Absolute error. Absolute and relative calculation error

X \u003d x cp 

Absolute

error

Relative

error

a +. b.

a +. b.

Let some random value a. Measured n. Once in the same conditions. Measurement results gave a set n.different numbers

Absolute error - Size size. Among N. The values \u200b\u200bof absolute errors are necessarily found both positive and negative.

For the most likely value of the magnitude but Usually accept average The value of measurement results

.

The greater the number of measurements, the closer the average value to the true one.

Absolute errori.

.

Relative errori.-Ho dimension called the magnitude

Relative error - the magnitude is dimensionless. Usually relevant error is expressed as a percentage, for this e I. Domingered by 100%. The value of the relative error characterizes the measurement accuracy.

Average absolute error determined like this:

.

We emphasize the need to summarize absolute values \u200b\u200b(modules) values \u200b\u200bof D and i.Otherwise, the identical zero result will be.

Medium relative error called the magnitude

.

With a large number of measurements.

The relative error can be considered as the meaning of the error on the unit of the measured value.

The accuracy of measurements is judged on the basis of comparing the errors of measurement results. Therefore, measurement errors are expressed in this form so that it is enough to match the accuracy to match only the results of the results, without comparing the dimensions of the measured objects or knowing these sizes is very approximately. From the practice it is known that the absolute error of the measurement of the angle does not depend on the value of the angle, and the absolute error of measuring length depends on the length value. The greater the length of the length, the this method And the measurement conditions absolute error will be greater. Consequently, according to the absolute error of the result, the accuracy of measurement of the angle can be judged, and the accuracy of measurement of length is impossible. The expression of the error in relative form allows you to compare in known cases Accuracy of angular and linear measurements.

The basic concepts of the theory of probability. Random error.

Random error call the component of measurement errors varying randomly when repeated measurements of the same value.

When conducting with the same care and in the same conditions of repeated measurements of the same constant unchanged value, we obtain measurement results - some of them differ from each other, and some coincide. Such discrepancies in the measurement results are indicated about the presence of random components of the error.

Random error occurs with the simultaneous impact of many sources, each of which itself has an inconspicuous effect on the measurement result, but the total impact of all sources can be quite strong.

Random errors are the inevitable consequence of any measurements and are conditioned:

a) the inaccuracy of samples on the scale of instruments and tools;

b) not identical to the conditions of repeated measurements;

c) disorderly changes in external conditions (temperature, pressure, power field, etc.), which cannot be monitored;

d) all other impacts on measurements, the reasons for which we are unknown. The magnitude of the random error can be minimized by repeatedly repetition of the experiment and the corresponding mathematical processing of the results obtained.

A random error can take various value in the absolute value, which cannot be predicted for this measurement act. This error equally can be both positive and negative. Random errors are always present in the experiment. In the absence of systematic errors, they serve as the scatter of repeated measurements relative to the true value.

Suppose that with the help of the stopwatch, the period of oscillations of the pendulum is measured, and the measurement repeated repeatedly. Starting errors and stopwatch stops, error in the size of the reference, a small non-uniformity of the movement of the pendulum - all this causes the scatter of the results of repeated measurements and therefore can be attributed to the category of random errors.

If there are no other errors, then the results will be somewhat overestimated, while others are somewhat low. But if, in addition, the clock is also lagging behind, then all the results will be understated. This is a systematic error.

Some factors can cause systematic and random errors at the same time. So, including and turning off the stopwatch, we can create a small irregular scatter of starting moments and stopping the clock relative to the movement of the pendulum and make a random error. But if we are also in a hurry to turn on the stopwatch and go to turn it off somewhat somewhat, then it will lead to a systematic error.

Random errors are caused by a parallax error at counting the division of the scale of the device, concussion the foundation of the building, the influence of a minor air movement, etc.

Although it is impossible to exclude accidental errors of individual measurements, the mathematical theory of random phenomena make it possible to reduce the influence of these errors on the final measurement result. Below will be shown that for this it is necessary to produce not one, but several measurements, and the lower the meaning of the error we want to get, the more measurements need to be carried out.

Due to the fact that the emergence of random errors is inevitable and unreasonably, the main task of every 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 are the same size, but of different signThose errors in the direction of increasing and decreasing the result are quite common.

2. Large in the absolute amount of errors are less common than small, therefore, the probability of an error is reduced with an increase in its magnitude.

The behavior of random variables describe statistical patterns that are the subject of probability theory. Statistical definition of probability w I. events i. is attitude

where n. - the total number of experiments n I. - the number of experiments in which the event i.occurred. In this case, the total number of experiments should be very large ( n. ® ¥). With a large number of measurements, random errors are subordinate to the normal distribution (Gauss distribution), the main signs of which are the following:

1. The larger the deviation of the measured value from the true, the less likelihood of such a result.

2. Deviations in both directions from the true value are equally possible.

Of the above assumptions, it follows that to reduce the effect of random errors, it is necessary to measure this value several times. Suppose we measure some value x. Let produced n.measurements: x 1, x 2, ... x n - the same method and with the same thoroughness. It can be expected that the number dN.the results obtained, which lie in some fairly narrow interval from X. before x + DX.must be proportional to:

The magnitude of the interval taken DX.;

The total number of measurements N..

Probability dW.(x.) what is some meaning x. lies in the interval from x. before x + dx, Determined as follows :

(with the number of measurements N. ®¥).

Function f.(h.) It 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 their large quantities are subject to the law of normal distribution.

Found by Gauss Function distribution of a continuous random variable X. It has the following form:

where MUS. - distribution parameters .

Parametermal distribution is equal to the average value of Á x.ñ random variable, which, with an arbitrary known distribution function, is determined by the integral

.

In this way, the value M is the most likely value of the measured value of X, i.e. Its best evaluation.

The parameter S 2 of the normal distribution is equal to the dispersion D of a random variable, which in the general case is determined by the following integral

.

Square root From the dispersion is called an average quadratic deviation of a random variable.

The average deviation (error) of the random variable Ásñ is determined using the distribution function as follows.

The average measurement error of Ásñ, calculated according to the function of the Gauss distribution, is correlated from the value of the average quadratic deviation s as follows:

< s. > \u003d 0.8s.

The parameters S and M are interconnected as follows:

.

This expression allows you to find the average quadratic deviation s if there is a normal distribution curve.

The graph of the Gauss function is presented in drawings. Function f.(x.) symmetrical about the ordinate spent at the point x \u003d.m; passes through maximum at point x \u003d.m and has a bend at the points M ± s. Thus, the dispersion characterizes the width of the distribution function, or indicates how widely the values \u200b\u200bof the random variable relative to its true value are scattered. More precisely, the measurement, the closer to the true value of the results of individual measurements, i.e. The value of S is less. Figure A shows the function F.(x.) for three values \u200b\u200bs .

Square Figure, limited curve f.(x.) and vertical straight, spent from points X. 1 I. x. 2 (Fig. B) , Numerically equal to the likelihood of the measurement result in the interval D x \u003d X. 1 - X. 2, which is called a trustful probability. Area under the entire curve f.(x.) equal to the likelihood of random variance in the interval from 0 to ¥, i.e.

,

since the probability of a reliable event is equal to one.

Using a normal distribution, error theory puts and solves two main tasks. The first is the assessment of the accuracy of the measurements. Second - Evaluation of the accuracy of the average arithmetic value Measurement results. Trust interval. Coefficient of a lifting.

Probability Theory allows you to determine the size of the interval in which with a certain probability w. There are results of individual measurements. This probability is called trust probability, and the corresponding interval (<x.\u003e ± D. x.) W. called confidential interval. The trusting probability is also equal to the relative proportion of the results inside the confidence interval.

If the number of measurements N. large enough, the trust probability expresses the share of total N. These measurements in which the measured value was within the confidence interval. Each trust probability w.it corresponds to your confidence interval.w 2 80%. The wider the confidential interval, the more likely to get the result inside this interval. In probability theory, a quantitative relationship between the value of the confidence interval, the trust probability and the number of measurements, is established.

If you choose the interval corresponding to the average error, that is, a \u003d.Ád. butС, then with a sufficiently large number of measurements correspond to the trust probability W. 60%. With a decrease in the number of measurements, the confidence probability corresponding to this trust interval (Á butñ ± Ád. butñ), decreases.

Thus, to assess the confidence interval of random variable, you can use the value of the average error. butñ .

To characterize the value of a random error, you must specify two numbers, namely, the value of the confidence interval and the value of the confidence probability . An indication of the magnitude of the error without the corresponding trust probability is largely devoid of meaning.

If the average measurement error of the Ásñ, the confidence interval recorded in the form (<x.\u003e ± Ásñ) W.defined with trust probability w.= 0,57.

If the average quadratic deviation s is known distribution of measurement results, the specified interval has the form (<x.>± t W.s) W.where T W. - The coefficient depending on the value of the confidence probability and is calculated on the distribution of Gauss.

The most commonly used values x. shown in Table 1.

In practice, it is usually numbers that computations are made are approximate values \u200b\u200bof certain quantities. For brevity speech, the approximate value of the magnitude is called an approximate number. The true value of magnitude is called the exact number. The approximate number has practical value only when we can determine which degree of accuracy is given, i.e. Rate his error. Recall the basic concepts from general course mathematics.

Denote: x. - accurate number (true value value), but -Cable number (approximate value of magnitude).

Definition 1.. The error (or true error) of an approximate number is the difference between the number x. and his approximate value but. Error of an approximate number but We will denote. So,

Exact number x. Most often it is unknown, so it is not possible to find true and absolute errors. On the other hand, it is necessary to estimate the absolute error, i.e. Specify the number that cannot exceed the absolute error. For example, measuring the length of the subject with this tool, we must be confident that the error of the resulting numerical value will not exceed a certain number, for example 0.1 mm. In other words, we need to know the boundary of the absolute error. We will call this border with an extreme absolute error.

Definition 3.. The utmost absolute error of the approximate number but Called a positive number such that, i.e.

It means h. By lack, - in excess. Apply such an entry:

It is clear that the limit absolute error is determined ambiguously: if a certain number is the limit absolute error, then any larger number is also the limit absolute error. In practice, it is trying to choose a smaller and easy-to-write (with 1-2 significant numbers) a number satisfying inequality (2.3).

Example. Determine the true, absolute and limit absolute error number A \u003d 0.17, taken as the approximate value of the number.

True error:

Absolute error:

For the limit absolute error you can take a number and any greater number. In the decimal record we will have: replacing this number large and possibly easier by recording, we will take:

Comment. If a but There is an approximate value of the number h., moreover, the maximum absolute error is equal to h., then they say that butthere is an approximate value of the number h. With accuracy h.

The knowledge of the absolute error is not enough to characterize the quality of measurement or calculation. Let, for example, obtained such results when measuring the length. Distance between two cities S 1\u003d 500 1 km and the distance between two buildings in the city S 2.\u003d 10 1 km. Although the absolute errors of both results are the same, but it has significant importance that in the first case the absolute error in 1 km falls on 500 km, in the second - 10 km. The measurement quality in the first case is better than in the second. The quality of the measurement or calculation is characterized by a relative error.

Definition 4. Relative error of approximate value but numbers h. called the ratio of the absolute error number butto the absolute value of the number h.:

Definition 5. The limit relative error of the approximate number but called a positive number such that.

Since, then from formula (2.7) it follows that can be calculated by the formula

For brevity speech in cases where this does not cause misunderstandings, instead of the "limit relative error" they say simply "relative error".

The limit relative error is often expressed as a percentage.

Example 1.. . Believing, we can accept \u003d. By producing division and rounded (necessarily upwards), we obtain \u003d 0.0008 \u003d 0.08%.

Example 2.When weighing the body, the result was obtained: p \u003d 23.4 0.2 g. We have \u003d 0.2. . By producing division and rounding, we obtain \u003d 0.9%.

Formula (2.8) determines the relationship between absolute and relative errors. From formula (2.8) follows:

Using formulas (2.8) and (2.9), we can, if the number is known but, on this absolute error to find a relative error and vice versa.

Note that formulas (2.8) and (2.9) often have to be used and then when we still do not know the approximate number butwith the required accuracy, and we know the rough approximate value but. For example, it is required to measure the length of the subject with a relative error not higher than 0.1%. It is asked: is it possible to measure the length with the desired accuracy with the help of a caliper, allowing you to measure length with an absolute error up to 0.1 mm? Let we have not yet measured the subject of the exact tool, but we know that the rough approximate length of length is about 12 cm. By formula (1.9) we find an absolute error:

It can be seen that with the help of a calipers, it is possible to measure with the required accuracy.

In the process of computational work, it is often necessary to move from absolute error to relative, and vice versa, which is done using formulas (1.8) and (1.9).

No dimension is free from errors, or, more precisely, the probability of measurement without errors is approaching zero. Rod and the causes of errors are very diverse and many factors affect them (Fig. 1.2).

The overall characteristics of influencing factors can be systematized from various points of view, for example, by influencing listed factors (Fig.1.2).

According to the results of measurement of the error, it is possible to divide into three types: systematic, random and misses.

Systematic errors, in turn, they are divided into groups due to their occurrence and character of manifestation. They can be eliminated in various ways, for example, the introduction of amendments.

fig. 1.2.

Random error associated with a complex combination of changing factors, usually unknown and difficult to analyze. Their effect on the measurement result can be reduced, for example, by repeated measurements with further statistical processing of the results obtained by the method of probability theory.

TO promacham these are gross errors that occur with sudden changes of the experimental condition. These errors are also random in nature, and after detection must be excluded.

The accuracy of measurements is assessed by measurement errors, which are divided by the nature of the occurrence of the instrumental and methodological and the method of calculations to absolute, relative and given.

Instrumental the error is characterized by the accuracy class of the measuring device, which is given in his passport in the form of normalized primary and additional errors.

Methodical the error is due to the imperfection of methods and measuring instruments.

Absolute the error is the difference between the measured G U and the true G values \u200b\u200bof the value determined by the formula:

Δ \u003d Δg \u003d g u -g

Note that the value has the dimension of the measured value.

Relative error find from equality

Δ \u003d ± Δg / g u · 100%

Led the error is calculated by the formula (the accuracy class of the measuring device)

Δ \u003d ± Δg / g norms · 100%

where G norms is the rational value of the measured value. It takes equal:

a) the final value of the device scale, if the zero mark is on the edge or outside the scale;

b) the amount of the end values \u200b\u200bof the scale excluding signs, if the zero mark is located inside the scale;

c) the length of the scale if the scale is uneven.

The accuracy class of the device is set when it is verified and is a normalized error calculated by formulas

γ \u003d ± Δg / g norms · 100% ifΔg m \u003d const

where Δg m is the highest possible absolute error of the device;

G k - the final value of the measurement limit of the device; C and D - coefficients that take into account the design parameters and properties of the measuring mechanism of the device.

For example, for a voltmeter with a constant relative error there is equality

Δ m \u003d ± C

Relative and reduced errors are associated with the following dependencies:

a) for any meaning of the above error

Δ \u003d ± γ · G norms / g u

b) for the greatest error

Δ \u003d ± γ m · g norms / g u

From these ratios, it follows that when measuring, for example, a voltmeter, in the chain at the same same value of the voltage, the relative error is the greater the less measureable voltage. And if this voltmeter is incorrectly selected, then the relative error may be commensurate with the valueG N. What is unacceptable. Note that in accordance with the terminology of the solved tasks, for example, when measuring the voltage G \u003d U, when measuring the current C \u003d i, the letter notation in the formulas for calculating the errors must be replaced with the corresponding characters.

Example 1.1. Voltmeter having values \u200b\u200bγ m \u003d 1.0%, U n \u003d g norms, g k \u003d 450 inThe voltage U u is measured, equal to 10 V. We estimate the measurement error.

Decision.

Answer. Measurement error is 45%. With such an error, the measured voltage cannot be considered reliable.

With limited device selection capabilities (voltmeter), a methodological error may be taken into account as amended by the formula

Example 1.2. Calculate the absolute error of the voltmeter B7-26 when measuring the voltage in the DC circuit. The accuracy class of the voltmeter is set as the maximally reduced error Γ m \u003d ± 2.5%. Used in the work limit of the Voltmeter scale U norms \u003d 30 V.

Decision.The absolute error is calculated according to the famous formulas:

(Since the error given, by definition, is expressed by the formula , From here you can find an absolute error:

Answer. ΔU \u003d ± 0.75 V.

Important stages in the measurement process are the processing of results and rounding rules. The theory of approximate calculations allows, knowing the degree of data accuracy, estimate the degree of accuracy of the results even before performing actions: to select data with the proper degree of accuracy sufficient to ensure the required accuracy of the result, but not too large to save the calculator from useless calculations; rationalize the calculation process itself, freeing it from the calculations that will not affect the exact figures results.

When processing results, rounding rules apply.

• Rule 1. If the first of the discarded numbers is greater than five, the latter of the stored digits is increased by one.

• Rule 2. If the first of the discarded numbers is less than five, then the increase is not done.

• Rule 3. If the discarded number is equal to five, and there are no meaningful numbers behind it, the rounding is made on the nearest even number. The last saved figure remains unchanged if it is even, and increases if it is not even.

If there are significant digits for five digits, the rounding is performed according to rule 2.

Applying rule 3 to rounding one number, we do not increase the accuracy of rounding. But with numerous roundings, excessive numbers will occur approximately as often as not enough. Mutual error compensation will provide the greatest accuracy of the result.

The number is obviously exceeding the absolute error (or in the worst case equal to it) is called limit absolute error.

The magnitude of the limit error is not quite defined. For each approximate number, its limiting error (absolute or relative) should be known.

When it is not directly indicated, it is understood that the limit absolute error is half the unit of the last discharge written. So, if an approximate number of 4.78 is given without the indication of the utmost error, it is understood that the limiting absolute error is 0.005. As a result of this agreement, you can always do without the indication of the marginal error number, rounded according to the rules 1-3, i.e., if the approximate number is indicated by the letter α, then

Where Δn is the limit absolute error; A Δ N is a limit relative error.

In addition, when processing results are used rules for finding error Amounts, differences, works and private.

• Rule 1. The limit absolute error of the amount is equal to the sum of the limit absolute errors of individual terms, but with a significant number of the errors of the terms usually occurs mutual compensation of errors, so the true error of the amount only in exceptional cases coincides with the utmost error or is close to it.

• Rule 2. The limit absolute difference in difference is equal to the sum of the limit absolute errors of the reduced or subtracted.

The limit relative error is easy to find by calculating the limit absolute error.

• Rule 3. The limit relative error of the amount (but not difference) lies between the smallest and most of the relative errors of the terms.

If all the components have the same limit relative error, the amount has the same limit relative error. In other words, in this case the accuracy of the amount (in percentage terms) is not inferior to the accuracy of the components.

In contrast, the amount of approximate numbers may be less accurate than the diminished and subtractable. The loss of accuracy is especially large in the case when the diminished and subtractable differ little from each other.

• Rule 4. The limit relative error of the work is approximately equal to the sum of the limit relative errors of the factors: δ \u003d δ 1 + δ 2, or, more precisely, δ \u003d δ 1 + δ 2 + δ 1 δ 2 where δ is the relative error of the work, δ 1 Δ 2 - relative errors in factory.

Notes:

1. If approximate numbers are multiplied with the same number of meaningful numbers, then in the work should be saved as many meaningful numbers. The latter of the saved digits will not be completely reliable.

2. If some factors have more meaningful numbers than others, then to multiplication, the first round should be rounded, while saving so much figures as it has the least accumulator or one more (as a spare), further digits are useless.

3. If it is required that the work of two numbers had a predetermined number is quite reliable, then in each of the factors the number of accurate digits (obtained by measurement or calculation) should be per unit. If the number of factors is more than two and less than ten, then in each of the factors, the number of accurate digits for a complete warranty should be two units larger than the desired number of accurate digits. Almost it is quite enough to take only one excessive figure.

• Rule 5. The limit relative error of the private is approximately equal to the sum of the limit relative errors of the division and divider. The exact value of the limit relative error always exceeds the approximate. The percentage of exceeding is approximately equal to the maximum relative error of the divider.

Example 1.3. Find the limit absolute error of the private 2,81: 0.571.

Decision.The limit relative dividery error is 0.005: 2.81 \u003d 0.2%; divider - 0.005: 0,571 \u003d 0.1%; Private - 0.2% + 0.1% \u003d 0.3%. The limit absolute error of the private approximately will be 2.81: 0.571 · 0.0030 \u003d 0.015

So, in the private 2,81: 0.571 \u003d 4.92, the third meaning figure is not reliable.

Answer.0,015.

Example 1.4. Calculate the relative error of the voltmeter readings included in the diagram (Fig. 1.3), which is obtained, assuming that the voltmeter has infinitely large resistance and does not make distortions into the measured chain. Classify measurement error for this task.

fig. 1.3

Decision.Denote by the readings of the real voltmeter through and, and the voltmeter with infinitely large resistance and ∞. Skewing relative error

notice, that

then we get

Since R and \u003e\u003e R and R\u003e R, the fraction in the denominator of the last equality is much less than one. Therefore, you can use the approximate formula Fair for λ≤1 for any α. Supposed that in this formula α \u003d -1 and λ \u003d RR (R + R) -1 R and -1, we obtain Δ ≈ RR / (R + R) R and.

The greater the resistance of the voltmeter compared to the external resistance of the chain, the smaller the error. But the condition R.<

Answer.Error systematic methodical.

Example 1.5. In the DC circuit (Fig.1.4), appliances are included: A - ammeter type M 330 accuracy class K a \u003d 1.5 with measurement limit I k \u003d 20 A; A 1 is an ammeter of type M 366 accuracy class to a1 \u003d 1.0 with a measurement limit I K1 \u003d 7.5 A. Find the largest possible relative error of measuring current i 2 and possible limits of its actual value if the devices have shown that i \u003d 8 , 0a. and I 1 \u003d 6,0A. Classify measurement.

fig. 1.4.

Decision.Determine the current I 2 according to the instrument readings (excluding their errors): i 2 \u003d i - i 1 \u003d 8.0-6.0 \u003d 2.0 A.

We find the modules of absolute errors of ammeters A and A 1

For a having equality For ammeter

We will find the amount of absolute error modules:

Consequently, the greatest possible and same value expressed in the shares of this value is equal to 1. 10 3 - for one device; 2 · 10 3 - for another instrument. Which of these devices will be the most accurate?

Decision.The accuracy of the device is characterized by a value, inverse errors (the more accurate the device, the smaller the error), i.e. For the first instrument, this will be 1 / (1. 10 3) \u003d 1000, for the second - 1 / (2. 10 3) \u003d 500. Note that 1000\u003e 500. Therefore, the first device is more accurately twice.

You can come to a similar output by checking the correspondence of errors: 2. 10 3/1. 10 3 \u003d 2.

Answer.The first device is two times more accurate.

Example 1.6. Find the sum of approximate measurements of the device. Find the number of faithful signs: 0.0909 + 0.0833 + 0.0769 + 0.0714 + 0.0667 + 0.0625 + 0.0588+ 0.0556 + 0.0526.

Decision.Folding all the results of measurements, we get 0.6187. The maximum largest amount of the amount of 0.00005 · 9 \u003d 0.00045. So, in the last fourth sum of the amount, an error is possible up to 5 units. Therefore, we round the amount to the third mark, i.e. Thousands, we get 0.619 - the result in which all signs are correct.

Answer.0,619. The number of faithful signs is three signs after the comma.

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