In physics and other sciences, it is very common to make measurements of various quantities (for example, length, mass, time, temperature, electrical resistance etc.).

Measurement– the process of finding a value physical quantity using special technical means– measuring instruments.

Measuring instrument is a device that is used to compare a measured quantity with a physical quantity of the same kind, taken as a unit of measurement.

There are direct and indirect measurement methods.

Direct measurement methods – methods in which the values ​​of the quantities being determined are found by direct comparison of the measured object with the unit of measurement (standard). For example, the length of a body measured by a ruler is compared with a unit of length - a meter, the mass of a body measured by scales is compared with a unit of mass - a kilogram, etc. Thus, as a result of direct measurement, the determined value is obtained immediately, directly.

Indirect measurement methods– methods in which the values ​​of the quantities being determined are calculated from the results of direct measurements of other quantities with which they are related by a known functional relationship. For example, determining the circumference from the results of measuring the diameter or determining the volume of a body from the results of measuring its linear dimensions.

Due to the imperfection of measuring instruments, our senses, the influence external influences on the measuring equipment and the object being measured, as well as other factors, all measurements can be made only with a certain degree of accuracy; therefore, the measurement results do not give the true value of the measured value, but only an approximate one. If, for example, body weight is determined with an accuracy of 0.1 mg, this means that the found weight differs from the true body weight by less than 0.1 mg.

Accuracy of measurements – characteristic of measurement quality, reflecting the closeness of measurement results to the true value of the measured quantity.

The smaller the measurement errors, the greater the measurement accuracy. The accuracy of measurements depends on the instruments used in the measurements and on common methods measurements. It is completely useless to strive to go beyond this limit of accuracy when making measurements under these conditions. It is possible to minimize the impact of reasons that reduce the accuracy of measurements, but it is impossible to completely get rid of them, that is, more or less significant errors (errors) are always made during measurements. To increase the accuracy of the final result, any physical dimension must be done not once, but several times under the same experimental conditions.

As a result of the i-th measurement (i – measurement number) of the value “X”, an approximate number X i is obtained, which differs from the true value of Xist by a certain amount ∆X i = |X i – X|, which is an error made or, in other words , error. The true error is not known to us, since we do not know the true value of the measured quantity. The true value of the measured physical quantity lies in the interval

Х i – ∆Х< Х i – ∆Х < Х i + ∆Х

where X i is the value of X obtained during the measurement (that is, the measured value); ∆X – absolute error in determining the value of X.

Absolute mistake (error) of measurement ∆Х is the absolute value of the difference between the true value of the measured quantity Hist and the measurement result X i: ∆Х = |Х source – X i |.

Relative error (error) of measurement δ (characterizing the accuracy of measurement) is numerically equal to the ratio of the absolute measurement error ∆X to the true value of the measured value X source (often expressed as a percentage): δ = (∆X / X source) 100%.

Errors or measurement errors can be divided into three classes: systematic, random and gross (misses).

Systematic they call such an error that remains constant or changes naturally (according to some functional dependence) with repeated measurements of the same quantity. Such errors arise as a result of the design features of the measuring instruments, shortcomings of the adopted measurement method, any omissions of the experimenter, the influence external conditions or a defect in the measurement object itself.

Any measuring instrument contains one or another systematic error, which cannot be eliminated, but the order of which can be taken into account. Systematic errors either increase or decrease the measurement results, that is, these errors are characterized by a constant sign. For example, if during weighing one of the weights has a mass 0.01 g greater than indicated on it, then the found value of body mass will be overestimated by this amount, no matter how many measurements are made. Sometimes systematic errors can be taken into account or eliminated, sometimes this cannot be done. For example, fatal errors include instrument errors, about which we can only say that they do not exceed a certain value.

Random errors are called errors that change their magnitude and sign in an unpredictable way from experiment to experiment. The appearance of random errors is due to many diverse and uncontrollable reasons.

For example, when weighing with scales, these reasons may be air vibrations, settled dust particles, different friction in the left and right suspension of cups, etc. Random errors manifest themselves in the fact that, having made measurements of the same value X under the same experimental conditions, we get several differing values: X1, X2, X3,..., Xi,..., Xn, where Xi is the result of the i-th measurement. It is not possible to establish any pattern between the results, therefore the result of the i -th measurement of X is considered a random variable. Random errors can have a certain impact on a single measurement, but with repeated measurements they obey statistical laws and their influence on the measurement results can be taken into account or significantly reduced.

Mistakes and gross errors– excessively large errors that clearly distort the measurement result. This class of errors is most often caused by incorrect actions of the experimenter (for example, due to inattention, instead of the instrument reading “212”, a completely different number is recorded - “221”). Measurements containing misses and gross errors should be discarded.

Measurements can be carried out in terms of their accuracy using technical and laboratory methods.

When using technical methods, the measurement is carried out once. In this case, they are satisfied with such accuracy that the error does not exceed a certain, predetermined value determined by the error of the measuring equipment used.

With laboratory measurement methods, it is necessary to more accurately indicate the value of the measured quantity than is allowed by its single measurement using a technical method. In this case, several measurements are made and the arithmetic mean of the obtained values ​​is calculated, which is taken as the most reliable (true) value of the measured value. Then the accuracy of the measurement result is assessed (taking into account random errors).

From the possibility of carrying out measurements using two methods, it follows that there are two methods for assessing the accuracy of measurements: technical and laboratory.

Relative error

Errors root mean square T, true A are called absolute errors.

In some cases, the absolute error is not sufficiently indicative, in particular with linear measurements. For example, a line is measured with an error of ±5 cm. For a line length of 1 meter, this accuracy is obviously low, but for a line length of 1 kilometer, the accuracy is certainly higher. Therefore, the measurement accuracy will be more clearly characterized by the ratio of the absolute error to the obtained value of the measured quantity. This ratio is called relative error. The relative error is expressed as a fraction, and the fraction is transformed so that its numerator is equal to one.

The relative error is determined by the corresponding absolute

error. Let X- the obtained value of a certain quantity, then - the mean square relative error of this quantity; - true relative error.

It is advisable to round the denominator of the relative error to two significant figures with zeros.

Example. In the above case, the root mean square relative error of line measurement will be equal to

Marginal error

The marginal error is called highest value random error that may appear under given conditions of equal precision measurements.

Probability theory has proven that random errors in only three cases out of 1000 can exceed the value Zt; 5 mistakes out of 100 can exceed 2t and 32 errors out of 100 can exceed T.

Based on this, in geodetic practice, measurement results containing errors 0>3t, are classified as measurements containing gross errors and are not accepted for processing.

Error values ​​0 = 2 T used as limits when compiling technical requirements for this type of work, i.e., all random measurement errors exceeding these values ​​in magnitude are considered unacceptable. Upon receipt of discrepancies exceeding the value 2t, take measures to improve measurement conditions, and repeat the measurements themselves.

Test questions and exercises:

• 1. List the types of measurements and give their definition.
• 2. List the types of measurement errors and give their definition.
• 3. List the criteria used to assess the accuracy of measurements.
• 4. Find the root mean square error of a number of measurements if the most probable errors are equal to: - 2.3; + 1.6; - 0.2; + 1.9; - 1.1.
• 5. Find the relative error in measuring the line length based on the results: 487.23 m and 486.91 m.

One of the most important issues in numerical analysis is the question of how an error that occurs at a certain location during a calculation propagates further, that is, whether its influence becomes larger or smaller as subsequent operations are performed. An extreme case is the subtraction of two almost equal numbers: even with very small errors in both of these numbers, the relative error of the difference can be very large. This relative error will propagate further during all subsequent arithmetic operations.

One of the sources of computational errors (errors) is the approximate representation of real numbers in a computer, due to the finiteness of the bit grid. Although the initial data is presented in a computer with great accuracy, the accumulation of rounding errors during the calculation process can lead to a significant resulting error, and some algorithms may turn out to be completely unsuitable for real calculation on a computer. You can find out more about the representation of real numbers in a computer.

Propagation of errors

As a first step in considering the issue of error propagation, it is necessary to find expressions for the absolute and relative errors of the result of each of the four arithmetic operations as a function of the quantities involved in the operation and their errors.

Absolute mistake

Addition

There are two approximations and to two quantities and , as well as the corresponding absolute errors and . Then as a result of addition we have

.

The error of the sum, which we denote by , will be equal to

.

Subtraction

In the same way we get

.

Multiplication

When multiplying we have

.

Since the errors are usually much smaller than the quantities themselves, we neglect the product of the errors:

.

The product error will be equal to

.

Division

.

Let's transform this expression to the form

.

The factor in parentheses can be expanded into a series

.

Multiplying and neglecting all terms that contain products of errors or degrees of error higher than the first, we have

.

Hence,

.

It must be clearly understood that the error sign is known only in very rare cases. It is not a fact, for example, that the error increases when adding and decreases when subtracting because in the formula for addition there is a plus, and for subtraction - a minus. If, for example, the errors of two numbers have opposite signs, then the situation will be just the opposite, that is, the error will decrease when adding and increase when subtracting these numbers.

Relative error

Once we have derived the formulas for the propagation of absolute errors in the four arithmetic operations, it is quite easy to derive the corresponding formulas for the relative errors. For addition and subtraction, the formulas were transformed so that they explicitly included the relative error of each original number.

Addition

.

Subtraction

.

Multiplication

.

Division

.

We begin an arithmetic operation with two approximate values ​​and with corresponding errors and . These errors can be of any origin. The quantities and may be experimental results containing errors; they may be the results of a pre-computation according to some infinite process and may therefore contain constraint errors; they may be the results of previous arithmetic operations and may contain rounding errors. Naturally, they can also contain all three types of errors in various combinations.

The above formulas give an expression for the error of the result of each of the four arithmetic operations as a function of ; rounding error in this arithmetic operation wherein not taken into account. If in the future it becomes necessary to calculate how the error of this result is propagated in subsequent arithmetic operations, then it is necessary to calculate the error of the result calculated using one of the four formulas add rounding error separately.

Computational process graphs

Now consider a convenient way to calculate the propagation of error in any arithmetic calculation. To this end, we will depict the sequence of operations in a calculation using graph and we will write coefficients near the arrows of the graph that will allow us to relatively easily determine the general error of the final result. This method is also convenient because it allows you to easily determine the contribution of any error that arises during the calculation process to the overall error.

Fig.1. Computational process graph

On Fig.1 a graph of a computational process is depicted. The graph should be read from bottom to top, following the arrows. First, operations located at some horizontal level are performed, after that operations located at a higher level high level, etc. From Fig. 1, for example, it is clear that x And y first added and then multiplied by z. The graph shown in Fig.1, is only an image of the computational process itself. To calculate the total error of the result, it is necessary to supplement this graph with coefficients, which are written next to the arrows according to the following rules.

Addition

Let two arrows that enter the addition circle come out of two circles with values ​​and . These values ​​can be either initial or the results of previous calculations. Then the arrow leading from to the + sign in the circle receives the coefficient, while the arrow leading from to the + sign in the circle receives the coefficient.

Subtraction

If the operation is performed, then the corresponding arrows receive coefficients and .

Multiplication

Both arrows included in the multiplication circle receive a coefficient of +1.

Division

If division is performed, then the arrow from to the slash in the circle receives a coefficient of +1, and the arrow from to the slash in the circle receives a coefficient of −1.

The meaning of all these coefficients is as follows: the relative error of the result of any operation (circle) is included in the result of the next operation, multiplied by the coefficients of the arrow connecting these two operations.

Examples

Fig.2. Computational process graph for addition, and

Let us now apply the graph technique to examples and illustrate what error propagation means in practical calculations.

Example 1

Consider the problem of adding four positive numbers:

, .

The graph of this process is shown in Fig.2. Let us assume that all initial quantities are specified accurately and have no errors, and let , and be the relative rounding errors after each subsequent addition operation. Successively applying the rule to calculate the total error of the final result leads to the formula

.

Reducing the sum in the first term and multiplying the entire expression by , we get

.

Considering that the rounding error is (in in this case it is assumed that a real number is represented in a computer in the form decimal With t in significant figures), we finally have

Absolute and relative errors

Errors such as mean (J), root mean square ( m), probable ( r), true (D) and limit (D etc) are absolute errors. They are always expressed in units of the quantity being measured, i.e. have the same dimension as the measured value.
Cases often arise when objects of different sizes are measured with the same absolute errors. For example, the root mean square error of measuring lines of length: l 1 = 100 m and l 2 = 1000 m, amounted to m= 5 cm. The question arises: which line was measured more accurately? To avoid uncertainty, the accuracy of measurements of a number of quantities is assessed as the ratio of the absolute error to the value of the measured quantity. The resulting ratio is called the relative error, which is usually expressed as a fraction with a numerator equal to one.
The name of the absolute error determines the name of the corresponding relative measurement error [1].

Let x- the result of measuring a certain quantity. Then
- mean square relative error;

Average relative error;

Probable relative error;

True relative error;

Limit relative error.

Denominator N relative error must be rounded to two significant figures with zeros:

m x= 0.3 m; x= 152.0 m;

m x= 0.25 m; x= 643.00 m; .

m x= 0.033 m; x= 795,000 m;

As can be seen from the example, the larger the denominator of the fraction, the more accurate the measurements are.

Rounding errors

When processing measurement results, an important role is played by rounding errors, which in their properties can be classified as random variables [2]:

1) the maximum error of one rounding is 0.5 units of the retained sign;

2) large and smaller rounding errors in absolute value are equally possible;
3) positive and negative rounding errors are equally possible;
4) the mathematical expectation of rounding errors is zero.
These properties make it possible to attribute rounding errors to random variables that have uniform distribution. Continuous random variable X has a uniform distribution over the interval [ a, b], if on this interval the distribution density random variable is constant, and outside it is equal to zero (Fig. 2), i.e.

j (x) . (1.32)

Distribution function F(x)

a b x(1.33)

Rice. 2 Expected value

(1.34)

Dispersion
(1.35)

Standard deviation

(1.36)

For rounding errors

Measurement error- assessment of the deviation of the measured value of a quantity from its true value. Measurement error is a characteristic (measure) of measurement accuracy.

Since it is impossible to determine with absolute accuracy the true value of any quantity, it is impossible to indicate the amount of deviation of the measured value from the true one. (This deviation is usually called measurement error. In a number of sources, for example, in the Great Soviet encyclopedia, terms measurement error And measurement error are used as synonyms, but according to RMG 29-99 the term measurement error Not recommended for use as less successful). It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. In practice, instead of the true value, they use actual value of quantity x d, that is, the value of a physical quantity obtained experimentally and so close to the true value that it can be used instead of it in the given measurement task. This value is usually calculated as the average value obtained from statistical processing of the results of a series of measurements. This obtained value is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, the measurement error is indicated along with the result obtained. For example, record T=2.8±0.1 c. means that the true value of the quantity T lies in the range from 2.7 s. before 2.9 s. with some specified probability

In 2004, a new document was adopted at the international level, dictating the conditions for carrying out measurements and establishing new rules for comparing state standards. The concept of “error” has become obsolete; instead, the concept of “measurement uncertainty” was introduced, however GOST R 50.2.038-2004 allows the use of the term error for documents used in Russia.

The following types of errors are distinguished:

· absolute error;

· relative error;

· reduced error;

· basic error;

· additional error;

· systematic error;

· random error;

· instrumental error;

· methodical error;

· personal error;

· static error;

· dynamic error.

Measurement errors are classified according to the following criteria.

· According to the method of mathematical expression, errors are divided into absolute errors and relative errors.

· According to the interaction of changes in time and the input value, errors are divided into static errors and dynamic errors.

· Based on the nature of their occurrence, errors are divided into systematic errors and random errors.

· According to the nature of the dependence of the error on the influencing quantities, errors are divided into basic and additional.

· Based on the nature of the error’s dependence on the input value, errors are divided into additive and multiplicative.

Absolute error– this is a value calculated as the difference between the value of a quantity obtained during the measurement process and the real (actual) value of this quantity. The absolute error is calculated using the following formula:

AQ n =Q n /Q 0 , where AQ n is the absolute error; Qn– the value of a certain quantity obtained during the measurement process; Q 0– the value of the same quantity taken as the basis of comparison (real value).

Absolute error of the measure– this is a value calculated as the difference between the number, which is the nominal value of the measure, and the real (real) value of the quantity reproduced by the measure.

Relative error is a number that reflects the degree of measurement accuracy. The relative error is calculated using the following formula:

Where ∆Q is the absolute error; Q 0– real (real) value of the measured quantity. The relative error is expressed as a percentage.

Reduced error is a value calculated as the ratio of the absolute error value to the normalizing value.

The standard value is determined as follows:

· for measuring instruments for which a nominal value is approved, this nominal value is taken as the standard value;

· for measuring instruments in which the zero value is located at the edge of the measurement scale or outside the scale, the normalizing value is taken equal to the final value from the measurement range. The exception is measuring instruments with a significantly uneven measurement scale;

· for measuring instruments whose zero mark is located inside the measurement range, the normalizing value is accepted equal to the amount finite numerical values ​​of the measurement range;

· for measuring instruments (measuring instruments) in which the scale is uneven, the normalizing value is taken equal to the whole length of the measurement scale or the length of that part of it that corresponds to the measurement range. The absolute error is then expressed in units of length.

Measurement error includes instrumental error, method error, and counting error. Moreover, the counting error arises due to the inaccuracy in determining the division fractions of the measurement scale.

Instrumental error– this is an error that arises due to errors made during the manufacturing process of functional parts of measuring instruments.

Methodological error is the error arising from the following reasons:

· inaccuracy of model construction physical process, on which the measuring instrument is based;

· incorrect use of measuring instruments.

Subjective error– this is an error arising due to the low degree of qualification of the operator of the measuring instrument, as well as due to the error visual organs human, i.e. the cause of subjective error is the human factor.

Errors in the interaction of changes over time and the input quantity are divided into static and dynamic errors.

Static error– this is an error that arises in the process of measuring a constant (not changing over time) quantity.

Dynamic error is an error, the numerical value of which is calculated as the difference between the error that occurs when measuring a non-constant (time-variable) quantity and the static error (the error in the value of the measured quantity at a certain point in time).

According to the nature of the dependence of the error on the influencing quantities, errors are divided into basic and additional.

Basic error– this is the error obtained under normal operating conditions of the measuring instrument (at normal values ​​of the influencing quantities).

Additional error– this is an error that arises in conditions of discrepancy between the values ​​of the influencing quantities normal values, or if the influencing quantity exceeds the boundaries of the normal range.

Normal conditions – these are conditions in which all values ​​of influencing quantities are normal or do not go beyond the boundaries of the normal range.

Working conditions– these are conditions in which the change in influencing quantities has a wider range (the influencing values ​​do not go beyond the boundaries of the working range of values).

Working range of influencing quantities– this is the range of values ​​in which the values ​​of the additional error are normalized.

Based on the nature of the error’s dependence on the input value, errors are divided into additive and multiplicative.

Additive error– this is an error that arises due to the summation of numerical values ​​and does not depend on the value of the measured quantity taken modulo (absolute).

Multiplicative bias is an error that changes with changes in the values ​​of the quantity being measured.

It should be noted that the value of the absolute additive error is not related to the value of the measured quantity and the sensitivity of the measuring instrument. Absolute additive errors are constant over the entire measurement range.

The value of the absolute additive error determines the minimum value of the quantity that can be measured by the measuring instrument.

The values ​​of multiplicative errors change in proportion to changes in the values ​​of the measured quantity. The values ​​of multiplicative errors are also proportional to the sensitivity of the measuring instrument. The multiplicative error arises due to the influence of influencing quantities on the parametric characteristics of the elements of the device.

Errors that may arise during the measurement process are classified according to the nature of their occurrence. Highlight:

· systematic errors;

· random errors.

Gross errors and errors may also occur during the measurement process.

Systematic error- This component the entire error of the measurement result, which does not change or changes naturally with repeated measurements of the same quantity. Usually they try to eliminate systematic error possible ways(for example, by using measurement methods that reduce the likelihood of its occurrence), if a systematic error cannot be excluded, then it is calculated before the start of measurements and appropriate corrections are made to the measurement result. In the process of normalizing the systematic error, the boundaries of its permissible values ​​are determined. Systematic error determines the accuracy of measurements of measuring instruments (metrological property). Systematic errors in some cases can be determined experimentally. The measurement result can then be clarified by introducing a correction.

Methods for eliminating systematic errors are divided into four types:

· elimination of the causes and sources of errors before the start of measurements;

· elimination of errors in the process of already started measurement by means of substitution, compensation of errors by sign, opposition, symmetrical observations;

· correction of measurement results by making corrections (elimination of errors by calculations);

· determination of the limits of systematic error in case it cannot be eliminated.

Elimination of causes and sources of errors before starting measurements. This method is the most optimal option, since its use simplifies the further course of measurements (there is no need to eliminate errors in the process of already started measurement or to make corrections to the result obtained).

To eliminate systematic errors in the process of already started measurements, various ways

Method of introducing amendments is based on knowledge of the systematic error and the current patterns of its change. When using this method, corrections are made to the measurement result obtained with systematic errors, equal in magnitude to these errors, but opposite in sign.

Substitution method consists in the fact that the measured quantity is replaced by a measure placed in the same conditions in which the object of measurement was located. The replacement method is used when measuring the following electrical parameters: resistance, capacitance and inductance.

Sign error compensation method consists in the fact that measurements are performed twice in such a way that an error of unknown magnitude is included in the measurement results with the opposite sign.

Method of opposition similar to the sign compensation method. This method consists of taking measurements twice so that the source of error in the first measurement has an opposite effect on the result of the second measurement.

Random error- this is a component of the error of the measurement result, changing randomly, irregularly when performing repeated measurements of the same quantity. The occurrence of a random error cannot be foreseen or predicted. Random error cannot be completely eliminated; it always distorts the final measurement results to some extent. But you can make the measurement result more accurate by taking repeated measurements. The cause of a random error may be, for example, a random change external factors, affecting the measurement process. A random error when carrying out repeated measurements with a sufficiently high degree of accuracy leads to scattering of the results.

Mistakes and gross errors– these are errors that far exceed the systematic and random errors expected under the given measurement conditions. Errors and gross errors can appear due to gross errors during the measurement process, technical malfunction of the measuring instrument, or unexpected changes in external conditions.