abstract

Absolute and relative error

Introduction

Absolute error - is an estimate of the absolute measurement error. It is calculated in 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 BUT- the exact value of a certain quantity, hereinafter called the exact number a.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.

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Measurement errors of physical quantities

1.Introduction(measurements and measurement errors)

2. Random and systematic errors

3. Absolute and relative errors

4. Errors of measuring instruments

5. Accuracy class of electrical measuring instruments

6.Reading error

7. Total absolute error of direct measurements

8. Recording the final result of direct measurement

9. Errors of indirect measurements

10.Example

1. Introduction(measurements and measurement errors)

Physics as a science was born more than 300 years ago, when Galileo essentially created the scientific study of physical phenomena: physical laws are established and verified experimentally by accumulating and comparing experimental data represented by a set of numbers, laws are formulated in the language of mathematics, i.e. with the help of formulas linking numerical values ​​of physical quantities by functional dependence. Therefore, physics is an experimental science, physics is a quantitative science.

Let's get acquainted with some characteristic features of any measurements.

Measurement is finding the numerical value of a physical quantity empirically using measuring instruments (rulers, voltmeters, watches, etc.).

Measurements can be direct and indirect.

Direct measurement is the determination of the numerical value of a physical quantity directly by measuring instruments. For example, length - with a ruler, atmospheric pressure - with a barometer.

Indirect measurement is the determination of the numerical value of a physical quantity according to a formula that relates the desired value with other quantities determined by direct measurements. For example, the resistance of a conductor is determined by the formula R=U/I, where U and I are measured by electrical measuring instruments.

Consider an example of measurement.

Measure the length of the bar with a ruler (division 1 mm). It can only be stated that the length of the bar is between 22 and 23 mm. The width of the “unknown” interval is 1 mm, that is, it is equal to the division value. Replacing the ruler with a more sensitive instrument, such as a caliper, will reduce this interval, resulting in an increase in measurement accuracy. In our example, the measurement accuracy does not exceed 1 mm.

Therefore, measurements can never be absolutely accurate. The result of any measurement is approximate. Uncertainty in measurement is characterized by an error - a deviation of the measured value of a physical quantity from its true value.

We list some of the reasons leading to the appearance of errors.

1. Limited accuracy in the manufacture of measuring instruments.

2. Influence on measurement of external conditions (temperature change, voltage fluctuation...).

3. Actions of the experimenter (delay in turning on the stopwatch, different position of the eye...).

4. Approximate nature of the laws used to find the measured quantities.

The listed reasons for the appearance of errors cannot be eliminated, although they can be minimized. To establish the reliability of the conclusions obtained as a result of scientific research, there are methods for assessing these errors.

2. Random and systematic errors

Errors arising from measurements are divided into systematic and random.

Systematic errors are errors corresponding to the deviation of the measured value from the true value of a physical quantity, always in one direction (increase or decrease). With repeated measurements, the error remains the same.

Causes of systematic errors:

1) non-compliance of measuring instruments with the standard;

2) incorrect installation of measuring instruments (tilt, unbalance);

3) non-coincidence of the initial indicators of devices with zero and ignoring the corrections that arise in connection with this;

4) discrepancy between the measured object and the assumption about its properties (presence of voids, etc.).

Random errors are errors that change their numerical value in an unpredictable way. Such errors are caused by a large number of uncontrollable causes that affect the measurement process (irregularities on the surface of the object, wind blowing, power surges, etc.). The influence of random errors can be reduced by repeated repetition of the experiment.

3. Absolute and relative errors

For a quantitative assessment of the quality of measurements, the concepts of absolute and relative measurement errors are introduced.

As already mentioned, any measurement gives only an approximate value of a physical quantity, but you can specify an interval that contains its true value:

A pr - D A< А ист < А пр + D А

D value A is called the absolute error in measuring the quantity A. The absolute error is expressed in units of the measured quantity. The absolute error is equal to the module of the maximum possible deviation of the value of a physical quantity from the measured value. A pr - the value of a physical quantity obtained experimentally, if the measurement was carried out repeatedly, then the arithmetic mean of these measurements.

But to assess the quality of the measurement, it is necessary to determine the relative error e. e \u003d D A / A pr or e \u003d (D A / A pr) * 100%.

If during the measurement a relative error of more than 10% is obtained, then they say that only an estimate of the measured value has been made. In the laboratories of a physical workshop, it is recommended to carry out measurements with a relative error of up to 10%. In scientific laboratories, some precise measurements (such as determining the wavelength of light) are performed with an accuracy of millionths of a percent.

4. Errors of measuring instruments

These errors are also called instrumental or instrumental. They are due to the design of the measuring device, the accuracy of its manufacture and calibration. Usually they are satisfied with the permissible instrumental errors reported by the manufacturer in the passport for this device. These permissible errors are regulated by GOSTs. This also applies to standards. Usually, the absolute instrumental error is denoted by D and A.

If there is no information about the permissible error (for example, for a ruler), then half the division price can be taken as this error.

When weighing, the absolute instrumental error is the sum of the instrumental errors of the scales and weights. The table shows the permissible errors most often

measuring instruments encountered in the school experiment.

Measuring	Measurement limit	Value of division	Allowable error
student's ruler
demonstration ruler
measuring tape
beaker
weights 10.20, 50 mg
weights 100.200 mg
weights 500 mg
calipers
micrometer
dynamometer
educational scales
Stopwatch			1s for 30 min
aneroid barometer	720-780 mmHg	1 mmHg	3 mmHg
laboratory thermometer	0-100 degrees C
school ammeter
voltmeter school

5. Accuracy class of electrical measuring instruments

According to the permissible error values, pointer electrical measuring instruments are divided into accuracy classes, which are indicated on the instrument scales by the numbers 0.1; 0.2; 0.5; 1.0; 1.5; 2.5; 4.0. Accuracy class g pr instrument shows how many percent is the absolute error of the entire scale of the instrument.

g pr \u003d (D and A / A max) * 100% .

For example, the absolute instrumental error of a class 2.5 instrument is 2.5% of its scale.

If the accuracy class of the device and its scale are known, then the absolute instrumental measurement error can be determined

D and A \u003d ( g pr * A max) / 100.

To improve the accuracy of measurement with a pointer electrical measuring device, it is necessary to choose a device with such a scale that during the measurement process they are located in the second half of the scale of the device.

6. Reading error

The reading error is obtained from insufficiently accurate reading of the readings of measuring instruments.

In most cases, the absolute reading error is taken equal to half the division value. Exceptions are measurements with analog clocks (hands move in jerks).

The absolute error of reading is usually denoted D oA

7. Total absolute error of direct measurements

When performing direct measurements of the physical quantity A, it is necessary to evaluate the following errors: D uA, D oA and D sA (random). Of course, other sources of errors associated with incorrect installation of instruments, misalignment of the initial position of the instrument pointer with 0, etc., should be excluded.

The total absolute error of direct measurement must include all three types of errors.

If the random error is small compared to the smallest value that can be measured by this measuring instrument (compared to the division value), then it can be neglected and then one measurement is sufficient to determine the value of the physical quantity. Otherwise, the probability theory recommends finding the measurement result as the arithmetic mean of the results of the entire series of multiple measurements, the result error is calculated by the method of mathematical statistics. Knowledge of these methods goes beyond the school curriculum.

8. Recording the final result of the direct measurement

The final result of the measurement of the physical quantity A should be written in this form;

A=A pr + D A, e \u003d (D A / A pr) * 100%.

A pr - the value of a physical quantity obtained experimentally, if the measurement was carried out repeatedly, then the arithmetic mean of these measurements. D A is the total absolute error of direct measurement.

Absolute error is usually expressed as one significant figure.

Example: L=(7.9 + 0.1) mm, e=13%.

9. Errors of indirect measurements

When processing the results of indirect measurements of a physical quantity that is functionally related to the physical quantities A, B and C, which are measured in a direct way, the relative error of the indirect measurement is first determined e= D X / X pr, using the formulas given in the table (without evidence).

The absolute error is determined by the formula D X \u003d X pr * e,

where e expressed as a decimal, not as a percentage.

The final result is recorded in the same way as in the case of direct measurements.

Function type	Formula
X=A+B+C
X=A-B
X=A*B*C
X=A n
X=A/B

Example: Let us calculate the error in measuring the friction coefficient using a dynamometer. The experience is that the bar is uniformly pulled along a horizontal surface and the applied force is measured: it is equal to the force of sliding friction.

Using a dynamometer, we weigh a bar with weights: 1.8 N. F tr \u003d 0.6 N

μ = 0.33. The instrumental error of the dynamometer (find from the table) is Δ and = 0.05N, Reading error (half of the scale division)

Δ o = 0.05 N. The absolute error in measuring the weight and friction force is 0.1 N.

Relative measurement error (5th line in the table)

, therefore, the absolute error of indirect measurement of μ is 0.22*0.33=0.074

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 in solving theoretical problems of 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;

In relation to the unit of physical quantity - the error 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 that retain their value for a long time (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(briefly - periodic error) - an error, the value of which is a function of time or a function of the movement of the pointer of the measuring device (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 instrument parts, excessive friction in the instrument mechanism, inaccurate strokes on the scale, discrepancy between the actual and nominal values ​​​​of the measure, 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, so 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

(1.5)

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.

(1.6)

For example, as a result of measurements, a series of electrical resistance values ​​\u200b\u200bare obtained (which are corrected for the effects of systematic errors): R 1 \u003d 15.5 Ohm, R 2 \u003d 15.6 Ohm, R 3 \u003d 15.4 Ohm, R 4 \u003d 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, it accidentally turned out to be zero), the random error of the measurement result is necessarily 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 values ​​​​of the quantity 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

(1.8)

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

(1.11)

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 = lim S (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

(1.14)

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

(1.15)

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

(1.16)

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

(1.17)

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

(1.18)

(1.19)

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

= 86 mm 2 at n = 10,

= 3.1 mm

= 0.7 mm or S = ±0.7 mm

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, the term "measurement reliability" has become widespread. 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 standard error of measurements - Su including the standard error S (determined by statistical methods) and the standard error u (determined by non-statistical methods), i.e.

(1.20)

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 adhere 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:

(1.21)

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).

The absolute calculation error is found by the formula:

The modulo sign shows that we don't care which value is larger and which is smaller. Important, how far the approximate result deviated from the exact value in one direction or another.

The relative calculation error is found by the formula:
, or, the same:

The relative error shows by what percentage the approximate result deviated from the exact value. There is a version of the formula without multiplying by 100%, but in practice I almost always see the above version with percentages.

After a short background, we return to our problem, in which we calculated the approximate value of the function using a differential.

Let's calculate the exact value of the function using a microcalculator:
, strictly speaking, the value is still approximate, but we will consider it exact. Such tasks do occur.

Calculate the absolute error:

Let's calculate the relative error:
, thousandths of a percent are obtained, so the differential provided just a great approximation.

Answer: , absolute calculation error , relative calculation error

The following example is for a standalone solution:

Example 4

at point . Calculate a more accurate value of the function at a given point, evaluate the absolute and relative calculation errors.

A rough example of finishing work and an answer at the end of the lesson.

Many have noticed that in all the examples considered, roots appear. This is not accidental; in most cases, in the problem under consideration, functions with roots are indeed proposed.

But for the suffering readers, I dug up a small example with the arcsine:

Example 5

Calculate approximately using the differential the value of the function at the point

This short but informative example is also for independent decision. And I rested a little in order to consider a special task with renewed vigor:

Example 6

Calculate approximately using the differential, round the result to two decimal places.

Solution: What's new in the task? By condition, it is required to round the result to two decimal places. But that's not the point, the school rounding problem, I think, is not difficult for you. The matter is that at us the tangent with argument which is expressed in degrees is given. What to do when you are asked to solve a trigonometric function with degrees? For example , etc.

The solution algorithm is fundamentally preserved, that is, it is necessary, as in the previous examples, to apply the formula

Write down the obvious function

The value must be represented as . Serious help will table of values ​​of trigonometric functions . By the way, if you haven't printed it, I recommend doing so, since you will have to look there throughout the entire course of studying higher mathematics.

Analyzing the table, we notice a “good” value of the tangent, which is close to 47 degrees:

In this way:

After preliminary analysis degrees must be converted to radians. Yes, and only so!

In this example, directly from the trigonometric table, you can find out that. The formula for converting degrees to radians is: (formulas can be found in the same table).

Further template:

In this way: (in calculations we use the value ). The result, as required by the condition, is rounded to two decimal places.

Answer:

Example 7

Calculate approximately using the differential, round the result to three decimal places.

This is a do-it-yourself example. Full solution and answer at the end of the lesson.

As you can see, nothing complicated, we translate the degrees into radians and adhere to the usual solution algorithm.

Approximate Calculations Using the Total Differential of a Function of Two Variables

Everything will be very, very similar, so if you came to this page with this particular task, then first I recommend looking at at least a couple of examples of the previous paragraph.

To study a paragraph, you need to be able to find second order partial derivatives , where without them. In the above lesson, I denoted the function of two variables with the letter . With regard to the task under consideration, it is more convenient to use the equivalent notation .

As in the case of a function of one variable, the condition of the problem can be formulated in different ways, and I will try to consider all the formulations encountered.

Example 8

Solution: No matter how the condition is written, in the solution itself, to designate the function, I repeat, it is better to use not the letter “Z”, but .

And here is the working formula:

Before us is actually the older sister of the formula of the previous paragraph. The variable just got bigger. What can I say, myself the solution algorithm will be fundamentally the same!

By condition, it is required to find the approximate value of the function at the point .

Let's represent the number 3.04 as . The gingerbread man asks to be eaten:
,

Let's represent the number 3.95 as . The turn has come to the second half of Kolobok:
,

And do not look at all sorts of fox tricks, there is a Gingerbread Man - you have to eat it.

Let's calculate the value of the function at the point :

The differential of a function at a point is found by the formula:

From the formula it follows that you need to find partial derivatives of the first order and calculate their values ​​at the point .

Let's calculate the partial derivatives of the first order at the point :

Total differential at point :

Thus, according to the formula, the approximate value of the function at the point :

Let's calculate the exact value of the function at the point :

This value is absolutely correct.

Errors are calculated using standard formulas, which have already been discussed in this article.

Absolute error:

Relative error:

Answer: , absolute error: , relative error:

Example 9

Calculate the approximate value of a function at a point using a full differential, evaluate the absolute and relative error.

This is a do-it-yourself example. Whoever dwells in more detail on this example will pay attention to the fact that the calculation errors turned out to be very, very noticeable. This happened for the following reason: in the proposed problem, the increments of the arguments are large enough: .

The general pattern is a - the greater these increments in absolute value, the lower the accuracy of calculations. So, for example, for a similar point, the increments will be small: , and the accuracy of approximate calculations will be very high.

This feature is also valid for the case of a function of one variable (the first part of the lesson).

Example 10

Solution: Let us calculate this expression approximately using the total differential of a function of two variables:

The difference from Examples 8-9 is that we first need to compose a function of two variables: . How the function is composed, I think, is intuitively clear to everyone.

The value 4.9973 is close to "five", therefore: , .
The value of 0.9919 is close to "one", therefore, we assume: , .

Let's calculate the value of the function at the point :

We find the differential at a point by the formula:

To do this, we calculate the partial derivatives of the first order at the point .

The derivatives here are not the simplest, and you should be careful:

;

.

Total differential at point :

Thus, the approximate value of this expression:

Let's calculate a more accurate value using a microcalculator: 2.998899527

Let's find the relative calculation error:

Answer: ,

Just an illustration of the above, in the considered problem, the increments of the arguments are very small, and the error turned out to be fantastically scanty.

Example 11

Using the total differential of a function of two variables, calculate approximately the value of this expression. Calculate the same expression using a microcalculator. Estimate in percent the relative error of calculations.

This is a do-it-yourself example. An approximate sample of finishing at the end of the lesson.

As already noted, the most common guest in this type of task is some kind of roots. But from time to time there are other functions. And a final simple example for relaxation:

Example 12

Using the total differential of a function of two variables, calculate approximately the value of the function if

The solution is closer to the bottom of the page. Once again, pay attention to the wording of the tasks of the lesson, in different examples in practice the wording may be different, but this does not fundamentally change the essence and algorithm of the solution.

To be honest, I got a little tired, because the material was boring. It was not pedagogical to say at the beginning of the article, but now it is already possible =) Indeed, the problems of computational mathematics are usually not very difficult, not very interesting, the most important thing, perhaps, is not to make a mistake in ordinary calculations.

May the keys of your calculator not be erased!

Solutions and answers:

Example 2:

Solution: We use the formula:
In this case: , ,

In this way:

Answer:

Example 4:

Solution: We use the formula:
In this case: , ,

In this way:

Let's calculate a more accurate value of the function using a microcalculator:

Absolute error:

Relative error:

Answer: , absolute calculation error , relative calculation error

Example 5:

Solution: We use the formula:

In this case: , ,

In this way:

Answer:

Example 7:

Solution: We use the formula:
In this case: , ,

In the process of measuring something, it must be taken into account that the result obtained is not yet final. To more accurately calculate the desired value, it is necessary to take into account the error. Calculating it is quite simple.

How to find the error - calculation

Types of errors:

• relative;
• absolute.

What you need to calculate:

• calculator;
• results of several measurements of the same quantity.

How to find an error - a sequence of actions

• Measure the value 3-5 times.
• Add up all the results and divide the resulting number by their number. This number is a real value.
• Calculate the absolute error by subtracting the value obtained in the previous step from the measurement results. Formula: ∆X = Hisl - Hist. During the calculations, you can get both positive and negative values. In either case, the modulus of the result is taken. If it is necessary to know the absolute error of the sum of two quantities, then the calculations are carried out according to the following formula: ∆(X + Y) = ∆X + ∆Y. It also works when it is necessary to calculate the error of the difference between two quantities: ∆(X-Y) = ∆X+∆Y.
• Find out the relative error for each of the measurements. In this case, you need to divide the obtained absolute error by the actual value. Then multiply the quotient by 100%. ε(x)=Δx/x0*100%. The value may or may not be converted to a percentage.
• To get a more accurate value of the error, it is necessary to find the standard deviation. It is looked for quite simply: calculate the squares of all values ​​​​of the absolute error, and then find their sum. The result obtained must be divided by the number (N-1), in which N is the number of all measurements. The last step is to extract the root from the result. After such calculations, the standard deviation will be obtained, which usually characterizes the measurement error.
• To find the limiting absolute error, it is necessary to find the smallest number, which in its value is equal to or exceeds the value of the absolute error.
• The limiting relative error is searched for by the same method, only it is necessary to find a number that is greater than or equal to the value of the relative error.

Measurement errors arise for various reasons and affect the accuracy of the obtained value. Knowing what the error is equal to, you can find out a more accurate value of the measurement.