Calculation of relative and absolute measurement errors. Relative and absolute error: concept, calculation and properties
Measurements of many quantities occurring in nature cannot be accurate. The measurement gives a number expressing a value with varying degrees of accuracy (length measurement with an accuracy of 0.01 cm, calculation of the value of a function at a point with an accuracy of up to, etc.), that is, approximately, with some error. The error can be set in advance, or, conversely, it needs to be found.
The theory of errors has the object of its study mainly of approximate numbers. When calculating instead of 
usually use approximate numbers: (if accuracy is not particularly important), (if accuracy is important). How to carry out calculations with approximate numbers, determine their errors - this is the theory of approximate calculations (error theory).
In the future, exact numbers will be denoted by capital letters, and the corresponding approximate numbers will be denoted by lowercase letters.
Errors arising at one or another stage of solving the problem can be divided into three types:
1) Problem error. This type of error occurs when constructing mathematical model phenomena. It is far from always possible to take into account all the factors and the degree of their influence on the final result. That is, the mathematical model of an object is not its exact image, its description is not accurate. Such an error is unavoidable.
2) Method error. This error arises as a result of replacing the original mathematical model with a more simplified one, for example, in some problems of correlation analysis, a linear model is acceptable. Such an error is removable, since at the stages of calculation it can be reduced to an arbitrarily small value.
3) Computational ("machine") error. Occurs when a computer performs arithmetic operations.
Definition 1.1. Let be - exact value quantities (numbers), - the approximate value of the same quantity (). True absolute error approximate number is the modulus of the difference between the exact and approximate values:
. (1.1)
Let, for example, =1/3. When calculating on the MK, they gave the result of dividing 1 by 3 as an approximate number = 0.33. Then 
.
However, in reality, in most cases, the exact value of the quantity is not known, which means that (1.1) cannot be applied, that is, the true absolute error cannot be found. Therefore, another value is introduced that serves as some estimate (upper bound for ).
Definition 1.2.
Limit absolute error approximate number, representing an unknown exact number, is called such a possibly smaller number, which does not exceed the true absolute error, i.e 
. (1.2)
For an approximate number of quantities satisfying inequality (1.2), there are infinitely many, but the most valuable of them will be the smallest of all those found. From (1.2), based on the definition of the modulus, we have , or abbreviated as the equality
. (1.3)
Equality (1.3) determines the boundaries within which an unknown exact number is located (they say that an approximate number expresses an exact number with a limiting absolute error). It is easy to see that the smaller , the more precisely these boundaries are determined.
For example, if measurements of a certain value gave the result cm, while the accuracy of these measurements did not exceed 1 cm, then the true (exact) length 
cm.
Example 1.1. Given a number. Find the limiting absolute error of the number by the number .
Decision: From equality (1.3) for the number ( =1.243; =0.0005) we have a double inequality , i.e.
Then the problem is posed as follows: to find for the number the limiting absolute error satisfying the inequality 
. Taking into account the condition (*), we obtain (in (*) we subtract from each part of the inequality)
Since in our case 
, then , whence =0.0035.
Answer: =0,0035.
The limiting absolute error often gives a poor idea of the accuracy of measurements or calculations. For example, \u003d 1 m when measuring the length of a building will indicate that they were not carried out accurately, and the same error \u003d\u003d 1 m when measuring the distance between cities gives very quality assessment. Therefore, another value is introduced.
Definition 1.3. True relative error number, which is an approximate value of the exact number, is the ratio of the true absolute error of the number to the modulus of the number itself:
. (1.4)
For example, if, respectively, the exact and approximate values, then
However, formula (1.4) is not applicable if the exact value of the number is not known. Therefore, by analogy with the limiting absolute error, the limiting relative error is introduced.
Definition 1.4. Limiting relative error a number that is an approximation of an unknown exact number is called the smallest possible number , which is not exceeded by the true relative error , i.e
.
(1.5)
From inequality (1.2) we have 
; whence, taking into account (1.5)
Formula (1.6) has a greater practical applicability compared to (1.5), since the exact value does not participate in it. Taking into account (1.6) and (1.3), one can find the boundaries that contain the exact value of the unknown quantity.
Let some random value 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
.
How more number 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
.
At large numbers 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 instruments;
b) not identical conditions for repeated measurements;
c) random changes 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, mathematical theory random phenomena allow 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 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 events i is the attitude
where n - total number 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
.
Thus, 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
.
Square root from 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. How precise measurement, 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.
The measurements are called straight, if the values of the quantities are determined directly by the instruments (for example, measuring the length with a ruler, determining the time with a stopwatch, etc.). The measurements are called indirect, if the value of the measured quantity is determined by direct measurements of other quantities that are associated with the measured specific relationship.
Random errors in direct measurements
Absolute and relative error. Let it be held N measurements of the same quantity x in the absence of systematic error. The individual measurement results look like: x 1 ,x 2 , …,x N. The average value of the measured quantity is chosen as the best:
Absolute error single measurement is called the difference of the form:
.
Average absolute error N single measurements:
(2)
called average absolute error.
Relative error is the ratio of the average absolute error to the average value of the measured quantity:
.
(3)
Instrument errors in direct measurements
If there are no special instructions, the error of the instrument is equal to half of its division value (ruler, beaker).
The error of instruments equipped with a vernier is equal to the division value of the vernier (micrometer - 0.01 mm, caliper - 0.1 mm).
The error of tabular values is equal to half the unit of the last digit (five units of the next order after the last significant digit).
The error of electrical measuring instruments is calculated according to the accuracy class With indicated on the instrument scale:
For example: 
and 
,
where U max and I max– measurement limit of the device.
The error of devices with digital indication is equal to the unit of the last digit of the indication.
After assessing the random and instrumental errors, the one whose value is greater is taken into account.
Calculation of errors in indirect measurements
Most measurements are indirect. In this case, the desired value X is a function of several variables a,b, c… , the values of which can be found by direct measurements: Х = f( a, b, c…).
The arithmetic mean of the result of indirect measurements will be equal to:
X = f( a, b, c…).
One of the ways to calculate the error is the way of differentiating the natural logarithm of the function X = f( a,
b,
c...). If, for example, the desired value X is determined by the relation X = 
, then after taking the logarithm we get: lnX = ln a+ln b+ln( c+
d).
The differential of this expression is:
.
With regard to the calculation of approximate values, it can be written for the relative error in the form:
=
.
(4)
The absolute error in this case is calculated by the formula:
Х = Х(5)
Thus, the calculation of errors and the calculation of the result for indirect measurements are carried out in the following order:
1) Carry out measurements of all quantities included in the original formula to calculate the final result.
2) Calculate the arithmetic mean values of each measured value and their absolute errors.
3) Substitute in the original formula the average values of all measured values and calculate the average value of the desired value:
X = f( a, b, c…).
4) Take the logarithm of the original formula X = f( a, b, c...) and write down the expression for the relative error in the form of formula (4).
5) Calculate the relative error = 
.
6) Calculate the absolute error of the result using the formula (5).
7) The final result is written as:
|
X \u003d X cf X |
The absolute and relative errors of the simplest functions are given in the table:
|
Absolute error |
Relative error |
|
|
a+ b |
a+ b |
|
|
a+ b |
|
|
|
Due to the errors inherent in the measuring instrument, the chosen method and measurement technique, the difference in the external conditions in which the measurement is performed from the established ones, and other reasons, the result of almost every measurement is burdened with an error. This error is calculated or estimated and attributed to the result obtained. Measurement error(briefly - measurement error) - deviation of the measurement result from the true value of the measured quantity. The true value of the quantity due to the presence of errors remains unknown. It is used to solve theoretical tasks metrology. In practice, the actual value of the quantity is used, which replaces the true value. The measurement error (Δx) is found by the formula: x = x meas. - x actual (1.3) where x meas. - the value of the quantity obtained on the basis of measurements; x actual is the value of the quantity taken as real. The real value for single measurements is often taken as the value obtained with the help of an exemplary measuring instrument, for repeated measurements - the arithmetic mean of the values of individual measurements included in this series. Measurement errors can be classified according to the following criteria: By the nature of the manifestation - systematic and random; By way of expression - absolute and relative; According to the conditions for changing the measured value - static and dynamic; According to the method of processing a number of measurements - arithmetic and root mean squares; According to the completeness of the coverage of the measuring task - private and complete; Relative to unit physical quantity— errors of reproduction of the unit, storage of the unit and transmission of the size of the unit. Systematic measurement error(briefly - systematic error) - a component of the error of the measurement result, which remains constant for a given series of measurements or regularly changes during repeated measurements of the same physical quantity. According to the nature of the manifestation, systematic errors are divided into constant, progressive and periodic. Permanent systematic errors(briefly - constant errors) - errors, long time retaining their value (for example, during the entire series of measurements). This is the most common type of error. Progressive systematic errors(briefly - progressive errors) - continuously increasing or decreasing errors (for example, errors from wear of measuring tips that come into contact during grinding with a part when it is controlled by an active control device). Periodic systematic error(shortly - periodic error) - an error whose value is a function of time or a function of pointer movement measuring instrument(for example, the presence of eccentricity in goniometers with a circular scale causes a systematic error that varies according to a periodic law). Based on the reasons for the appearance of systematic errors, there are instrumental errors, method errors, subjective errors and errors due to the deviation of external measurement conditions from established methods. Instrumental measurement error(briefly - instrumental error) is the result of a number of reasons: wear of parts of the device, excessive friction in the mechanism of the device, inaccurate strokes on the scale, discrepancy between the actual and nominal values measures, etc. Measurement method error(briefly - the error of the method) may arise due to the imperfection of the measurement method or its simplifications, established by the measurement procedure. For example, such an error may be due to the insufficient speed of the measuring instruments used when measuring the parameters of fast processes or unaccounted for impurities when determining the density of a substance based on the results of measuring its mass and volume. Subjective measurement error(briefly - subjective error) is due to the individual errors of the operator. Sometimes this error is called personal difference. It is caused, for example, by a delay or advance in the acceptance of a signal by the operator. Deviation error(in one direction) external measurement conditions from those established by the measurement procedure leads to the occurrence of a systematic component of the measurement error. Systematic errors distort the measurement result, therefore, they must be eliminated, as far as possible, by introducing corrections or adjusting the instrument to bring the systematic errors to an acceptable minimum. Non-excluded systematic error(briefly - non-excluded error) - this is the error of the measurement result due to the error in calculating and introducing a correction for the effect of a systematic error, or a small systematic error, the correction for which is not introduced due to smallness. This type of error is sometimes referred to as non-excluded bias residuals(briefly - non-excluded balances). For example, when measuring the length of a line meter in the wavelengths of the reference radiation, several non-excluded systematic errors were revealed (i): due to inaccurate temperature measurement - 1 ; due to the inaccurate determination of the refractive index of air - 2, due to the inaccurate value of the wavelength - 3. Usually, the sum of non-excluded systematic errors is taken into account (their boundaries are set). With the number of terms N ≤ 3, the boundaries of non-excluded systematic errors are calculated by the formula When the number of terms is N ≥ 4, the formula is used for calculations
where k is the coefficient of dependence of non-excluded systematic errors on the chosen confidence probability P with their uniform distribution. At P = 0.99, k = 1.4, at P = 0.95, k = 1.1. Random measurement error(briefly - random error) - a component of the error of the measurement result, changing randomly (in sign and value) in a series of measurements of the same size of a physical quantity. Causes of random errors: rounding errors when reading readings, variation in readings, changes in measurement conditions of a random nature, etc. Random errors cause dispersion of measurement results in a series. The theory of errors is based on two provisions, confirmed by practice: 1. With a large number of measurements, random errors of the same numerical value, but of a different sign, occur equally often; 2. Large (in absolute value) errors are less common than small ones. An important conclusion for practice follows from the first position: with an increase in the number of measurements, the random error of the result obtained from a series of measurements decreases, since the sum of the errors of individual measurements of this series tends to zero, i.e.
For example, as a result of measurements, a series of values was obtained electrical resistance(which are corrected for the effects of systematic errors): R 1 = 15.5 ohms, R 2 = 15.6 ohms, R 3 = 15.4 ohms, R 4 = 15.6 ohms and R 5 = 15.4 ohms . Hence R = 15.5 ohms. Deviations from R (R 1 \u003d 0.0; R 2 \u003d +0.1 Ohm, R 3 \u003d -0.1 Ohm, R 4 \u003d +0.1 Ohm and R 5 \u003d -0.1 Ohm) are random errors of individual measurements in a given series. It is easy to see that the sum R i = 0.0. This indicates that the errors of individual measurements of this series are calculated correctly. Despite the fact that with an increase in the number of measurements, the sum of random errors tends to zero (in this example she happened to be zero), the random error of the measurement result must be estimated. In the theory of random variables, the dispersion of o2 serves as a characteristic of the dispersion of the values of a random variable. "| / o2 \u003d a is called the standard deviation of the general population or standard deviation. It is more convenient than dispersion, since its dimension coincides with the dimension of the measured quantity (for example, the value of the quantity is obtained in volts, the standard deviation will also be in volts). Since in the practice of measurements one deals with the term “error”, the term “rms error” derived from it should be used to characterize a number of measurements. A number of measurements can be characterized by the arithmetic mean error or the range of measurement results. The range of measurement results (briefly - range) is the algebraic difference between the largest and smallest results of individual measurements that form a series (or sample) of n measurements: R n \u003d X max - X min (1.7) where R n is the range; X max and X min - the largest and smallest value values in a given series of measurements. For example, out of five measurements of the hole diameter d, the values R 5 = 25.56 mm and R 1 = 25.51 mm turned out to be its maximum and minimum values. In this case, R n \u003d d 5 - d 1 \u003d 25.56 mm - 25.51 mm \u003d 0.05 mm. This means that the remaining errors of this series are less than 0.05 mm. Average arithmetic error of a single measurement in a series(briefly - the arithmetic mean error) - the generalized scattering characteristic (due to random reasons) of individual measurement results (of the same value), included in a series of n equally accurate independent measurements, is calculated by the formula
where X i is the result of the i-th measurement included in the series; x is the arithmetic mean of n values of the quantity: |X i - X| is the absolute value of the error of the i-th measurement; r is the arithmetic mean error. The true value of the arithmetic mean error p is determined from the ratio p = lim r, (1.9) With the number of measurements n > 30, between the arithmetic mean (r) and the mean square (s) there are correlations s = 1.25r; r and = 0.80 s. (1.10) The advantage of the arithmetic mean error is the simplicity of its calculation. But still more often determine the mean square error. Root mean square error individual measurement in a series (briefly - root mean square error) - a generalized scattering characteristic (due to random reasons) of individual measurement results (of the same value) included in a series of P equally accurate independent measurements, calculated by the formula
The root mean square error for the general sample o, which is the statistical limit of S, can be calculated for /i-mx > by the formula: Σ = limS (1.12) In reality, the number of dimensions is always limited, so it is not σ that is calculated , and its approximate value (or estimate), which is s. The more P, the closer s is to its limit σ . With a normal distribution, the probability that the error of a single measurement in a series will not exceed the calculated root mean square error is small: 0.68. Therefore, in 32 cases out of 100 or 3 cases out of 10, the actual error may be greater than the calculated one.
Figure 1.2 Decrease in the value of the random error of the result of multiple measurements with an increase in the number of measurements in a series In a series of measurements, there is a relationship between the rms error of an individual measurement s and the rms error of the arithmetic mean S x: which is often called the "rule of Y n". It follows from this rule that the measurement error due to the action of random causes can be reduced by n times if n measurements of the same size of any quantity are performed, and the arithmetic mean value is taken as the final result (Fig. 1.2). Performing at least 5 measurements in a series makes it possible to reduce the effect of random errors by more than 2 times. With 10 measurements, the effect of random error is reduced by a factor of 3. A further increase in the number of measurements is not always economically feasible and, as a rule, is carried out only for critical measurements requiring high accuracy. The root mean square error of a single measurement from a series of homogeneous double measurements S α is calculated by the formula
where x" i and x"" i are i-th results of measurements of the same size quantity in the forward and reverse directions by one measuring instrument. With unequal measurements, the root mean square error of the arithmetic mean in the series is determined by the formula
where p i is the weight of the i-th measurement in a series of unequal measurements. The root mean square error of the result of indirect measurements of the quantity Y, which is a function of Y \u003d F (X 1, X 2, X n), is calculated by the formula
where S 1 , S 2 , S n are root-mean-square errors of measurement results for X 1 , X 2 , X n . If, for greater reliability of obtaining a satisfactory result, several series of measurements are carried out, the root-mean-square error of an individual measurement from m series (S m) is found by the formula
Where n is the number of measurements in the series; N is the total number of measurements in all series; m is the number of series. With a limited number of measurements, it is often necessary to know the RMS error. To determine the error S, calculated by formula (2.7), and the error S m , calculated by formula (2.12), you can use the following expressions
where S and S m are the mean square errors of S and S m , respectively. For example, when processing the results of a series of measurements of the length x, we obtained
The value S = ±0.7 mm means that due to the calculation error, s is in the range from 2.4 to 3.8 mm, therefore, tenths of a millimeter are unreliable here. In the considered case it is necessary to write down: S = ±3 mm. In order to have greater confidence in the estimation of the error of the measurement result, the confidence error or confidence limits of the error are calculated. With a normal distribution law, the confidence limits of the error are calculated as ±t-s or ±t-s x , where s and s x are the root mean square errors, respectively, of a single measurement in a series and the arithmetic mean; t is a number depending on the confidence level P and the number of measurements n. An important concept is the reliability of the measurement result (α), i.e. the probability that the desired value of the measured quantity falls within a given confidence interval. For example, when processing parts on machine tools in a stable technological mode, the distribution of errors obeys the normal law. Assume that the part length tolerance is set to 2a. In this case, the confidence interval in which the desired value of the length of the part a is located will be (a - a, a + a). If 2a = ±3s, then the reliability of the result is a = 0.68, i.e., in 32 cases out of 100, the part size should be expected to go beyond the tolerance of 2a. When evaluating the quality of the part according to the tolerance 2a = ±3s, the reliability of the result will be 0.997. In this case, only three parts out of 1000 can be expected to go beyond the established tolerance. However, an increase in reliability is possible only with a decrease in the error in the length of the part. So, to increase reliability from a = 0.68 to a = 0.997, the error in the length of the part must be reduced by a factor of three. Recently received wide use the term "measurement reliability". In some cases, it is unreasonably used instead of the term "measurement accuracy". For example, in some sources you can find the expression "establishing the unity and reliability of measurements in the country." Whereas it would be more correct to say “establishment of unity and the required accuracy of measurements”. Reliability is considered by us as a qualitative characteristic, reflecting the proximity to zero of random errors. Quantitatively, it can be determined through the unreliability of measurements. Uncertainty of measurements(briefly - unreliability) - an assessment of the discrepancy between the results in a series of measurements due to the influence of the total impact of random errors (determined by statistical and non-statistical methods), characterized by the range of values in which the true value of the measured quantity is located. In accordance with the recommendations of the International Bureau of Weights and Measures, the uncertainty is expressed as the total rms measurement error - Su including the rms error S (determined by statistical methods) and the rms error u (determined by non-statistical methods), i.e.
Limit measurement error(briefly - marginal error) - the maximum measurement error (plus, minus), the probability of which does not exceed the value of P, while the difference 1 - P is insignificant. For example, with a normal distribution, the probability of a random error of ±3s is 0.997, and the difference 1-P = 0.003 is insignificant. Therefore, in many cases, the confidence error ±3s is taken as the limit, i.e. pr = ±3s. If necessary, pr can also have other relationships with s for sufficiently large P (2s, 2.5s, 4s, etc.). In connection with the fact that in the GSI standards, instead of the term "root mean square error", the term "root mean square deviation" is used, in further reasoning we will stick to this term. Absolute measurement error(briefly - absolute error) - measurement error, expressed in units of the measured value. So, the error X of measuring the length of the part X, expressed in micrometers, is an absolute error. The terms “absolute error” and “absolute error value” should not be confused, which is understood as the value of the error without taking into account the sign. So, if the absolute measurement error is ±2 μV, then the absolute value of the error will be 0.2 μV. Relative measurement error(briefly - relative error) - measurement error, expressed as a fraction of the value of the measured value or as a percentage. The relative error δ is found from the ratios:
For example, there is a real value of the part length x = 10.00 mm and an absolute value of the error x = 0.01 mm. The relative error will be Static error is the error of the measurement result due to the conditions of the static measurement. Dynamic error is the error of the measurement result due to the conditions of dynamic measurement. Unit reproduction error- error of the result of measurements performed when reproducing a unit of physical quantity. So, the error in reproducing a unit using the state standard is indicated in the form of its components: a non-excluded systematic error, characterized by its boundary; random error characterized by the standard deviation s and yearly instability ν. Unit Size Transmission Error is the error in the result of measurements performed when transmitting the size of the unit. The unit size transmission error includes non-excluded systematic errors and random errors of the method and means of unit size transmission (for example, a comparator). abstract Absolute and relative error Introduction Absolute error - is an estimate of the absolute measurement error. Computed different ways. The calculation method is determined by the distribution of the random variable. Accordingly, the magnitude of the absolute error depending on the distribution of the random variable may be different. If a is the measured value, and is the true value, then the inequality must be satisfied with some probability close to 1. If the random variable distributed according to the normal law, then usually its standard deviation is taken as the absolute error. Absolute error is measured in the same units as the value itself. There are several ways to write a quantity along with its absolute error. · Usually signed notation is used ± . For example, the 100m record set in 1983 is 9.930±0.005 s. · To record values measured with very high accuracy, another notation is used: the numbers corresponding to the error of the last digits of the mantissa are added in brackets. For example, the measured value of the Boltzmann constant is 1,380 6488 (13)×10?23 J/K, which can also be written much longer as 1.380 6488×10?23 ± 0.000 0013×10?23 J/K. Relative error- measurement error, expressed as the ratio of the absolute measurement error to the actual or average value of the measured quantity (RMG 29-99):. Relative error is a dimensionless quantity, or is measured as a percentage. 1. What is called an approximate value? Too much and too little? In the process of calculations, one often has to deal with approximate numbers. Let be BUT- the exact value of a certain quantity, hereinafter called exact number BUT.Under the approximate value of the quantity BUT,or approximate numberscalled a number a, which replaces the exact value of the quantity BUT.If a a< BUT,then ais called the approximate value of the number And for lack.If a a> BUT,- then in excess.For example, 3.14 is an approximation of the number ? by deficiency, and 3.15 by excess. To characterize the degree of accuracy of this approximation, the concept is used errors or errors. error ?aapproximate number ais called the difference of the form ?a = A - a, where BUTis the corresponding exact number. The figure shows that the length of the segment AB is between 6 cm and 7 cm. This means that 6 is the approximate value of the length of the segment AB (in centimeters)\u003e with a deficiency, and 7 is with an excess. Denoting the length of the segment with the letter y, we get: 6< у < 1. Если a < х < b, то а называют приближенным значением числа х с недостатком, a b - приближенным значением х с избытком. Длина segmentAB (see Fig. 149) is closer to 6 cm than to 7 cm. It is approximately equal to 6 cm. They say that the number 6 was obtained by rounding the length of the segment to integers. . What is an approximation error? A) absolute? B) Relative? A) The absolute error of approximation is the modulus of the difference between the true value of a quantity and its approximate value. |x - x_n|, where x is the true value, x_n is the approximate value. For example: The length of a sheet of A4 paper is (29.7 ± 0.1) cm. And the distance from St. Petersburg to Moscow is (650 ± 1) km. The absolute error in the first case does not exceed one millimeter, and in the second - one kilometer. The question is to compare the accuracy of these measurements. If you think that the length of the sheet is measured more precisely because the absolute error does not exceed 1 mm. Then you are wrong. These values cannot be directly compared. Let's do some reasoning. When measuring the length of a sheet, the absolute error does not exceed 0.1 cm by 29.7 cm, that is, as a percentage, it is 0.1 / 29.7 * 100% = 0.33% of the measured value. When we measure the distance from St. Petersburg to Moscow, the absolute error does not exceed 1 km per 650 km, which is 1/650 * 100% = 0.15% of the measured value as a percentage. We see that the distance between cities is measured more accurately than the length of an A4 sheet. B) The relative error of approximation is the ratio of the absolute error to the modulus of the approximate value of the quantity. mathematical error fraction where x is the true value, x_n is the approximate value. Relative error is usually called as a percentage. Example. Rounding the number 24.3 to units results in the number 24. The relative error is equal. They say that the relative error in this case is 12.5%. ) What kind of rounding is called rounding? A) with a disadvantage? b) Too much? A) rounding down When rounding a number expressed as a decimal fraction to within 10^(-n), with a deficiency, the first n digits after the decimal point are retained, and the subsequent ones are discarded. For example, rounding 12.4587 to the nearest thousandth with a demerit results in 12.458. B) Rounding up When rounding a number expressed as a decimal fraction, up to 10^(-n), the first n digits after the decimal point are retained with an excess, and the subsequent ones are discarded. For example, rounding 12.4587 to the nearest thousandth with a demerit results in 12.459. ) The rule for rounding decimals. Rule. To round a decimal to a certain digit of the integer or fractional part, all smaller digits are replaced by zeros or discarded, and the digit preceding the digit discarded during rounding does not change its value if it is followed by the numbers 0, 1, 2, 3, 4, and increases by 1 (one) if the numbers are 5, 6, 7, 8, 9. Example. Round the fraction 93.70584 to: ten-thousandths: 93.7058 thousandths: 93.706 hundredths: 93.71 tenths: 93.7 integer: 94 tens: 90 Despite the equality of absolute errors, since measured quantities are different. The larger the measured size, the smaller the relative error at a constant absolute. TutoringNeed help learning a topic?
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= 86 mm 2 at n = 10,
= 3.1 mm
= 0.7 mm or S = ±0.7 mm
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