Absolute and relative error expediency consideration. Measurement errors

Instruction

First of all, take several measurements with the instrument of the same value in order to be able to get the actual value. The more measurements you take, the more accurate the result will be. For example, weigh on an electronic scale. Let's say you got results of 0.106, 0.111, 0.098 kg.

Now calculate the actual value of the quantity (valid, since the true value cannot be found). To do this, add the results and divide them by the number of measurements, that is, find the arithmetic mean. In the example, the actual value would be (0.106+0.111+0.098)/3=0.105.

Sources:

• how to find measurement error

An integral part of any measurement is some error. She represents qualitative characteristic the accuracy of the study. According to the form of representation, it can be absolute and relative.

You will need

• - calculator.

Instruction

The second arise from the influence of causes, and random nature. These include incorrect rounding when counting readings and influence. If such errors are much smaller than the divisions of the scale of this measuring instrument, then it is advisable to take half a division as an absolute error.

Slip or rough error is the result of observation, which differs sharply from all the others.

Absolute error approximate numerical value is the difference between the result, during the measurement, and the true value of the measured quantity. The true or actual value reflects the investigated physical quantity. This error is the simplest quantitative measure errors. It can be calculated using the following formula: ∆X = Hisl - Hist. It can take positive and negative values. For a better understanding, consider. The school has 1205 students, when rounded to 1200 absolute error equals: ∆ = 1200 - 1205 = 5.

There are certain calculation of error values. First, absolute error the sum of two independent quantities is equal to the sum of their absolute errors: ∆(Х+Y) = ∆Х+∆Y. A similar approach is applicable for the difference of two errors. You can use the formula: ∆(X-Y) = ∆X+∆Y.

Sources:

• how to determine the absolute error

measurements physical quantities are always accompanied by one or another error. It represents the deviation of the measurement results from the true value of the measured quantity.

You will need

• -measuring device:
• -calculator.

Instruction

Errors may result from the influence various factors. Among them, one can single out the imperfection of means or methods of measurement, inaccuracies in their manufacture, non-compliance special conditions when conducting research.

There are several classifications. According to the form of presentation, they can be absolute, relative and reduced. The first are the difference between the calculated and actual value of the quantity. They are expressed in units of the measured phenomenon and are found according to the formula: ∆x = chisl-hist. The latter are determined by the ratio of absolute errors to the value of the true value of the indicator. The calculation formula is: δ = ∆х/hist. It is measured in percentages or shares.

The reduced error of the measuring device is found as the ratio of ∆x to the normalizing value хн. Depending on the type of device, it is taken either equal to the measurement limit, or referred to their certain range.

According to the conditions of occurrence, basic and additional are distinguished. If the measurements were taken in normal conditions, then the first kind arises. Deviations due to the output of values ​​outside the normal range is additional. To evaluate it, the documentation usually establishes norms within which the value can change if the measurement conditions are violated.

Also, the errors of physical measurements are divided into systematic, random and rough. The former are caused by factors that act upon repeated repetition of measurements. The second arise from the influence of causes, and character. A miss is a result of an observation that differs sharply from all others.

Depending on the nature of the measured quantity, various ways error measurement. The first of these is the Kornfeld method. It is based on the calculation of a confidence interval ranging from the minimum to the maximum result. The error in this case will be half the difference between these results: ∆х = (хmax-xmin)/2. Another way is to calculate the root mean square error.

Measurements can be taken with varying degrees accuracy. At the same time, even precision instruments are not absolutely accurate. Absolute and relative errors may be small, but in reality they are almost always present. The difference between the approximate and exact values ​​of a certain quantity is called absolute. error. In this case, the deviation can be both up and down.

You will need

• - measurement data;
• - calculator.

Instruction

Before calculating the absolute error, take several postulates as initial data. Eliminate gross errors. Assume that the necessary corrections have already been calculated and applied to the result. Such an amendment can be a transfer of the initial measurement point.

Take as a starting point the fact that random errors are taken into account. This implies that they are less systematic, that is, absolute and relative, characteristic of this particular device.

Random errors affect the result of even high-precision measurements. Therefore, any result will be more or less close to the absolute, but there will always be discrepancies. Define this interval. It can be expressed by the formula (Xmeas- ΔX) ≤ Xism ≤ (Xism + ΔX).

Determine the value closest to the value. In measurements, arithmetic is taken, which can be obtained from the formula in the figure. Accept the result as the true value. In many cases, the reading of a reference instrument is taken as accurate.

Knowing the true value, you can find the absolute error, which must be taken into account in all subsequent measurements. Find the value of X1 - the data of a particular measurement. Determine the difference ΔX by subtracting the smaller from the larger. When determining the error, only the modulus of this difference is taken into account.

note

As a rule, it is not possible to carry out an absolutely accurate measurement in practice. Therefore, the marginal error is taken as the reference value. She represents maximum value modulus of absolute error.

Helpful advice

AT practical measurements The absolute error is usually taken to be half lowest price division. When operating with numbers, the absolute error is taken to be half the value of the digit that is in the next exact numbers discharge.

To determine the accuracy class of the device, the ratio of the absolute error to the measurement result or to the length of the scale is more important.

Measurement errors are associated with the imperfection of devices, tools, methods. Accuracy also depends on the attentiveness and condition of the experimenter. Errors are divided into absolute, relative and reduced.

Instruction

Let a single measurement of the value give the result x. The true value is indicated by x0. Then the absolute errorΔx=|x-x0|. She evaluates the absolute . Absolute error consists of three components: random errors, systematic errors and misses. Usually, when measuring with an instrument, half the division value is taken as an error. For a millimeter ruler, this would be 0.5 mm.

The true value of the measured value in the interval (x-Δx; x+Δx). In short, this is written as x0=x±Δx. It is important to measure x and Δx in the same units and write in the same format, for example, whole part and three commas. So the absolute error gives the boundaries of the interval in which the true value lies with some probability.

Measurements are direct and indirect. In direct measurements, the desired value is immediately measured with the appropriate instrument. For example, bodies with a ruler, voltage with a voltmeter. With indirect measurements, the value is found according to the formula of the relationship between it and the measured values.

If the result is a dependence on three directly measured quantities with errors Δx1, Δx2, Δx3, then error indirect measurement ΔF=√[(Δx1 ∂F/∂x1)²+(Δx2 ∂F/∂x2)²+(Δx3 ∂F/∂x3)²]. Here ∂F/∂x(i) are the partial derivatives of the function with respect to each of the directly measured quantities.

Helpful advice

Misses are gross inaccuracies in measurements that occur when the instruments malfunction, the experimenter's inattention, and the experimental methodology is violated. To reduce the likelihood of such misses, be careful when taking measurements and describe the result in detail.

Sources:

• Guidelines to laboratory work in physics
• how to find relative error

The result of any measurement is inevitably accompanied by a deviation from the true value. There are several ways to calculate the measurement error, depending on its type, for example, statistical methods determination of the confidence interval, standard deviation, etc.

Page 1

The absolute error of determination does not exceed 0 01 μg of phosphorus. This method was used by us to determine phosphorus in nitric, acetic, hydrochloric and sulfuric acids and acetone with their preliminary evaporation.

The absolute error of determination is 0 2 - 0 3 mg.

The absolute error in the determination of zinc in zinc-manganese ferrites by the proposed method does not exceed 0 2 % rel.

The absolute error in the determination of hydrocarbons C2 - C4, when their content in the gas is 0 2 - 50%, is 0 01 - 0 2%, respectively.

Here au - - absolute error definition of r /, which is obtained as a result of the error Yes in the definition of a. For example, the relative error of the square of a number is twice the error in determining the number itself, and the relative error of the number under cube root, is just one third of the error in determining the number.

More complex considerations are necessary when choosing a measure of comparison of absolute errors in determining the time of the beginning of the accident TV - Ts, where Tv and Ts are the time of the restored and real accident, respectively. By analogy, here we can use the average time to reach the pollution peak from a real discharge to those monitoring points that recorded an accident during the pollution transit time Tsm. Calculation of the reliability of determining the power of accidents is based on the calculation of the relative error MV - Ms / Mv, where Mv and Ms are the restored and real powers, respectively. Finally, the relative error in determining the duration emergency release characterized by the value rv - rs / re, where rv and rs are the restored and real durations of accidents, respectively.

More complex considerations are necessary when choosing a measure of comparison of absolute errors in determining the time of the beginning of the accident TV - Ts, where Tv and Ts are the time of the restored and real accident, respectively. By analogy, here we can use the average time to reach the pollution peak from a real discharge to those monitoring points that recorded an accident during the pollution transit time Tsm. Calculation of the reliability of determining the power of accidents is based on the calculation of the relative error Mv - Ms / Ms, where Mv and Ms are the restored and real powers, respectively. Finally, the relative error in determining the duration of an emergency release is characterized by the value rv - rs / rs, where rv and rs are the reconstructed and real durations of accidents, respectively.

With the same absolute measurement error ay, the absolute error in determining the amount of ax decreases with increasing sensitivity of the method.

Since errors are based not on random, but on systematic errors, the total absolute error in determining suction cups can theoretically reach 10%. required amount air. Only with unacceptably loose fireboxes (A a0 25) accepted method gives more or less satisfactory results. What has been described is well known to adjusters, who, when reducing the air balance of dense furnaces, often get negative values suction cups.

An analysis of the error in determining the value of pet showed that it consists of 4 components: the absolute error in determining the mass of the matrix, the capacity of the sample, weighing, and the relative error due to fluctuations in the mass of the sample around the equilibrium value.

Subject to all the rules for the selection, counting of volumes and analysis of gases using the GKhP-3 gas analyzer, the total absolute error in determining the content of CO2 and O2 should not exceed 0 2 - 0 4% of their true value.

From Table. 1 - 3, we can conclude that the data we use for the starting substances, taken from different sources, have relatively small differences that lie within the absolute errors in determining these quantities.

Random errors can be absolute or relative. Random error, which has the dimension of the measured value, is called the absolute error of determination. The average arithmetic value The absolute errors of all individual measurements are called the absolute error of the analysis method.

Tolerance value, or confidence interval, is not set arbitrarily, but is calculated from specific measurement data and characteristics of the instruments used. The deviation of the result of an individual measurement from the true value of a quantity is called the absolute error of determination or simply error. The ratio of the absolute error to the measured value is called the relative error, which is usually expressed as a percentage. Knowledge of individual measurement error has no independent value, and in any serious experiment, several parallel measurements must be carried out, according to which the error of the experiment is calculated. Measurement errors, depending on the causes of their occurrence, are divided into three types.

In physics and other sciences, it is very often necessary to measure 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 devices.

Measuring device called a device by which a measured quantity is compared 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 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 dependence. For example, determining the circumference of a circle based on the results of measuring the diameter or determining the volume of a body based on the results of measuring its linear dimensions.

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

Accuracy of measurements - a characteristic of the quality of measurements, reflecting the proximity of the measurement results to the true value of the measured quantity.

The smaller the measurement errors, the greater the measurement accuracy. The measurement accuracy depends on the instruments used in the measurements and on common methods measurements. It is absolutely useless to try to go beyond this limit of accuracy when making measurements under given conditions. It is possible to minimize the influence of causes 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 any physical dimension it is necessary to do not one, but several times under the same experimental conditions.

As a result of the i-th measurement (i is the measurement number) of the value "X", an approximate number X i is obtained, which differs from the true value Xist by some value ∆X i = |X i - X|, which is a mistake 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 the X value obtained during the measurement (that is, the measured value); ∆X - absolute error determining the value of X.

Absolute error (error) of measurement ∆X is the absolute value of the difference between the true value of the measured quantity Xist and the measurement result X i: ∆X = |X ist - X i |.

Relative error (error) measurement δ (characterizing the measurement accuracy) is numerically equal to the ratio of the absolute measurement error ∆X to the true value of the measured value X sist (often expressed as a percentage): δ \u003d (∆X / X sist) 100% .

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

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

In any measuring device, one or another systematic error is inherent, 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 of 0.01 g more than indicated on it, then the found value of the body weight 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, which we can only say that they do not exceed a certain value.

Random mistakes called errors that change their magnitude and sign in an unpredictable way from experience to experience. The appearance of random errors is due to the action of many diverse and uncontrollable causes.

For example, when weighing with a balance, these reasons can be air vibrations, dust particles that have settled, different friction in the left and right suspension of the cups, etc. different values: X1, X2, X3,…, X i ,…, X n , where X i is the result of the i-th measurement. It is not possible to establish any regularity between the results, therefore the result of the i -th measurement of X is considered random variable. Random errors can certain influence to 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.

Misses and blunders- 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 reading of the device “212”, a completely different number is written - “221”). Measurements containing misses and gross errors should be discarded.

Measurements can be made in terms of their accuracy by technical and laboratory methods.

When using technical methods, the measurement is carried out once. In this case, they are satisfied with such an accuracy at which the error does not exceed some predetermined set value determined by the error of the applied measuring equipment.

At laboratory methods measurements, it is required 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 (accounting for random errors).

From the possibility of carrying out measurements by two methods, the existence of two methods for assessing the accuracy of measurements follows: technical and laboratory.

The measurement of a quantity is an operation, as a result of which we find out how many times the measured value is greater (or less) than the corresponding value, taken as a 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 change, 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 dependence. For example, determining the speed uniform motion by measurements of the distance traveled over a period of time, measurement of body density by measurements of body mass and volume, etc.

A common feature of measurements is the impossibility of obtaining the true value of the measured quantity, the measurement result always contains some kind of error (error). 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 result obtained is to the true value, the measurement error is indicated along with the result obtained.

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 is between 254 and 258 mm. But in fact this equality (1) has a probabilistic meaning. We cannot say with complete certainty that the value lies within the indicated limits, there is only a certain probability of this, therefore equality (1) must be supplemented with an indication of the probability with which this ratio makes sense (below we will formulate this statement more precisely).

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

Usually calculate the absolute and relative error. 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 value ε = (μ - x)/μ is called the relative error.

The absolute error characterizes the error of the method that has been chosen for the 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 LOSS is caused by a sharp violation of the measurement conditions in individual observations. This is an error associated with a shock or breakage of the device, a gross miscalculation of the experimenter, unforeseen interference, 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 skew the result containing the miss. The easiest way is to establish the cause of the slip and eliminate it during the measurement process. If a slip 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 distinguish in each series of observations blunder if it exists.

A systematic error is a component of the measurement error that remains constant and regularly changes during repeated measurements of the same value. Systematic errors arise if one does not take into account, for example, thermal expansion when measuring the volume of a liquid or gas produced at a slowly changing temperature; if, when measuring the mass, the effect of the buoyancy force of air on the weighed body and on weights is not taken into account, etc.

Systematic errors are observed if the scale of the ruler is applied inaccurately (unevenly); the capillary of the thermometer in different parts has a different cross section; with absence electric current through the ammeter, the arrow of the device is not at zero, etc.

As can be seen from the examples, the systematic error is caused by certain reasons, its value remains constant (zero shift of the scale of the instrument, uneven scales), or changes according to a certain (sometimes quite complex) law (nonuniformity of the scale, uneven cross section of the thermometer capillary, etc.).

We can say that a systematic error is a softened expression that replaces the words "experimenter's error."

These errors occur because:

1. inaccurate measuring instruments;
2. the real installation is somewhat different from the ideal;
3. the theory of the phenomenon is not entirely correct, i.e. no effects were taken into account.

We know what to do in the first case - calibration or graduation is needed. In two other cases ready recipe does not exist. The better you know physics, the more experience you have, the more likely you are to detect such effects, and therefore eliminate them. General rules, there are no recipes for identifying and eliminating systematic errors, but some classification can be made. We distinguish four types of systematic errors.

1. Systematic errors, the nature of which is known to you, and the value can be found, therefore, excluded by the introduction of amendments. Example. Weighing on unequal scales. Let the difference in shoulder lengths be 0.001 mm. With a rocker length of 70 mm and weighed body weight 200 G the systematic error will be 2.86 mg. The systematic error of this measurement can be eliminated by applying special methods weighing (Gauss method, Mendeleev method, etc.).
2. Systematic errors, which are known to be less than a certain certain value. In this case, when recording the answer, their maximum value can be indicated. Example. The passport attached to the micrometer says: “The permissible error is ± 0.004 mm. The temperature is +20 ± 4 ° C. This means that when measuring the dimensions of a body with this micrometer at the temperatures indicated in the passport, we will have an absolute error not exceeding ± 0.004 mm for any measurement results.

   Often, the maximum absolute error given by a given instrument is indicated by the accuracy class of the instrument, which is depicted on the instrument's scale by the corresponding number, most often taken in a circle.

   The number indicating the accuracy class indicates the maximum absolute error of the instrument, expressed as a percentage of the greatest value measured value at the upper limit of the scale.

   Let a voltmeter be used in the measurements, having a scale from 0 to 250 AT, its accuracy class is 1. This means that the maximum absolute error that can be made when measuring with this voltmeter will not be more than 1% of the highest voltage value that can be measured on this instrument scale, in other words:

   δ = ±0.01 250 AT= ±2.5 AT.

   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 dramatically, because the instruments provide good accuracy when the arrow deviates almost to the entire scale and does not give it when measuring at the beginning of the scale. Hence the recommendation: select the instrument (or the scale of the multirange instrument) so that the arrow of the instrument during measurements goes beyond the middle of the scale.

   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 applied 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 practically equal to the reading error by eye (≤0.5 mm). It is better not to use wooden and plastic rulers, their errors can turn out to 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 a reading can be made, i.e. vernier accuracy (usually 0.1 mm or 0.05 mm).

3. Systematic errors due to the properties of the measured object. These errors can often be reduced to random ones. Example.. The electrical conductivity of some 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 measurements gives the same value, i.e. there is some systematic error. Let us measure the resistance of several segments of such a 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.
4. Systematic errors, the existence of which is not known. Example.. Determine the density of any metal. First, find the volume and mass of the sample. Inside the sample there is an emptiness about which we know nothing. 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 detected with the help of additional studies, by carrying out measurements by a completely different method and under different conditions.

RANDOM is the component of the measurement error that changes randomly with repeated measurements of the same value.

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

Random error arises from the simultaneous action of many sources, each of which in itself has an imperceptible effect on the measurement result, but the total effect of all sources can be quite strong.

Random error may take various absolute value values ​​that cannot be predicted for a given act of measurement. This error can equally be both positive and negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause repeated measurements to scatter about the true value ( fig.14).

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

Rice. 14 Fig. fifteen

Let us assume that with the help of a stopwatch we measure the period of oscillation of the pendulum, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the value of the reference, a small uneven movement of the pendulum - all this causes a scatter in the results of repeated measurements and therefore can be classified as random errors.

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

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

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

Although it is impossible to exclude random errors of individual measurements, 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 need 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 instrument, then, obviously, there is no point in trying to further reduce the magnitude of the random error - all the same, the measurement results will not become more accurate from this.

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 one order of magnitude with the instrument error.

Measurement error

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

• Reduced error- relative error, expressed as the ratio of the absolute error of the measuring instrument to the conditionally accepted value of the quantity, which is constant over the entire measurement range or in part of the range. Calculated according to the formula

where X n- normalizing value, which depends on the type of measuring instrument scale and is determined by its graduation:

If the scale of the device is one-sided, i.e. the lower measurement limit is zero, then X n is determined equal to upper limit measurements;
- if the scale of the device is two-sided, then the normalizing value is equal to the width of the measurement range of the device.

The given error is a dimensionless value (it can be measured as a percentage).

Due to the occurrence

• Instrumental / Instrumental Errors- errors that are determined by the errors of the measuring instruments used and are caused by the imperfection of the operating principle, the inaccuracy of the scale graduation, and the lack of visibility of the device.
• Methodological errors- errors due to the imperfection of the method, as well as simplifications underlying the methodology.
• Subjective / operator / personal errors- errors due to the degree of attentiveness, concentration, preparedness and other qualities of the operator.

In engineering, devices are used to measure only with a certain predetermined accuracy - the main error allowed by the normal under normal operating conditions for this device.

If the device is operated under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by temperature deviation environment from normal, installation, due to the deviation of the position of the device from the normal operating position, etc. 20°C is taken as normal ambient temperature, Atmosphere pressure 01.325 kPa.

A generalized characteristic of measuring instruments is an accuracy class determined by the limit values ​​of the permissible basic and additional errors, as well as other parameters that affect the accuracy of measuring instruments; the parameter value is set by the standards to certain types measuring instruments. The accuracy class of measuring instruments characterizes their accuracy properties, but is not a direct indicator of the accuracy of measurements performed using these instruments, since the accuracy also depends on the measurement method and the conditions for their implementation. Measuring instruments, the limits of the permissible basic error of which are given in the form of the reduced basic (relative) errors, are assigned accuracy classes selected from the series following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0; 5.0; 6.0) * 10n, where n = 1; 0; -one; -2 etc.

According to the nature of the manifestation

• random error- error, changing (in magnitude and in sign) from measurement to measurement. Random errors can be associated with the imperfection of devices (friction in mechanical devices, etc.), shaking in urban conditions, with the imperfection of the object of measurement (for example, when measuring the diameter of a thin wire, which may not have a completely round cross section as a result of the imperfection of the manufacturing process ), with features of the measured quantity itself (for example, when measuring the amount elementary particles passing per minute through a Geiger counter).
• Systematic error- an error that changes over time according to a certain law (a special case is a constant error that does not change over time). Systematic errors can be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
• Progressive (drift) error is an unpredictable error that changes slowly over time. It is a non-stationary random process.
• Gross error (miss)- an error resulting from an oversight of the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the division number on the scale of the device, if there was a short circuit in the electrical circuit).

According to the method of measurement

• Accuracy of direct measurements
• Uncertainty of indirect measurements- error of the calculated (not measured directly) value:

If a F = F(x 1 ,x 2 ...x n) , where x i- directly measured independent variables, having an error Δ x i, then:

see also

• Measurement of physical quantities
• System for automated data collection from meters over the air

Literature

• Nazarov N. G. Metrology. Basic concepts and mathematical models. M.: graduate School, 2002. 348 p.

• Laboratory classes in physics. Textbook / Goldin L. L., Igoshin F. F., Kozel S. M. and others; ed. Goldina L. L. - M .: Science. Main edition of physical and mathematical literature, 1983. - 704 p.

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STANDARD ERRORS OF MEASUREMENT- Evaluation of the extent to which a certain set of measurements obtained in a given situation (for example, in a test or in one of several parallel forms of a test) can be expected to deviate from true values. Designated as a (M) ...

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ERROR OPTIONS- The size of the variance, which cannot be explained by controllable factors. The error of variance is offset by sampling errors, measurement errors, experimental errors, etc… Dictionary in psychology