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 determine with absolute accuracy the true value of any quantity, it is impossible to indicate the amount of deviation of the measured value from the true one. (This deviation is usually called 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 for use 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, they use actual value of quantity 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 given measurement task. This value is usually calculated as the average value obtained from statistical processing of the results of a series of measurements. This obtained value is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, the measurement error is indicated along with the result obtained. For example, record T=2.8±0.1 c. means that the true value of the quantity T lies in the range 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” has become obsolete; instead, the concept of “measurement uncertainty” was introduced, however GOST R 50.2.038-2004 allows the use of the term error for documents used in Russia.
The following types of errors are distinguished:
· absolute error;
· relative error;
· reduced error;
· basic 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, errors are divided into absolute errors and relative errors.
· According to the interaction of changes in time and the input value, errors are divided into static errors and dynamic errors.
· Based on the nature of their occurrence, errors are divided into systematic errors and random errors.
· According to the nature of the dependence of the error on the influencing quantities, errors are divided into basic and additional.
· Based on the nature of the error’s dependence on the input value, errors are divided into additive and multiplicative.
Absolute error– this is a value calculated as the difference between the value of a quantity obtained during the measurement process and the real (actual) value of this 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 during the measurement process; Q 0– the value of the same quantity taken as the basis of comparison (real value).
Absolute error of the measure– this is a value calculated as the difference between the number, which is the nominal value of the measure, and the real (real) value of the quantity reproduced by the measure.
Relative error is a number that reflects the degree of measurement accuracy. The relative error is calculated using the following formula:
Where ∆Q is the absolute error; Q 0– real (real) value of the measured quantity. The relative error is expressed as a percentage.
Reduced error is a value calculated as the ratio of the absolute error value to the normalizing value.
The standard value is determined as follows:
· for measuring instruments for which a nominal value is approved, this nominal value is taken as the standard value;
· for measuring instruments in which the zero value is located at 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 whose zero mark is located inside 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) in which the scale is uneven, the normalizing value is taken equal to the whole 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, method error, and counting error. Moreover, the counting error arises due to the inaccuracy in determining the division fractions of the measurement scale.
Instrumental error– this is an error that arises due to errors made during the manufacturing process of functional parts of measuring instruments.
Methodological error is an error that occurs for the following reasons:
· inaccuracy in constructing 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 cause of the subjective error is the human factor.
Errors in the interaction of changes over time and the input quantity are divided into static and dynamic errors.
Static error– this is an error that arises in the process of measuring a constant (not changing over time) quantity.
Dynamic error is an error, the numerical value of which is calculated as the difference between the error that occurs when measuring a non-constant (time-variable) quantity and the 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, errors are divided into basic and additional.
Basic error– this is the error obtained under normal operating conditions of the measuring instrument (at normal values of the influencing quantities).
Additional error– this is an error that occurs when the values of influencing quantities do not correspond to their normal values, or if the influencing quantity exceeds the boundaries of the region of normal values.
Normal conditions– these are conditions in which all values of influencing quantities are normal or do not go beyond the boundaries of the normal range.
Working conditions– these are conditions in which the change in influencing quantities has a wider range (the influencing values do not go beyond the boundaries of the working range of values).
Working range of influencing quantities– this is the range of values in which the values of the additional error are normalized.
Based on the nature of the error’s dependence on the input value, errors are divided into additive and multiplicative.
Additive error– this is an error that arises due to the summation of numerical values and does not depend on the value of the measured quantity taken modulo (absolute).
Multiplicative bias is an error that changes with changes 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 constant 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 elements of the device.
Errors that may arise during the measurement process are classified according to the nature of their occurrence. Highlight:
· systematic errors;
· random errors.
Gross errors and errors may also occur during the measurement process.
Systematic error- this is a component of the entire error of the measurement result, which does not change or changes naturally with repeated measurements of the same quantity. Usually, a systematic error is tried to be eliminated in possible ways (for example, by using measurement methods that reduce the likelihood of its occurrence), but if the systematic error cannot be eliminated, 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 permissible values are determined. Systematic error determines the accuracy of measurements of measuring instruments (metrological property). Systematic errors in some cases can be determined experimentally. The measurement result can then be clarified 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 started measurement by means of substitution, compensation of errors by sign, opposition, symmetrical observations;
· correction of measurement results by making corrections (elimination of errors by calculations);
· determination of the limits of systematic error in case it cannot be eliminated.
Elimination of causes and sources of errors before starting 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 already started measurement or make corrections to the result obtained).
To eliminate systematic errors in the process of already started measurement, various methods are used
Method of introducing amendments is based on knowledge of the systematic error and the current patterns of its change. When using this method, corrections are made to the measurement result obtained with systematic errors, equal in magnitude to these errors, but opposite in sign.
Substitution method consists in the fact that the measured quantity is replaced by a measure placed in the same conditions in which the object of measurement was located. The replacement method is used when measuring the following electrical parameters: resistance, capacitance and inductance.
Sign error compensation method consists in the fact that measurements are performed twice in such a way that an error of unknown magnitude is included in the measurement results with the opposite sign.
Method of opposition similar to the sign compensation method. This method consists of taking measurements twice so that the source of error in the first measurement has an opposite effect on the result of the second measurement.
Random error- this is a component of the error of the measurement result, changing randomly, irregularly when performing repeated measurements of the same quantity. The occurrence of a random error cannot be foreseen or 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 when carrying out repeated measurements with a sufficiently high degree of accuracy leads to scattering of the results.
Mistakes and gross errors– these are errors that far exceed the systematic and random errors expected under the given measurement conditions. Errors and gross errors can appear due to gross errors during the measurement process, technical malfunction of the measuring instrument, or unexpected changes in external conditions.
Let some random variable a measured n times under the same conditions. The measurement results gave a set n different numbers
Absolute error- dimensional value. Among n Absolute error values are necessarily both positive and negative.
For the most probable value of the quantity A usually taken average value of measurement results
.
The greater the number of measurements, the closer the average value is to the true value.
Absolute errori
.
Relative errori-th measurement is called quantity
Relative error is a dimensionless quantity. Usually the relative error is expressed as a percentage, for this e i multiply by 100%. The magnitude of the relative error characterizes the accuracy of the measurement.
Average absolute error is defined like this:
.
We emphasize the need to sum the absolute values (modules) of the quantities D and i. Otherwise, the result will be identically zero.
Average relative error is called the quantity
.
For a large number of measurements.
Relative error can be considered as the error value per unit of the measured value.
The accuracy of measurements is judged by comparing the errors of the measurement results. Therefore, measurement errors are expressed in such a form that to assess the accuracy it is enough to compare only the errors of the results, without comparing the sizes of the objects being measured or knowing these sizes very approximately. It is known from practice that the absolute error in measuring an angle does not depend on the value of the angle, and the absolute error in measuring length depends on the value of the length. The larger the length, the greater the absolute error for a given method and measurement conditions. Consequently, the absolute error of the result can be used to judge the accuracy of the angle measurement, but the accuracy of the length measurement cannot be judged. Expressing the error in relative form makes it possible to compare the accuracy of angular and linear measurements in known cases.
Basic concepts of probability theory. Random error.
Random error called the component of measurement error that changes randomly during repeated measurements of the same quantity.
When repeated measurements of the same constant, unchanging quantity are carried out with the same care and under the same conditions, we obtain measurement results - some of them differ from each other, and some of them coincide. Such discrepancies in measurement results indicate the presence of random error components in them.
Random error arises from the simultaneous influence of many sources, each of which in itself has an imperceptible effect on the measurement result, but the total influence of all sources can be quite strong.
Random errors are an inevitable consequence of any measurements and are caused by:
a) inaccuracy of readings on the scale of instruments and instruments;
b) non-identity of conditions for repeated measurements;
c) random changes in external conditions (temperature, pressure, force field, etc.), which cannot be controlled;
d) all other influences on measurements, the causes of which are unknown to us. The magnitude of random error can be minimized by repeating the experiment many times and corresponding mathematical processing of the results obtained.
A random error can take on different absolute values, which are impossible to predict for a given measurement. This error can be equally positive or negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause scatter of repeated measurements relative to the true value.
Let us assume that the period of oscillation of a pendulum is measured using a stopwatch, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the reading value, a slight unevenness in the movement of the pendulum - all this causes scattering of the results of repeated measurements and therefore can be classified as random errors.
If there are no other errors, then some results will be somewhat overestimated, while others will be somewhat underestimated. But if, in addition to this, the clock is also behind, then all the results will be underestimated. This is already a systematic error.
Some factors can cause both systematic and random errors at the same time. So, by turning the stopwatch on and off, we can create a small irregular spread in the starting and stopping times of the clock relative to the movement of the pendulum and thereby introduce a random error. But if, moreover, we are in a hurry to turn on the stopwatch every time and are somewhat late to turn it off, then this will lead to a systematic error.
Random errors are caused by parallax error when counting instrument scale divisions, shaking of the foundation of a building, the influence of slight air movement, etc.
Although it is impossible to eliminate random errors in 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 made.
Due to the fact that the occurrence of random errors is inevitable and unavoidable, the main task of any measurement process is to reduce errors to a minimum.
The theory of errors is based on two main assumptions, confirmed by experience:
1. With a large number of measurements, random errors of the same magnitude, but of different signs, that is, errors in the direction of increasing and decreasing the result occur quite often.
2. Errors that are large in absolute value are less common than small ones, thus, the probability of an error occurring decreases as its magnitude increases.
The behavior of random variables is described by statistical patterns, which are the subject of probability theory. Statistical definition of probability w i events i is the attitude
Where n- total number of experiments, n i- the number of experiments in which the event i happened. In this case, the total number of experiments should be very large ( n®¥). With a large number of measurements, random errors obey a normal distribution (Gaussian distribution), the main features of which are the following:
1. The greater the deviation of the measured value from the true value, the less likely it is for such a result.
2. Deviations in both directions from the true value are equally probable.
From the above assumptions it follows that in order to reduce the influence of random errors it is necessary to measure this value several times. Suppose we are measuring some quantity x. Let it be produced n measurements: x 1 , x 2 , ... x n- using the same method and with the same care. It can be expected that the number dn obtained results, which lie in some fairly narrow interval from x before x + dx, must be proportional:
The size of the interval taken dx;
Total number of measurements n.
Probability dw(x) that some value x lies in the range from x before x + dx, is defined as follows :
(with the number of measurements n ®¥).
Function f(X) is called the distribution function or probability density.
As a postulate of the error theory, it is accepted that the results of direct measurements and their random errors, when there are a large number of them, obey the law of normal distribution.
The 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 b xñ a random variable, which, for an arbitrary known distribution function, is determined by the integral
.
Thus, the value m is the most probable value of the measured quantity x, i.e. her best estimate.
The parameter s 2 of the normal distribution is equal to the variance D of the random variable, which in the general case is determined by the following integral
.
The square root of the variance is called the standard deviation of the random variable.
The average deviation (error) of the random variable ásñ is determined using the distribution function as follows
The average measurement error ásñ, calculated from the Gaussian distribution function, is related to the value of the standard deviation s as follows:
< s > = 0.8s.
The parameters s and m are related to each other as follows:
.
This expression allows you to find the standard deviation s if there is a normal distribution curve.
The graph of the Gaussian function is presented in the figures. Function f(x) is symmetrical about the ordinate drawn at the point x = m; passes through a maximum at the point x = m and has an inflection at points m ±s. Thus, variance characterizes the width of the distribution function, or shows how widely the values of a random variable are scattered relative to its true value. The more accurate the measurements, the closer to the true value the results of individual measurements, i.e. the value s is less. Figure A shows the function f(x) for three values of s .
Area of a figure enclosed by a curve f(x) and vertical lines drawn from points x 1 and x 2 (Fig. B) , numerically equal to the probability of the measurement result falling into the interval D x = x 1 - x 2, which is called the confidence probability. Area under the entire curve f(x) is equal to the probability of a random variable falling into the interval from 0 to ¥, i.e.
,
since the probability of a reliable event is equal to one.
Using the normal distribution, error theory poses and solves two main problems. The first is an assessment of the accuracy of the measurements taken. The second is an assessment of the accuracy of the arithmetic mean value of the measurement results.5. Confidence interval. Student's coefficient.
Probability theory allows us to determine the size of the interval in which, with a known probability w the results of individual measurements are found. This probability is called confidence probability, and the corresponding interval (<x>±D x)w called confidence interval. The confidence probability is also equal to the relative proportion of results that fall within the confidence interval.
If the number of measurements n is sufficiently large, then the confidence probability expresses the proportion of the total number n those measurements in which the measured value was within the confidence interval. Each confidence probability w corresponds to its confidence interval. w 2 80%. The wider the confidence interval, the greater the likelihood of getting a result within that interval. In probability theory, a quantitative relationship is established between the value of the confidence interval, confidence probability and the number of measurements.
If we choose as a confidence interval the interval corresponding to the average error, that is, D a =áD Añ, then for a sufficiently large number of measurements it corresponds to the confidence probability w 60%. As the number of measurements decreases, the confidence probability corresponding to such a confidence interval (á Añ ± áD Añ), decreases.
Thus, to estimate the confidence interval of a random variable, one can use the value of the average error áD Añ .
To characterize the magnitude of the random error, it is necessary to specify two numbers, namely, the value of the confidence interval and the value of the confidence probability . Indicating only the magnitude of the error without the corresponding confidence probability is largely meaningless.
If the average measurement error ásñ is known, the confidence interval written as (<x> ± ásñ) w, determined with confidence probability w= 0,57.
If the standard deviation s is known distribution of measurement results, the specified interval has the form (<x>± t w s) w, Where t w- coefficient depending on the confidence probability value and calculated using the Gaussian distribution.
Most commonly used quantities D x are given in table 1.
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The absolute error of determination does not exceed 0 01 μg of phosphorus. We used this method 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 determining zinc in zinc-manganese ferrites using the proposed method does not exceed 0 2% rel.
The absolute error in determining hydrocarbons C2 - C4, when their content in the gas is 0 2 - 5 0%, is 0 01 - 0 2%, respectively.
Here Ау is the absolute error in determining r/, which results from the error Yes in determining 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 simply one-third of the error in determining the number.
More complex considerations are necessary when choosing a measure for comparisons of absolute errors in determining the time of the start of the accident TV - Ts, where Tv and Ts are the time of the reconstructed and real accident, respectively. By analogy, the average travel time of the pollution peak from the actual discharge to those monitoring points that recorded the accident during the passage of the pollution Tsm can be used here. 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 power, 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, respectively, the reconstructed and real duration of the accidents.
More complex considerations are necessary when choosing a measure for comparisons of absolute errors in determining the time of the start of the accident TV - Ts, where Tv and Ts are the time of the reconstructed and real accident, respectively. By analogy, the average travel time of the pollution peak from the actual discharge to those monitoring points that recorded the accident during the passage of the pollution Tsm can be used here. 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 power, 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, respectively, the reconstructed and real duration of the accidents.
For the same absolute measurement error ay, the absolute error in determining the quantity ax decreases with increasing sensitivity of the method.
Since the errors are based not on random, but on systematic errors, the final absolute error in determining the suction cups can reach 10% of the theoretically required amount of air. Only with unacceptably leaky fireboxes (A a0 25) does the generally accepted method give more or less satisfactory results. This is well known to service technicians who, when balancing the air balance of dense fireboxes, often receive 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 sample mass around the equilibrium value.
If all the rules for selecting, measuring volumes and analyzing gases using the GKhP-3 gas analyzer are observed, the total absolute error in determining the content of CO2 and O2 should not exceed 0 2 - 0 4% of their true value.
From the table 1 - 3 we can conclude that the data we use for the starting substances, taken from different sources, have relatively small differences, which lie within the absolute errors in determining these quantities.
Random errors can be absolute and relative. A random error having 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 analytical 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 has no independent meaning, and in any seriously conducted experiment several parallel measurements must be carried out, from which the experimental error is calculated. Measurement errors, depending on the reasons for their occurrence, are divided into three types.
It is almost impossible to determine the true value of a physical quantity absolutely accurately, because any measurement operation is associated with a number of errors or, in other words, inaccuracies. The reasons for errors can be very different. Their occurrence may be associated with inaccuracies in the manufacture and adjustment of the measuring device, due to the physical characteristics of the object under study (for example, when measuring the diameter of a wire of non-uniform thickness, the result randomly depends on the choice of the measurement site), random reasons, etc.
The experimenter’s task is to reduce their influence on the result, and also to indicate how close the result obtained is to the true one.
There are concepts of absolute and relative error.
Under absolute error measurements 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 and is the true value of the measured value.
The result of any physical measurement is usually written in the form:
where is the arithmetic mean value of the measured value, 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 quantity lies in the interval [ - , + ].
Absolute error is a dimensional quantity; it has the same dimension as the measured quantity.
The absolute error does not fully characterize the accuracy of the measurements taken. In fact, if we measure segments 1 m and 5 mm long with the same absolute error ± 1 mm, the accuracy of the measurements 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 occurrence of errors, they can be divided into the following classes: instrumental, systematic, random, and misses (gross errors).
Errors are caused either by a malfunction of the device, or 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 occurrence, it is necessary to be careful and thorough when working with devices. Results containing errors must be excluded from consideration (discarded).
Instrument errors. If the measuring device is in good working order and adjusted, then measurements can be made on it with limited accuracy determined by the type of device. It is customary to consider the instrument error of a pointer instrument to be equal to half the smallest division of its scale. In instruments with digital readout, the instrument error is equated to the value of one smallest digit of 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 ensure their elimination.
Systematic errors are conventionally 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 the 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: a malfunction of the device itself, a scale that does not correspond to the zero value, or the accuracy class of the device;
3) errors, the existence of which may not be suspected, but their magnitude can often be significant. Such errors occur most often in complex measurements. A simple example of such an error is the measurement of the density of some sample containing a cavity inside;
4) errors caused by 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 of repeated measurements of the same quantity.
Related information.
Absolute measurement error is a quantity 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 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 figures. If the absolute error of the value a does not exceed one place unit of the last digit of the number a, then the number is said to have all the correct signs. Approximate numbers should be written down, keeping only the correct signs. If, for example, the absolute error of the number 52400 is 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 correct significant digits it contains. When counting significant figures, the zeros on the left side of the number are not counted.
For example, the number 0.0283 has three valid significant figures, and 2.5400 has five valid significant figures.
Rules for rounding numbers. If the approximate number contains extra (or incorrect) digits, then it should be rounded. When rounding, an additional error occurs that does not exceed half a unit of the place of the last significant digit ( d) rounded number. When rounding, only the correct digits are retained; extra characters are discarded, and if the first discarded digit is greater than or equal to d/2, then the last digit stored is increased by one.
Extra digits in integers are replaced by zeros, and in decimals they are discarded (as are extra zeros). For example, if the measurement error is 0.001 mm, then the result 1.07005 is rounded to 1.070. If the first of the digits modified by zeros and discarded is less than 5, the remaining digits are not changed. For example, the number 148935 with a measurement precision of 50 has a rounding value of 148900. If the first of the digits replaced by zeros or discarded is 5, and there are no digits or zeros following it, then it is rounded to the nearest even number. For example, the number 123.50 is rounded to 124. If the first zero or drop digit is greater than 5 or equal to 5 but is followed by a significant digit, then the last remaining digit is incremented by one. For example, the number 6783.6 is rounded to 6784.
Example 2.2. When rounding 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. A seller weighs a watermelon on a scale. The smallest weight in the set 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 called truncation errors, rounding errors, and propagation errors. For example, when using iterative methods for searching for the roots of nonlinear equations, the results are approximate, in contrast to direct methods that provide an exact solution.
Truncation errors
This type of error is associated with the error inherent in the task itself. It may be due to inaccuracy in determining the source data. For example, if any dimensions are specified in the problem statement, then in practice for real objects these dimensions are always known with some accuracy. The same applies to any other physical parameters. This also includes the inaccuracy of 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 a problem. During calculations, error accumulation or, in other words, 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 error depends on the nature and number of arithmetic operations used in the calculation.
Rounding errors
This type of error occurs because the true value of a number is not always accurately stored by the computer. When a real number is stored in computer memory, it is written as a mantissa and exponent in much the same way as a number is displayed on a calculator.
