What is the absolute error of the measured quantity. Absolute and relative errors

Measurement errors are classified according to the following types:

Absolute and relative.

Positive and negative.

Constant and proportional.

Rough, random and systematic.

Absolute mistake single measurement result (A y) is defined as the difference of the following values:

A y = y i- y ist. » y i -` y.

Relative error single measurement result (V y) is calculated as the ratio of the following quantities:

From this formula it follows that the magnitude of the relative error depends not only on the magnitude of the absolute error, but also on the value of the measured quantity. If the measured value remains unchanged ( y) the relative measurement error can be reduced only by reducing the absolute error (A y). If the absolute measurement error is constant, the technique of increasing the value of the measured quantity can be used to reduce the relative measurement error.

Example. Let’s assume that a store’s commercial scales have a constant absolute error in measuring mass: A m = 10 g. If you weigh 100 g of candy (m 1) on such a scale, then the relative error in measuring the mass of candy will be:

When weighing 500 g of sweets (m2) on the same scales, the relative error will be five times less:

Thus, if you weigh 100 g of sweets five times, then due to an error in measuring the mass, you will not receive a total of 50 g of product out of 500 g. When weighing a larger mass (500 g) once, you will lose only 10 g of candy, i.e. five times less.

Considering the above, it can be noted that first of all it is necessary to strive to reduce relative measurement errors. Absolute and relative errors can be calculated only after determining the average arithmetic value measurement result.

The sign of the error (positive or negative) is determined by the difference between the single and the actual measurement result:

y i -` y > 0 (error is positive);

y i -` y < 0 (error is negative).

If the absolute measurement error does not depend on the value of the measured quantity, then such an error is called constant. Otherwise the error will be proportional. The nature of the measurement error (constant or proportional) is determined after special studies.

Gross mistake measurement (miss) is a measurement result that is significantly different from others, which usually occurs when the measurement technique is violated. The presence of gross measurement errors in the sample is established only by methods mathematical statistics(for n>2). Get to know the methods for detecting gross errors yourself in.

The division of errors into random and systematic is quite arbitrary.

TO random errors include errors that do not have constant value and a sign. Such errors occur due to the following factors: unknown to the researcher; known but unregulated; constantly changing.

Random errors can only be assessed after measurements have been taken.

A quantitative estimate of the magnitude of the random measurement error can be following parameters: and etc.

Random measurement errors cannot be eliminated, they can only be reduced. One of the main ways to reduce the magnitude of a random measurement error is to increase the number of single measurements (increase the value of n). This is explained by the fact that the magnitude of random errors is inversely proportional to the value of n, for example:

Systematic errors- these are errors with unchanged magnitude and sign or varying according to a known law. These errors are caused by constant factors. Systematic errors can be quantified, reduced, and even eliminated.

Systematic errors are classified into errors of types I, II and III.

Towards systematic Type I errors include errors known origin, which can be estimated by calculation before the measurement. These errors can be eliminated by introducing them into the measurement result in the form of corrections. An example of an error of this type is an error in the titrimetric determination of the volumetric concentration of a solution if the titrant was prepared at one temperature and the concentration was measured at another. Knowing the dependence of the titrant density on temperature, it is possible to calculate, before the measurement, the change in the volume concentration of the titrant associated with a change in its temperature, and this difference can be taken into account as a correction as a result of the measurement.

Systematic type II errors- these are errors of known origin that can only be assessed during an experiment or as a result of special research. This type of errors includes instrumental (instrumental), reactive, reference, and other errors. Get to know the features of such errors yourself in .

Any device, when used in a measurement procedure, introduces its own instrument errors into the measurement result. Moreover, some of these errors are random, and the other part are systematic. Random instrument errors are not assessed separately; they are assessed in totality with all other random measurement errors.

Each instance of any device has its own personal systematic error. In order to evaluate this error, it is necessary to conduct special studies.

The most reliable way to assess type II instrument systematic error is to verify the operation of instruments against standards. For measuring glassware (pipettes, burettes, cylinders, etc.), a special procedure is carried out - calibration.

In practice, what is most often required is not to estimate, but to reduce or eliminate type II systematic error. The most common methods for reducing systematic errors are relativization and randomization methods.Discover these methods for yourself at .

TO mistakes Type III include errors of unknown origin. These errors can be detected only after eliminating all systematic errors of types I and II.

TO other errors let us include all other types of errors not discussed above (permissible, possible marginal errors and etc.). The concept of possible limiting errors is used in cases of using measuring instruments and assumes the maximum possible value of the instrumental measurement error (the actual value of the error may be less than the value of the possible limiting error).

When using measuring instruments, you can calculate the possible absolute limit (P` y, etc.) or relative (E` y, etc.) measurement errors. So, for example, the possible maximum absolute measurement error is found as the sum of the possible maximum random (x ` y, random, etc.) and non-excluded systematic (d` y, etc.) errors:

P` y,ex.= x ` y, random, etc. + d` y, etc.

For small samples (n £ 20) unknown population, subordinate normal law distributions, random possible maximum measurement errors can be estimated as follows:

x` y, random, etc. = D` y=S` y½t P, n ½,
where t P,n is the quantile of the Student’s distribution (criterion) for probability P and sample size n. The absolute possible maximum measurement error in this case will be equal to:

P` y,ex.= S ` y½t P, n ½+ d` y, etc.

If the measurement results do not obey the normal distribution law, then the errors are assessed using other formulas.

Determination of d` value y,etc. depends on whether the measuring instrument has an accuracy class. If the measuring instrument does not have an accuracy class, then for the value d ` y,etc. can be accepted minimum scale division price measuring . For a measuring instrument with a known accuracy class for the value d ` y, for example, you can take the absolute permissible systematic error of the measuring instrument (d y, additional):

d` y,etc." .

Value d y, add. calculated based on the formulas given in Table 5.

For many measuring instruments, the accuracy class is indicated in the form of numbers a×10 n, where a is equal to 1; 1.5; 2; 2.5; 4; 5; 6 and n is 1; 0; -1; -2, etc., which show the value of the possible maximum permissible systematic error (E y, additional) and special signs indicating its type (relative, reduced, constant, proportional).

Table 5

Examples of designation of accuracy classes of measuring instruments

Absolute measurement error is a quantity determined by the difference between the measurement result x and the true value of the measured quantity x 0:

Δ x = |x - x 0 |.

The value δ, equal to the ratio of the absolute measurement error to the measurement result, is called the relative error:

Example 2.1. The approximate value of π is 3.14. Then its error is 0.00159. The absolute error can be considered equal to 0.0016, and the relative error equal to 0.0016/3.14 = 0.00051 = 0.051%.

Significant figures. If the absolute error of the value a does not exceed one place unit of the last digit of the number a, then the number is said to have all the signs correct. Approximate numbers should be written down, keeping only sure signs. If, for example, the absolute error of the number 52400 is 100, then this number should be written, for example, as 524·10 2 or 0.524·10 5. You can estimate the error of an approximate number by indicating how many correct significant digits it contains. When counting significant figures, the zeros on the left side of the number are not counted.

For example, the number 0.0283 has three valid significant figures, and 2.5400 has five valid significant figures.

Rules for rounding numbers. If the approximate number contains extra (or incorrect) digits, then it should be rounded. When rounding, an additional error occurs that does not exceed half a unit of the place of the last significant digit ( d) rounded number. When rounding, only the correct digits are retained; extra characters are discarded, and if the first discarded digit is greater than or equal to d/2, then the last digit stored is increased by one.

Extra digits in integers are replaced by zeros, and in decimals they are discarded (as are extra zeros). For example, if the measurement error is 0.001 mm, then the result 1.07005 is rounded to 1.070. If the first of the digits modified by zeros and discarded is less than 5, the remaining digits are not changed. For example, the number 148935 with a measurement precision of 50 has a rounding value of 148900. If the first of the digits replaced by zeros or discarded is 5, and there are no digits or zeros following it, then it is rounded to the nearest even number. For example, the number 123.50 is rounded to 124. If the first zero or drop digit is greater than 5 or equal to 5 but is followed by a significant digit, then the last remaining digit is incremented by one. For example, the number 6783.6 is rounded to 6784.

Example 2.2. When rounding 1284 to 1300, the absolute error is 1300 - 1284 = 16, and when rounding to 1280, the absolute error is 1280 - 1284 = 4.

Example 2.3. When rounding the number 197 to 200, the absolute error is 200 - 197 = 3. The relative error is 3/197 ≈ 0.01523 or approximately 3/200 ≈ 1.5%.

Example 2.4. A seller weighs a watermelon on a scale. The smallest weight in the set is 50 g. Weighing gave 3600 g. This number is approximate. The exact weight of the watermelon is unknown. But the absolute error does not exceed 50 g. The relative error does not exceed 50/3600 = 1.4%.

Errors in solving the problem on PC

Three types of errors are usually considered as the main sources of error. These are called truncation errors, rounding errors, and propagation errors. For example, when using iterative methods searching for roots nonlinear equations the results are approximate, in contrast to direct methods that provide an exact solution.

Truncation errors

This type of error is associated with the error inherent in the task itself. It may be due to inaccuracy in determining the source data. For example, if any dimensions are specified in the problem statement, then in practice for real objects these dimensions are always known with some accuracy. The same goes for any other physical parameters. This also includes inaccuracy calculation formulas and the numerical coefficients included in them.

Propagation errors

This type of error is associated with the use of one or another method of solving a problem. During calculations, error accumulation or, in other words, propagation inevitably occurs. In addition to the fact that the original data themselves are not accurate, a new error arises when they are multiplied, added, etc. The accumulation of error depends on the nature and number of arithmetic operations used in the calculation.

Rounding errors

This type of error occurs because the true value of a number is not always accurately stored by the computer. When a real number is stored in computer memory, it is written as a mantissa and exponent in much the same way as a number is displayed on a calculator.

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 the value of a 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 a scale is compared with a unit of mass - a kilogram, etc. Thus, as a result 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 measurement object, as well as other factors, all measurements can be made only with to a certain extent 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 accuracy final result all sorts of things 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, different from true meaning Hist by a certain value ∆Х i = |Х i – Х|, which is an error made or, in other words, an error. The true error is not known to us, since we do not know the true value of the measured value. 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 design features measuring instruments, shortcomings of the adopted measurement method, any omissions of the experimenter, the influence of 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 X is considered random variable. Random errors can have an impact certain influence for a single measurement, but with multiple 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 big mistakes, clearly distorting 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 set value, determined by the error of the measuring equipment used.

At laboratory methods measurements, it is necessary to indicate the value of the measured quantity more accurately than its single measurement allows 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.

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 in 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 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 arose 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 both initial and results 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 real number in a computer it is represented in the form decimal With t in significant figures), we finally have

The measurement of a quantity is an operation as a result of which we find out how many times the measured quantity is greater (or less) than the corresponding value taken as the standard (unit of measurement). All measurements can be divided into two types: direct and indirect.

DIRECT these are measurements in which the directly interesting us is measured physical quantity(mass, length, time intervals, temperature changes, etc.).

INDIRECT these are measurements in which the quantity of interest to us is determined (calculated) from the results of direct measurements of other quantities associated with it by a certain functional relationship. For example, determining speed uniform motion by measuring the distance traveled, measuring the density of a body by measuring the mass and volume of the body, etc.

A common feature of measurements is the impossibility of obtaining the true value of the measured value; the measurement result always contains some kind of error (inaccuracy). This is explained both by the fundamentally limited measurement accuracy and by the nature of the measured objects themselves. Therefore, to indicate how close the obtained result is to the true value, the measurement error is indicated along with the obtained result.

For example, we measured the focal length of a lens f and wrote that

f = (256 ± 2) mm (1)

This means that the focal length ranges from 254 to 258 mm. But in fact, this equality (1) has a probabilistic meaning. We cannot say with complete confidence that the value lies within the specified limits; there is only a certain probability of this, therefore equality (1) must be supplemented with an indication of the probability with which this relationship makes sense (we will formulate this statement more precisely below).

An assessment of errors is necessary because, without knowing what they are, it is impossible to draw certain conclusions from the experiment.

Typically, absolute and relative error are calculated. The absolute error Δx is the difference between the true value of the measured quantity μ and the measurement result x, i.e. Δx = μ - x

The ratio of the absolute error to the true value of the measured quantity ε = (μ - x)/μ is called the relative error.

The absolute error characterizes the error of the method that was chosen for measurement.

The relative error characterizes the quality of measurements. The measurement accuracy is the reciprocal of the relative error, i.e. 1/ε.

§ 2. Classification of errors

All measurement errors are divided into three classes: misses (gross errors), systematic and random errors.

A MISS is caused by a sharp violation of measurement conditions during individual observations. This is an error associated with a shock or breakdown of the device, a gross miscalculation by the experimenter, unforeseen intervention, etc. a gross error usually appears in no more than one or two dimensions and differs sharply in magnitude from other errors. The presence of a miss can greatly distort the result containing the miss. The easiest way is to establish the cause of the mistake and eliminate it during the measurement process. If a mistake was not excluded during the measurement process, then this should be done when processing the measurement results, using special criteria that make it possible to objectively identify in each series of observations gross mistake

, if available. SYSTEMATIC ERROR is a component of measurement error that remains constant and changes naturally with repeated measurements of the same quantity. Systematic errors arise if you do not take into account, for example, thermal expansion

when measuring the volume of a liquid or gas made at a slowly changing temperature; if, when measuring mass, one does not take into account the effect of the buoyant force of air on the body being weighed and on the weights, etc. Systematic errors are observed if the ruler scale is applied inaccurately (unevenly); the capillary of the thermometer in different areas has a different cross-section; Without electric current

through the ammeter the instrument needle is not at zero, etc. As can be seen from the examples, systematic error

caused by certain reasons, its value remains constant (zero shift of the instrument scale, unevenness of the scales), or changes according to a certain (sometimes quite complex) law (unevenness of the scale, uneven cross-section of the thermometer capillary, etc.).

We can say that systematic error is a softened expression that replaces the words “experimenter error.”

Such errors occur because:
measuring instruments are inaccurate;
the actual installation differs in some way from the ideal;

The theory of the phenomenon is not entirely correct, i.e. some effects are not taken into account. We know what to do in the first case; calibration or calibration is needed. In two other cases ready-made recipe does not exist. The better you know physics, the more experience you have, the more likely it is that you will discover such effects, and therefore eliminate them. General rules

Systematic errors, the nature of which is known to you, and the value can be found, therefore, eliminated by introducing corrections. Example. Weighing on unequal-arm scales. Let the difference in arm lengths be 0.001 mm. With a rocker length of 70 mm and weight of the weighed body 200 G systematic error will be 2.86 mg. The systematic error in this measurement can be eliminated by using special methods weighing (Gauss method, Mendeleev method, etc.).
Systematic errors that are known to be less than a certain amount certain value. In this case, when recording the answer, their maximum value. Example. The data sheet supplied with the micrometer states: “the permissible error is ±0.004 mm. Temperature +20 ± 4° C. This means that, measuring the dimensions of any body with this micrometer at the temperatures indicated in the passport, we will have absolute error, not exceeding ± 0.004 mm for any measurement results.
Often the maximum absolute error given by a given device is indicated using the accuracy class of the device, which is depicted on the device scale by the corresponding number, most often circled.

The number indicating the accuracy class shows the maximum absolute error of the device, expressed as a percentage of highest value measured value on upper limit scales.

Let a voltmeter be used in the measurements, having a scale from 0 to 250 IN, its accuracy class is 1. This means that the maximum absolute error that can be made when measuring with this voltmeter will be no more than 1% of the highest voltage value that can be measured on this instrument scale, in other words:

δ = ±0.01·250 IN= ±2.5 IN.

The accuracy class of electrical measuring instruments determines the maximum error, the value of which does not change when moving from the beginning to the end of the scale. In this case, the relative error changes sharply, because the instruments provide good accuracy when the needle deflects almost the entire scale and does not provide it when measuring at the beginning of the scale. This is the recommendation: select a device (or the scale of a multi-range device) so that the arrow of the device goes beyond the middle of the scale during measurements.

If the accuracy class of the device is not specified and there is no passport data, then half the price of the smallest scale division of the device is taken as the maximum error of the device.

A few words about the accuracy of the rulers. Metal rulers are very accurate: millimeter divisions are marked with an error of no more than ±0.05 mm, and centimeter ones are no worse than with an accuracy of 0.1 mm. The error of measurements made with the accuracy of such rulers is almost equal to the error of reading by eye (≤0.5 mm). It is better not to use wooden and plastic rulers; their errors can be unexpectedly large.

A working micrometer provides an accuracy of 0.01 mm, and the measurement error with a caliper is determined by the accuracy with which the reading can be made, i.e. vernier accuracy (usually 0.1 mm or 0.05 mm).
Systematic errors caused by the properties of the measured object. These errors can often be reduced to chance. Example.. The electrical conductivity of a certain material is determined. If for such a measurement a piece of wire is taken that has some kind of defect (thickening, crack, inhomogeneity), then an error will be made in determining the electrical conductivity. Repeating the measurements gives the same value, i.e. some systematic error was made.
Let's measure the resistance of several pieces of such wire and find the average value of the electrical conductivity of this material, which may be greater or less than the electrical conductivity of individual measurements; therefore, the errors made in these measurements can be attributed to the so-called random errors. Example. Systematic errors that are not known to exist.

. Determine the density of any metal. First, we find the volume and mass of the sample. There is a void inside the sample that we know nothing about. An error will be made in determining the density, which will be repeated for any number of measurements. The example given is simple; the source of the error and its magnitude can be determined without much difficulty. Errors of this type can be identified with the help of additional research, by taking measurements using a completely different method and under different conditions.

RANDOM is the component of measurement error that changes randomly during repeated measurements of the same quantity.

When repeated measurements of the same constant, unchanging quantity are carried out with the same care and under the same conditions, we obtain measurement results - some of them differ from each other, and some of them coincide. Such discrepancies in measurement results indicate the presence of random error components in them.

A random error can take on values of different absolute values, which are impossible to predict for a given measurement act. This error can be equally positive or negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause scatter of repeated measurements relative to the true value ( Fig.14).

If, in addition, there is a systematic error, then the measurement results will be scattered relative to not the true, but the biased value ( Fig.15).

Rice. 14 Fig. 15

Let us assume that the period of oscillation of a pendulum is measured using a stopwatch, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the reading value, a slight unevenness in the movement of the pendulum - all this causes scattering of the results of repeated measurements and therefore can be classified as random errors.

If there are no other errors, then some results will be somewhat overestimated, while others will be somewhat underestimated. But if, in addition to this, the clock is also behind, then all the results will be underestimated. This is already a systematic error.

Some factors can cause both systematic and random errors at the same time. So, by turning the stopwatch on and off, we can create a small irregular spread in the starting and stopping times of the clock relative to the movement of the pendulum and thereby introduce a random error. But if, moreover, we are in a hurry to turn on the stopwatch every time and are somewhat late to turn it off, then this will lead to a systematic error.

Random errors are caused by parallax error when counting instrument scale divisions, shaking of the foundation of a building, the influence of slight air movement, etc.

Although it is impossible to eliminate random errors in individual measurements, the mathematical theory of random phenomena allows us to reduce the influence of these errors on the final measurement result. It will be shown below that for this it is necessary to make not one, but several measurements, and the smaller the error value we want to obtain, the more dimensions needs to be carried out.

It should be borne in mind that if the random error obtained from the measurement data turns out to be significantly less than the error determined by the accuracy of the device, then, obviously, there is no point in trying to further reduce the value of the random error; anyway, the measurement results will not become more accurate.

On the contrary, if the random error is greater than the instrumental (systematic) error, then the measurement should be carried out several times in order to reduce the error value for a given series of measurements and make this error less than or of the same order of magnitude as the instrument error.