Absolute and relative measurement errors. Absolute measurement error
Measurement error- assessment of the deviation of the measured value of a quantity from its true value. Measurement error is a characteristic (measure) of measurement accuracy.
Since it is impossible to find out with absolute accuracy the true value of any quantity, it is also impossible to indicate the magnitude of the deviation of the measured value from the true one. (This deviation is usually called the measurement error. In a number of sources, for example, in the Great Soviet Encyclopedia, the terms measurement error and measurement error are used as synonyms, but according to RMG 29-99 the term measurement error not recommended as less successful). It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. In practice, instead of the true value, we use actual value x d, that is, the value of a physical quantity obtained experimentally and so close to the true value that it can be used instead of it in the set measurement task. Such a value is usually calculated as the average value obtained by statistical processing of the results of a series of measurements. This value obtained is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, along with the result obtained, the measurement error is indicated. For example, the entry T=2.8±0.1 c. means that the true value of the quantity T lies in the interval from 2.7 s before 2.9 s with some specified probability
In 2004, 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, the concept of "measurement uncertainty" was introduced instead, however, GOST R 50.2.038-2004 allows the use of the term error for documents used in Russia.
There are the following types of errors:
The absolute error
Relative error
the reduced error;
The main error
Additional error
· systematic error;
Random error
Instrumental error
· methodical error;
· personal error;
· static error;
dynamic error.
Measurement errors are classified according to the following criteria.
· According to the method of mathematical expression, the errors are divided into absolute errors and relative errors.
· According to the interaction of changes in time and the input value, the errors are divided into static errors and dynamic errors.
By the nature of the occurrence of errors are divided into systematic errors and random errors.
· According to the nature of the dependence of the error on the influencing values, the errors are divided into basic and additional.
· According to the nature of the dependence of the error on the input value, the errors are divided into additive and multiplicative.
Absolute error is the value calculated as the difference between the value of the quantity obtained during the measurement process and the real (actual) value of the given quantity. The absolute error is calculated using the following formula:
AQ n =Q n /Q 0 , where AQ n is the absolute error; Qn- the value of a certain quantity obtained in the process of measurement; Q0- the value of the same quantity, taken as the base of comparison (real value).
Absolute error of measure is the value calculated as the difference between the number, which is the nominal value of the measure, and the real (actual) value of the quantity reproduced by the measure.
Relative error is a number that reflects the degree of accuracy of the measurement. The relative error is calculated using the following formula:
Where ∆Q is the absolute error; Q0 is the real (actual) value of the measured quantity. Relative error is expressed as a percentage.
Reduced error is the value calculated as the ratio of the absolute error value to the normalizing value.
The normalizing value is defined as follows:
For measuring instruments for which a nominal value is approved, this nominal value is taken as a normalizing value;
· for measuring instruments, in which the zero value is located on the edge of the measurement scale or outside the scale, the normalizing value is taken equal to the final value from the measurement range. The exception is measuring instruments with a significantly uneven measurement scale;
· for measuring instruments, in which the zero mark is located within the measurement range, the normalizing value is taken equal to the sum of the final numerical values of the measurement range;
For measuring instruments (measuring instruments) with an uneven scale, the normalizing value is taken equal to the entire length of the measurement scale or the length of that part of it that corresponds to the measurement range. The absolute error is then expressed in units of length.
Measurement error includes instrumental error, methodological error and reading error. Moreover, the reading error arises due to the inaccuracy in determining the division fractions of the measurement scale.
Instrumental error- this is the error arising due to the errors made in the manufacturing process of the functional parts of the error measuring instruments.
Methodological error is an error due to the following reasons:
· inaccuracy in building a model of the physical process on which the measuring instrument is based;
Incorrect use of measuring instruments.
Subjective error- this is an error arising due to the low degree of qualification of the operator of the measuring instrument, as well as due to the error of the human visual organs, i.e. the human factor is the cause of the subjective error.
Errors in the interaction of changes in time and the input value are divided into static and dynamic errors.
Static error- this is the error that occurs in the process of measuring a constant (not changing in time) value.
Dynamic error- this is an error, the numerical value of which is calculated as the difference between the error that occurs when measuring a non-constant (variable in time) quantity, and a static error (the error in the value of the measured quantity at a certain point in time).
According to the nature of the dependence of the error on the influencing quantities, the errors are divided into basic and additional.
Basic error is the error obtained under normal operating conditions of the measuring instrument (at normal values of the influencing quantities).
Additional error- this is the error that occurs when the values of the influencing quantities do not correspond to their normal values, or if the influencing quantity goes beyond the boundaries of the area of normal values.
Normal conditions are the conditions under which all values of the influencing quantities are normal or do not go beyond the boundaries of the range of normal values.
Working conditions- these are conditions in which the change in the influencing quantities has a wider range (the values of the influencing ones do not go beyond the boundaries of the working range of values).
Working range of values of the influencing quantity is the range of values in which the values of the additional error are normalized.
According to the nature of the dependence of the error on the input value, the errors are divided into additive and multiplicative.
Additive error- this is the error that occurs due to the summation of numerical values and does not depend on the value of the measured quantity, taken modulo (absolute).
Multiplicative error- this is an error that changes along with a change in the values of the quantity being measured.
It should be noted that the value of the absolute additive error is not related to the value of the measured quantity and the sensitivity of the measuring instrument. Absolute additive errors are unchanged over the entire measurement range.
The value of the absolute additive error determines the minimum value of the quantity that can be measured by the measuring instrument.
The values of multiplicative errors change in proportion to changes in the values of the measured quantity. The values of multiplicative errors are also proportional to the sensitivity of the measuring instrument. The multiplicative error arises due to the influence of influencing quantities on the parametric characteristics of the instrument elements.
Errors that may occur during the measurement process are classified according to the nature of their occurrence. Allocate:
systematic errors;
random errors.
Gross errors and misses may also appear in the measurement process.
Systematic error- this is an integral part of the entire error of the measurement result, which does not change or changes naturally with repeated measurements of the same value. Usually, a systematic error is tried to be eliminated by possible means (for example, by using measurement methods that reduce the likelihood of its occurrence), but if a systematic error cannot be excluded, then it is calculated before the start of measurements and appropriate corrections are made to the measurement result. In the process of normalizing the systematic error, the boundaries of its admissible values are determined. The systematic error determines the correctness of measurements of measuring instruments (metrological property). Systematic errors in some cases can be determined experimentally. The measurement result can then be refined by introducing a correction.
Methods for eliminating systematic errors are divided into four types:
elimination of the causes and sources of errors before the start of measurements;
· Elimination of errors in the process of already begun measurement by methods of substitution, compensation of errors in sign, oppositions, symmetrical observations;
Correction of measurement results by making an amendment (elimination of errors by calculations);
Determining the limits of systematic error in case it cannot be eliminated.
Elimination of the causes and sources of errors before the start of measurements. This method is the best option, since its use simplifies the further course of measurements (there is no need to eliminate errors in the process of an already started measurement or to amend the result obtained).
To eliminate systematic errors in the process of an already started measurement, various methods are used.
Amendment method is based on knowledge of the systematic error and the current patterns of its change. When using this method, the measurement result obtained with systematic errors is subject to corrections equal in magnitude to these errors, but opposite in sign.
substitution method consists in the fact that the measured value is replaced by a measure placed in the same conditions in which the object of measurement was located. The substitution method is used when measuring the following electrical parameters: resistance, capacitance and inductance.
Sign error compensation method consists in the fact that the measurements are performed twice in such a way that the error, unknown in magnitude, is included in the measurement results with the opposite sign.
Contrasting method similar to sign-based compensation. This method consists in the fact that measurements are performed twice in such a way that the source of error in the first measurement has the opposite effect on the result of the second measurement.
random error- this is a component of the error of the measurement result, which changes randomly, irregularly during repeated measurements of the same value. The occurrence of a random error cannot be foreseen and predicted. Random error cannot be completely eliminated; it always distorts the final measurement results to some extent. But you can make the measurement result more accurate by taking repeated measurements. The cause of a random error can be, for example, a random change in external factors affecting the measurement process. A random error during multiple measurements with a sufficiently high degree of accuracy leads to scattering of the results.
Misses and blunders are errors that are much larger than the systematic and random errors expected under the given measurement conditions. Slips and gross errors may appear due to gross errors in the measurement process, a technical malfunction of the measuring instrument, and unexpected changes in external conditions.
Let some random variable a measured n times under the same conditions. The measurement results gave a set n various numbers
Absolute error- dimensional value. Among n values of absolute errors necessarily meet both positive and negative.
For the most probable value of the quantity a usually take average the meaning of the measurement results
.
The larger the number of measurements, the closer the mean value is to the true value.
Absolute errori
.
Relative errori th dimension is called the quantity
Relative error is a dimensionless quantity. Usually, the relative error is expressed as a percentage, for this e i multiply by 100%. The value of the relative error characterizes the measurement accuracy.
Average absolute error is defined like this:
.
We emphasize the need to sum the absolute values (modules) of the quantities D and i . Otherwise, the identical zero result will be obtained.
Average relative error is called the quantity
.
For a large number of measurements.
Relative error can be considered as the value of the error per unit of the measured quantity.
The accuracy of measurements is judged on the basis of a comparison of the errors of the measurement results. Therefore, the measurement errors are expressed in such a form that, in order to assess the accuracy, it would be sufficient to compare only the errors of the results, without comparing the sizes of the measured objects or knowing these sizes very approximately. It is known from practice that the absolute error of measuring the angle does not depend on the value of the angle, and the absolute error of measuring the length depends on the value of the length. The larger the length value, the greater the absolute error for this method and measurement conditions. Therefore, according to the absolute error of the result, it is possible to judge the accuracy of the angle measurement, but it is impossible to judge the accuracy of the length measurement. The expression of the error in relative form makes it possible to compare, in certain cases, the accuracy of angular and linear measurements.
Basic concepts of probability theory. Random error.
Random error called the component of the measurement error, which changes randomly with repeated measurements of the same quantity.
When repeated measurements of the same constant, unchanging quantity are carried out with the same care and under the same conditions, we get measurement results - some of them differ from each other, and some of them coincide. Such discrepancies in the measurement results indicate the presence of random error components in them.
Random error arises from the simultaneous action of many sources, each of which in itself has an imperceptible effect on the measurement result, but the total effect of all sources can be quite strong.
Random errors are an inevitable consequence of any measurement and are due to:
a) inaccurate readings on the scale of instruments and tools;
b) not identical conditions for repeated measurements;
c) random changes in external conditions (temperature, pressure, force field, etc.) that cannot be controlled;
d) all other influences on measurements, the causes of which are unknown to us. The magnitude of the random error can be minimized by repeated repetition of the experiment and appropriate mathematical processing of the results.
A random error can take on different absolute values, which cannot be predicted for a given measurement act. This error can equally be both positive and negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause repeated measurements to scatter about the true value.
Let us assume that with the help of a stopwatch we measure the period of oscillation of the pendulum, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the value of the reference, a small uneven movement of the pendulum - all this causes a scatter in the results of repeated measurements and therefore can be classified as random errors.
If there are no other errors, then some results will be somewhat overestimated, while others will be slightly underestimated. But if, in addition to this, the clock is also behind, then all the results will be underestimated. This is already a systematic error.
Some factors can cause both systematic and random errors at the same time. So, by turning the stopwatch on and off, we can create a small irregular spread in the moments of starting and stopping the clock relative to the movement of the pendulum and thereby introduce a random error. But if, in addition, every time we rush to turn on the stopwatch and are somewhat late turning it off, then this will lead to a systematic error.
Random errors are caused by a parallax error when reading the divisions of the instrument scale, shaking of the building foundation, the influence of slight air movement, etc.
Although it is impossible to exclude random errors of individual measurements, the mathematical theory of random phenomena allows us to reduce the influence of these errors on the final measurement result. It will be shown below that for this it is necessary to make not one, but several measurements, and the smaller the error value we want to obtain, the more measurements need to be taken.
Due to the fact that the occurrence of random errors is inevitable and unavoidable, the main task of any measurement process is to bring the errors to a minimum.
The theory of errors is based on two main assumptions, confirmed by experience:
1. With a large number of measurements, random errors of the same magnitude, but of a different sign, i.e. errors in the direction of increasing and decreasing the result are quite common.
2. Large absolute errors are less common than small ones, so the probability of an error decreases as its value increases.
The behavior of random variables is described by statistical regularities, which are the subject of probability theory. Statistical definition of probability w i developments i is the attitude
where n- total number of experiments, n i- the number of experiments in which the event i happened. In this case, the total number of experiments should be very large ( n®¥). With a large number of measurements, random errors obey a normal distribution (Gaussian distribution), the main features of which are the following:
1. The greater the deviation of the value of the measured value from the true value, the less the probability of such a result.
2. Deviations in both directions from the true value are equally probable.
From the above assumptions, it follows that in order to reduce the influence of random errors, it is necessary to measure this quantity several times. Suppose we are measuring some value x. Let produced n measurements: x 1 , x 2 , ... x n- by the same method and with the same care. It can be expected that the number dn obtained results, which lie in a fairly narrow interval from x before x + dx, should be proportional to:
The value of the taken interval dx;
Total number of measurements n.
Probability dw(x) that some value x lies in the interval from x before x+dx, defined as follows :
(with the number of measurements n ®¥).
Function f(X) is called the distribution function or probability density.
As a postulate of the theory of errors, it is assumed that the results of direct measurements and their random errors, with a large number of them, obey the law of normal distribution.
The distribution function of a continuous random variable found by Gauss x has the following form:
, where mis -
distribution parameters .
The parameter m of the normal distribution is equal to the mean value á xñ a random variable, which, for an arbitrary known distribution function, is determined by the integral
.
In this way, the value m is the most probable value of the measured value x, i.e. her best estimate.
The parameter s 2 of the normal distribution is equal to the variance D of the random variable, which is generally determined by the following integral
.
The square root of the variance is called the standard deviation of the random variable.
The mean deviation (error) of the random variable ásñ is determined using the distribution function as follows
The average measurement error ásñ, calculated from the Gaussian distribution function, is related to the value of the standard deviation s as follows:
< s > = 0.8s.
The parameters s and m are related as follows:
.
This expression allows you to find the standard deviation s if there is a normal distribution curve.
The graph of the Gaussian function is shown in the figures. Function f(x) is symmetrical with respect to the ordinate drawn at the point x= m; passes through the maximum at the point x= m and has an inflection at the points m ±s. Thus, the dispersion characterizes the width of the distribution function, or shows how widely the values of a random variable are scattered relative to its true value. The more accurate the measurements, the closer to the true value the results of individual measurements, i.e. the value of s is less. Figure A shows the function f(x) for three values s .
Area of a figure bounded by a curve f(x) and vertical lines drawn from points x 1 and x 2 (Fig. B) , is numerically equal to the probability that the measurement result falls within the interval D x = x 1 - x 2 , which is called the confidence level. Area under the entire curve f(x) is equal to the probability of a random variable falling into the interval from 0 to ¥, i.e.
,
since the probability of a certain event is equal to one.
Using the normal distribution, error theory poses and solves two main problems. The first is an assessment of the accuracy of the measurements. The second is an assessment of the accuracy of the arithmetic mean of the measurement results.5. Confidence interval. Student's coefficient.
Probability theory allows you to determine the size of the interval in which with a known probability w are the results of individual measurements. This probability is called confidence level, and the corresponding interval (<x>±D x)w called confidence interval. The confidence level is also equal to the relative proportion of results that fall within the confidence interval.
If the number of measurements n is large enough, then the confidence probability expresses the proportion of the total number n those measurements in which the measured value was within the confidence interval. Each confidence level w corresponds to its confidence interval. w 2 80%. The wider the confidence interval, the more likely it is to get a result within that interval. In probability theory, a quantitative relationship is established between the value of the confidence interval, the confidence probability, and the number of measurements.
If we choose the interval corresponding to the average error as the confidence interval, that is, D a = AD añ, then for a sufficiently large number of measurements it corresponds to the confidence probability w 60%. As the number of measurements decreases, the confidence probability corresponding to such a confidence interval (á añ ± AD añ) decreases.
Thus, to estimate the confidence interval of a random variable, one can use the value of the average erroráD añ .
To characterize the magnitude of a random error, it is necessary to set two numbers, namely, the magnitude of the confidence interval and the magnitude of the confidence probability . Specifying only the magnitude of the error without the corresponding confidence probability is largely meaningless.
If the average measurement error ásñ is known, the confidence interval written as (<x> ±asñ) w, determined with confidence probability w= 0,57.
If the standard deviation s is known distribution of measurement results, the indicated interval has the form (<x>± tw s) w, where tw- coefficient depending on the value of the confidence probability and calculated according to the Gaussian distribution.
The most commonly used quantities D x are shown in table 1.
Page 1
The absolute error of determination does not exceed 0 01 μg of phosphorus. This method was used by us to determine phosphorus in nitric, acetic, hydrochloric and sulfuric acids and acetone with their preliminary evaporation.
The absolute error of determination is 0 2 - 0 3 mg.
The absolute error in the determination of zinc in zinc-manganese ferrites by the proposed method does not exceed 0 2 % rel.
The absolute error in the determination of hydrocarbons C2 - C4, when their content in the gas is 0 2 - 50%, is 0 01 - 0 2%, respectively.
Here Ay is the absolute error in the definition of r/, which results from the error Yes in the definition of a. For example, the relative error of the square of a number is twice the error in determining the number itself, and the relative error of the number under the cube root is just one third of the error in determining the number.
More complex considerations are necessary when choosing a measure of comparison of absolute errors in determining the time of the beginning of the accident TV - Ts, where Tv and Ts are the time of the restored and real accident, respectively. By analogy, here we can use the average time to reach the pollution peak from a real discharge to those monitoring points that recorded an accident during the passage of pollution Tsm. Calculation of the reliability of determining the power of accidents is based on the calculation of the relative error MV - Ms / Mv, where Mv and Ms are the restored and real powers, respectively. Finally, the relative error in determining the duration of an emergency release is characterized by the value rv - rs / re, where rv and rs are the reconstructed and real durations of accidents, respectively.
More complex considerations are necessary when choosing a measure of comparison of absolute errors in determining the time of the beginning of the accident TV - Ts, where Tv and Ts are the time of the restored and real accident, respectively. By analogy, here we can use the average time to reach the pollution peak from a real discharge to those monitoring points that recorded an accident during the passage of pollution Tsm. Calculation of the reliability of determining the power of accidents is based on the calculation of the relative error Mv - Ms / Ms, where Mv and Ms are the restored and real powers, respectively. Finally, the relative error in determining the duration of an emergency release is characterized by the value rv - rs / rs, where rv and rs are the reconstructed and real durations of accidents, respectively.
With the same absolute measurement error ay, the absolute error in determining the amount of ax decreases with increasing sensitivity of the method.
Since errors are based not on random, but on systematic errors, the final absolute error in determining suction cups can reach 10% of the theoretically required amount of air. Only with unacceptably loose furnaces (A 0 25) does the generally accepted method give more or less satisfactory results. The described is well known to adjusters, who, when reducing the air balance of dense furnaces, often get negative suction values.
An analysis of the error in determining the value of pet showed that it consists of 4 components: the absolute error in determining the mass of the matrix, the sample capacity, weighing, and the relative error due to fluctuations in the mass of the sample around the equilibrium value.
Subject to all the rules for the selection, counting of volumes and analysis of gases using the GKhP-3 gas analyzer, the total absolute error in determining the content of CO2 and O2 should not exceed 0 2 - 0 4% of their true value.
From Table. 1 - 3, we can conclude that the data we use for the starting substances, taken from different sources, have relatively small differences that lie within the absolute errors in determining these quantities.
Random errors can be absolute or relative. Random error, which has the dimension of the measured value, is called the absolute error of determination. The arithmetic mean of the absolute errors of all individual measurements is called the absolute error of the analysis method.
The value of the permissible deviation, or confidence interval, is not set arbitrarily, but is calculated from specific measurement data and the characteristics of the instruments used. The deviation of the result of an individual measurement from the true value of a quantity is called the absolute error of determination or simply error. The ratio of the absolute error to the measured value is called the relative error, which is usually expressed as a percentage. Knowing the error of an individual measurement is of no independent significance, and in any serious experiment, several parallel measurements must be carried out, from which the error of the experiment is calculated. Measurement errors, depending on the causes of their occurrence, are divided into three types.
It is practically impossible to determine the true value of a physical quantity absolutely exactly, because any measurement operation is associated with a number of errors or, otherwise, errors. The reasons for the errors can be very different. Their occurrence may be due to inaccuracies in the manufacture and adjustment of the measuring device, due to the physical features of the object under study (for example, when measuring the diameter of a wire of inhomogeneous thickness, the result randomly depends on the choice of the measurement area), random reasons, etc.
The task of the experimenter is to reduce their influence on the result, and also to indicate how close the result is to the true one.
There are concepts of absolute and relative error.
Under absolute error measurement will understand the difference between the measurement result and the true value of the measured quantity:
∆x i =x i -x and (2)
where ∆x i is the absolute error of the i-th measurement, x i _ is the result of the i-th measurement, x i is the true value of the measured value.
The result of any physical measurement is usually written as:
where is the arithmetic mean value of the measured quantity closest to the true value (the validity of x and ≈ will be shown below), is the absolute measurement error.
Equality (3) should be understood in such a way that the true value of the measured value lies in the interval [ - , + ].
Absolute error is a dimensional value, it has the same dimension as the measured value.
The absolute error does not fully characterize the accuracy of the measurements made. Indeed, if we measure with the same absolute error of ± 1 mm segments 1 m and 5 mm long, the measurement accuracy will be incomparable. Therefore, along with the absolute measurement error, the relative error is calculated.
Relative error measurements is the ratio of the absolute error to the measured value itself:
Relative error is a dimensionless quantity. It is expressed as a percentage:
In the example above, the relative errors are 0.1% and 20%. They differ markedly from each other, although the absolute values are the same. Relative error gives information about accuracy
Measurement errors
According to the nature of the manifestation and the reasons for the appearance of the error, it can be conditionally divided into the following classes: instrumental, systematic, random, and misses (gross errors).
Misses are due either to a malfunction of the device, or to a violation of the methodology or experimental conditions, or are of a subjective nature. In practice, they are defined as results that differ sharply from others. To eliminate their appearance, it is necessary to observe accuracy and thoroughness in working with devices. Results containing misses must be excluded from consideration (discarded).
instrumental errors. If the measuring device is serviceable and adjusted, then measurements can be taken on it with limited accuracy, determined by the type of device. It is accepted that the instrumental error of the pointer instrument is considered equal to half of the smallest division of its scale. In devices with a digital readout, the instrument error is equated to the value of one smallest digit on the instrument scale.
Systematic errors are errors whose magnitude and sign are constant for the entire series of measurements carried out by the same method and using the same measuring instruments.
When carrying out measurements, it is important not only to take into account systematic errors, but it is also necessary to achieve their elimination.
Systematic errors are conditionally divided into four groups:
1) errors, the nature of which is known and their magnitude can be determined quite accurately. Such an error is, for example, a change in the measured mass in air, which depends on temperature, humidity, air pressure, etc.;
2) errors, the nature of which is known, but the magnitude of the error itself is unknown. Such errors include errors caused by the measuring device: malfunction of the device itself, non-compliance of the scale with the zero value, the accuracy class of this device;
3) errors, the existence of which may not be suspected, but their magnitude can often be significant. Such errors occur most often with complex measurements. A simple example of such an error is the measurement of the density of some sample containing a cavity inside;
4) errors due to the characteristics of the measurement object itself. For example, when measuring the electrical conductivity of a metal, a piece of wire is taken from the latter. Errors can occur if there is any defect in the material - a crack, thickening of the wire or inhomogeneity that changes its resistance.
Random errors are errors that change randomly in sign and magnitude under identical conditions for repeated measurements of the same quantity.
Similar information.
Absolute measurement error called the value determined by the difference between the measurement result x and the true value of the measured quantity x 0:
Δ x = |x - x 0 |.
The value δ, equal to the ratio of the absolute measurement error to the measurement result, is called the relative error:
Example 2.1. The approximate value of the number π is 3.14. Then its error is 0.00159. The absolute error can be considered equal to 0.0016, and the relative error equal to 0.0016/3.14 = 0.00051 = 0.051%.
Significant numbers. If the absolute error of the value a does not exceed one unit of the last digit of the number a, then they say that the number has all the signs correct. Approximate numbers should be written down, keeping only the correct signs. If, for example, the absolute error of the number 52400 is equal to 100, then this number should be written, for example, as 524·10 2 or 0.524·10 5 . You can estimate the error of an approximate number by indicating how many true significant digits it contains. When counting significant digits, zeros on the left side of the number are not counted.
For example, the number 0.0283 has three valid significant digits, and 2.5400 has five valid significant digits.
Number Rounding Rules. If the approximate number contains extra (or incorrect) characters, then it should be rounded. When rounding, an additional error occurs, not exceeding half the unit of the last significant digit ( d) rounded number. When rounding, only correct signs are preserved; extra characters are discarded, and if the first discarded digit is greater than or equal to d/2, then the last stored digit is increased by one.
Extra digits in integers are replaced by zeros, and in decimal fractions they are discarded (as well as extra zeros). For example, if the measurement error is 0.001 mm, then the result 1.07005 is rounded up to 1.070. If the first of the zero-modified and discarded digits is less than 5, the remaining digits are not changed. For example, the number 148935 with a measurement precision of 50 has a rounding of 148900. If the first digit to be replaced with zeros or discarded is 5, and it is followed by no digits or zeros, then rounding is performed to the nearest even number. For example, the number 123.50 is rounded up to 124. If the first digit to be replaced with zeros or discarded is greater than 5 or equal to 5, but followed by a significant digit, then the last remaining digit is increased by one. For example, the number 6783.6 is rounded up to 6784.
Example 2.2. When rounding the number 1284 to 1300, the absolute error is 1300 - 1284 = 16, and when rounding to 1280, the absolute error is 1280 - 1284 = 4.
Example 2.3. When rounding the number 197 to 200, the absolute error is 200 - 197 = 3. The relative error is 3/197 ≈ 0.01523 or approximately 3/200 ≈ 1.5%.
Example 2.4. The seller weighs the watermelon on a scale. In the set of weights, the smallest is 50 g. Weighing gave 3600 g. This number is approximate. The exact weight of the watermelon is unknown. But the absolute error does not exceed 50 g. The relative error does not exceed 50/3600 = 1.4%.
Errors in solving the problem on PC
Three types of errors are usually considered as the main sources of error. These are the so-called truncation errors, rounding errors, and propagation errors. For example, when using iterative methods for finding the roots of nonlinear equations, the results are approximate, in contrast to direct methods that give an exact solution.
Truncation errors
This type of error is associated with the error inherent in the problem itself. It may be due to inaccuracy in the definition of the initial data. For example, if any dimensions are specified in the condition of the problem, then in practice for real objects these dimensions are always known with some accuracy. The same goes for any other physical parameters. This also includes the inaccuracy of the calculation formulas and the numerical coefficients included in them.
Propagation errors
This type of error is associated with the use of one or another method of solving the problem. In the course of calculations, an accumulation or, in other words, error propagation inevitably occurs. In addition to the fact that the original data themselves are not accurate, a new error arises when they are multiplied, added, etc. The accumulation of the error depends on the nature and number of arithmetic operations used in the calculation.
Rounding errors
This type of error is due to the fact that the true value of a number is not always accurately stored by the computer. When a real number is stored in the computer's memory, it is written as a mantissa and exponent in much the same way as a number is displayed on a calculator.
