize the accuracy of a series of observations: 
.
The considered normal distribution of random variables, including random errors, is theoretical, therefore the described normal distribution should be considered as “ideal”, that is, as a theoretical basis for the study of random errors and their influence on the measurement result.
The following describes how to apply this distribution in practice with varying degrees of approximation. Another distribution (Student distribution), used for small numbers of observations, is also considered.
Estimates of errors in the results of direct measurements. Let it be carried out P direct measurements of the same quantity. In general, in each measurement act the error will be different:
Di =ai-A,
where Di is the error of the i-th measurement; ai- the result of the i-th measurement.
Since the true value of the measured quantity A unknown, the random absolute error cannot be directly calculated. In practical calculations, instead of A use his assessment. It is usually assumed that the true value is the arithmetic average of a number of measurements:
. (4.2)
Where Ai- results of individual measurements; P - number of measurements.
Now, similarly to expression (4.1), we can determine the deviation of the result of each measurement from the average value :
(4.3)
Where v
i- deviation of the result of a single measurement from the average value. It should be remembered that the sum of deviations of the measurement result from the average value is zero, and the sum of their squares is minimal, i.e.
and min.
These properties are used when processing measurement results to control the correctness of calculations.
Then calculate the value estimate mean square error for a given series of measurements 
. (4.4)
According to probability theory, with a sufficiently large number of measurements having independent random errors, the estimate S converges in probability to s. Thus, 
. (4.5)
Due to the fact that the arithmetic mean
is also a random variable, the concept of the standard deviation of the arithmetic mean makes sense. We denote this value by the symbol sav. It can be shown that for independent errors 
. (4.6)
The sр value characterizes the degree of scatter .
As stated above,
acts as an estimate of the true value of the measured quantity, i.e. is the final result of the measurements performed. Therefore, sр is also called the mean square error of the measurement result.
In practice, the value of s, calculated using formula (4.5), is used if it is necessary to characterize the accuracy of the measurement method used: if the method is accurate, then the scatter of the results of individual measurements is small, i.e. small value s .
The value of sр ,
calculated by (4.6), is used to characterize the accuracy of the measurement result of a certain quantity, i.e. a result obtained through mathematical processing of the results of a number of individual direct measurements.
When assessing measurement results, the concept is sometimes used maximum or maximum permissible error, the value of which is determined in fractions s or S. Currently, there are different criteria for establishing the maximum error, i.e., the limits of the tolerance field ±D, within which random errors must fit. The generally accepted definition for the maximum error is D = 3s (or 3 S). Recently, based on the information theory of measurements, Professor P. V. Novitsky recommends using the value D = 2s.
Let us now introduce important concepts confidence probability And confidence interval. As stated above, the arithmetic mean ,
obtained as a result of a certain series of measurements is an estimate of the true value A and, as a rule, does not coincide with it, but differs by the error value. Let Rd there is a possibility that
differs from A by no more than D, i.e. R(-D<
A<
+
D)=Рд. Probability Rd called confidence probability, and the range of values of the measured quantity is from -
D to +
D- confidence interval.
The above inequalities mean that with probability Rd confidence interval from -
D to +
D contains the true meaning A. Thus, in order to characterize a random error quite fully, it is necessary to have two numbers - the confidence probability and the corresponding confidence interval. If the law of error probability distribution is known, then a confidence interval can be determined from a given confidence probability. In particular, with a sufficiently large number of measurements it is often justified to use the normal law, while with a small number of measurements (P<
20), the results of which belong to the normal distribution, the Student distribution should be used. This distribution has a probability density that practically coincides with the normal one at large P, but significantly different from normal at small P.
In table 4.1 shows the so-called quantiles of the Student distribution ½ t(n)½ Rd for number of measurements P= 2 - 20 and confidence probabilities R = 0,5 - 0,999.
We point out, however, that Student distribution tables are usually not given for the values P And Rd, and for values m =n-1 And a =1 - Рд, what should be taken into account when using them. To determine the confidence interval, it is necessary for the data P And Rd find the ½ quantile t(n)½Рд and calculate the values An =
-
sр×
½ t(n)½Rdi Av =
+
sр×
½ t(n)½Рд, which will be the lower and upper limits of the confidence interval.
After finding confidence intervals for a given confidence probability according to the above method, record the measurement result in the form ;
D=Dн¸
Dв; Rd,
Where
- assessment of the true value of the measurement result in units of the measured value; D - measurement error; Dв = + sр×
½ t(n)½Рд and Dн = - sр×
½ t(n)½Рд - upper and lower limits of measurement error; Рд - confidence probability.
Values of quantiles of Student's distribution t(n) with confidenceprobabilities Rd |
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Estimation of errors in the results of indirect measurements. In indirect measurements, the desired quantity A functionally related to one or more directly measured quantities: X,y,...,
t.
Let us consider the simplest case of determining the error with one variable, when A=
F(x).
Having designated the absolute measurement error of a quantity X through ±Dx, we get A+ D A= F(x± D x).
Expanding the right-hand side of this equality into a Taylor series and neglecting terms of the expansion containing Dx to a power higher than the first, we obtain
A+DA » F(x) ± Dx or DA » ± Dx.
The relative measurement error of the function is determined from the expression 
.
If the measured quantity A is a function of several variables: A=F(x,y,...,t), then the absolute error of the result of indirect measurements
.
Partial relative errors of indirect measurement are determined by the formulas 
; 
etc. Relative error of measurement result
.
Let us also dwell on the features of assessing the result of an indirect measurement in the presence of a random error.
To assess the random error of the results of indirect measurements of the quantity A we will assume that systematic errors in measuring quantities x, y,…, t are excluded, and random errors in measuring the same quantities do not depend on each other.
In indirect measurements, the value of the measured quantity is found using the formula 
,
where are the average or weighted average values of the quantities x, y,…, t.
To calculate the standard deviation of the measured value A it is advisable to use standard deviations obtained from measurements x, y,…, t.
In general, to determine the standard deviation s of an indirect measurement, the following formula is used: 
, (4.7)
Where Dx ;Dy ;…;Dt— so-called partial errors of indirect measurement 
; 
; …; 
; ; ; … ; —
partial derivatives A By x, y,…, t ;sx; sy ,…,st , …— standard deviations of measurement results x, y,…, t.
Let us consider some special cases of application of equation (4.7), when the functional relationship between indirectly and directly measured quantities is expressed by the formula A=k×
xa×
yb×
zg, Where k- numerical coefficient (dimensionless).
In this case, formula (4.7) will take the following form: 
.
If a =b =g = 1 And A=k×
x×
y×
z, then the relative error formula simplifies to the form 
.
This formula is applicable, for example, to calculate the standard deviation of the volume measurement result from the results of measuring the height, width and depth of a tank shaped like a rectangular parallelepiped.
4.5. Rules for summing random and systematic errors
The error of complex measuring instruments depends on the errors of its individual components (blocks). Errors are summed up according to certain rules.
Let, for example, a measuring device consist of m blocks, each of which has random errors independent of each other. In this case, the absolute values of the mean square sk or maximum Mk errors of each block.
Arithmetic summation or gives the maximum error of the device, which has a negligibly small probability and therefore is rarely used to assess the accuracy of the device as a whole. According to error theory, the resulting error sres and Mrez determined by addition according to the quadratic law 
or 
.
The resulting relative measurement error is determined similarly: 
. (4.8)
Equation (4.8) can be used to determine the permissible errors of individual units of devices being developed with a given total measurement error. When designing a device, equal errors are usually specified for the individual blocks included in it. If there are several sources of errors that affect the final measurement result differently (or the device consists of several blocks with different errors), weighting coefficients should be introduced into formula (4.8) ki :
, (4.9)
where d1, d2, …, dm are the relative errors of individual units (blocks) of the measuring device; k1,k2, … ,km- coefficients that take into account the degree of influence of the random error of a given block on the measurement result.
If the measuring device (or its units) also has systematic errors, the total error is determined by their sum:. The same approach is valid for a larger number of components.
When assessing the influence of particular errors, it should be taken into account that the accuracy of measurements mainly depends on errors that are large in absolute value, and some of the smallest errors can not be taken into account at all. The partial error is estimated based on the so-called criterion of negligible error, which is as follows. Let us assume that the total error dres is determined by formula (4.8) taking into account all m private errors, among which some error di is of small importance. If the total error d¢res, calculated without taking into account the error di, differs from dres by no more than 5%, i.e. drez-d¢rez< 0,05×dрез или 0,95×dрез
A measurement result has value only when its uncertainty interval can be estimated, i.e. degree of confidence. Therefore, the measurement result must contain the value of the measured quantity and the accuracy characteristics of this value, which are systematic and random errors. Quantitative indicators of errors, methods of their expression, as well as forms of presentation of measurement results are regulated by GOST 8.011-72 “Indicators of measurement accuracy and forms of presentation of measurement results.” Let's consider the main forms of presenting measurement results.
The error of the result of a direct single measurement depends on many factors, but is primarily determined by the error of the measuring instruments used. Therefore, to a first approximation, the error of the measurement result can be taken equal to
the error that characterizes the measuring instrument used at a given point in the measurement range.
The errors of measuring instruments vary over the measurement range. Therefore, in each case, for each measurement, it is necessary to calculate the error of the measurement result using formulas (3.19) - (3.21) for normalizing the error of the corresponding measuring instrument. Both absolute and relative errors of the measurement result must be calculated, since the first of them is needed to round the result and record it correctly, and the second - for an unambiguous comparative description of its accuracy.
For different normalization characteristics of SI errors, these calculations are performed differently, so we will consider three typical cases.
1. The device class is indicated as a single number q, enclosed in a circle. Then the relative error of the result (in percent) g = q, and its absolute error D x =q×
x/ 100.
2. The device class is indicated by one number p(without a circle). Then the absolute error of the measurement result D x =p×
xk/ 100, where xk is the measurement limit at which it was carried out, and the relative measurement error (in percent) is found by the formula 
,
i.e. in this case, when measuring, in addition to reading the measured value X The measurement limit must also be fixed xk, otherwise, it will be impossible to subsequently calculate the error of the result.
3. The class of the device is indicated by two numbers in the form c/d. In this case, it is more convenient to calculate the relative error d result using formula (3.21), and only then find the absolute error as Dx =d×
x/100.
After calculating the error, use one of the forms of presenting the measurement result in the following form: X;±
D And d, Where X- measured value; D- absolute measurement error; d-relative measurement error. For example, the following entry is made: “The measurement was made with a relative error d= …%. Measured value x = (A±
D), Where A- result of measurements.”
However, it is more clear to indicate the limits of the uncertainty interval of the measured value in the form: x = (A-D)¸(A+D) or (A-D)< х
< (A+D) indicating units of measurement.
Another form of presenting the measurement result is set as follows: X; D from Dн before Dв; R, Where X- measurement result in units of the measured quantity; DDn,Dв- respectively, the measurement error with its lower and upper boundaries in the same units; R- the probability with which the measurement error is within these limits.
GOST 8.011-72 allows other forms of presentation of measurement results that differ from the given forms in that they indicate separately the characteristics of the systematic and random components of the measurement error. At the same time, for a systematic error, its probabilistic characteristics are indicated. In this case, the main characteristics of the systematic error are the mathematical expectation M [
Dxc], standard deviation s[ Dxc] and its confidence interval. Isolating the systematic and random components of the error is advisable if the measurement result will be used in further data processing, for example, when determining the result of indirect measurements and assessing its accuracy, when summing up errors, etc.
Any form of presentation of a measurement result provided for by GOST 8.011-72 must contain the necessary data on the basis of which a confidence interval for the error of the measurement result can be determined. In general, a confidence interval can be established if the type of error distribution law and the main numerical characteristics of this law are known.
