The error is expressed by the attitude of the absolute error. Error measuring instrumentation sensors

Absolute measurement error called the value determined by the difference between the measurement result x. and the true meaning of the measured value x. 0:

Δ x. = |x. - x. 0 |.

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

Example 2.1. The approximate value of the number π is 3.14. Then the error is equal to 0.00159. The absolute error can be considered equal to 0.0016, and the relative error is 0.0016 / 3.14 \u003d 0.00051 \u003d 0.051%.

Meaning digits.If the absolute error of the value A does not exceed one unit of the discharge of the last figure of A, then they say that the number of all signs are correct. Approximated numbers should be recorded, while maintaining only correct signs. If, for example, the absolute error of the number 52400 is 100, then this number should be recorded, for example, in the form 524 · 10 2 or 0.524 · 10 5. It is possible to estimate the error of an approximate number by specifying how much faithful digits it contains. When calculating significant digits, zeros are not considered from the left side of the number.

For example, the number 0.0283 has three faithful meaning digits, and 2.5400 - five faithful meaningful digits.

Rules rounding numbers. If the approximate number contains extra (or incorrect) signs, then it should be rounded. When rounding, an additional error occurs that does not exceed half the unit of discharge of the last significant digit ( d.) Rounded number. When rounding, only true signs are saved; Excess signs are discarded, and if the first discarded number is greater than or equal d./ 2, the last saved digit increases by one.

Excess numbers are replaced by zeros in integers, and in decimal fractions discarded (as well as extra zeros). For example, if the measurement error is 0.001 mm, then the result of 1.07005 is rounded to 1.070. If the first of the variable zeros and the details of less than 5, the remaining numbers do not change. For example, the number 148935 with an accuracy of measurement 50 has rounding 148900. If the first of the numbers replaced by zeros or discarded is 5, and no digits follow it or go zero, the rounding is made to the nearest former number. For example, the number 123.50 is rounded up to 124. If the first of the replaced by zeros or discarded numbers is greater than 5 or equal to 5, but there should be a significant digit after it, the last remaining digit increases by one. For example, the number 6783.6 is rounded up to 6784.

Example 2.2. When rounding the number 1284 to 1300, the absolute error is 1300 - 1284 \u003d 16, and when rounding to 1280, the absolute error is 1280 - 1284 \u003d 4.

Example 2.3. When rounding the number 197 to 200, the absolute error is 200 - 197 \u003d 3. The relative error is 3/197 ≈ 0.01523 or approximately 3/200 ≈ 1.5%.

Example 2.4. The seller plays watermelon on cup of scales. In the set of weights, the smallest - 50 g. Weighing gave 3600. This number is approximate. The exact weight of watermelon is unknown. But the absolute error does not exceed 50 g. The relative error does not exceed 50/3600 \u003d 1.4%.

Errors of solving the problem on PC.

Three types of errors usually consider the main sources of error. These are so-called truncation errors, rounding errors and distribution errors. For example, when using iterative methods for finding roots of nonlinear equations, the results are approximate in contrast to direct methods that give an exact solution.

Error truncation

This type of errors is associated with the error laid in the task itself. It may be due to the inaccuracy of determining the source data. For example, if any sizes are specified in the task condition, then in practice for real objects, these dimensions are always known with some accuracy. The same applies to any other physical parameters. This can also include the inaccuracy of the calculated formulas and the numerical coefficients included in them.

Distribution errors

This type of errors is associated with the use of one or another way to solve the problem. During the calculations, accumulation inevitably occurs or, in other words, the spread of the error. In addition to the fact that the initial data themselves are not accurate, the new error occurs when they are multiplied, addition, etc. The accumulation of error depends on the nature and number of arithmetic actions used in the calculation.

Rounding errors

This error type is associated with the fact that the true value of the number is not always accurately saved by the computer. When saving a real number in the computer's memory, it is written as a mantissa and order about the same way as the number on the calculator is displayed.

In our age, a person came up with and uses a huge set of all sorts of measuring instruments. But whatever perfect technology of their manufacturing, they all have a greater or less error. This parameter is usually indicated on the tool itself, and to estimate the accuracy of the determined value you need to be able to understand what the figures indicated on the marking. In addition, the relative and absolute error inevitably occurs with complex mathematical calculations. It is widely used in statistics, industry (quality control) and in a number of other regions. How this value is calculated and how to interpret its value - this will be discussed in this article.

Absolute error

Denote by x the approximate value of any value obtained, for example, by means of a single measurement, and through x 0 is its exact value. Now we calculate the difference module between these two numbers. The absolute error is just the value that we happen from us as a result of this simple operation. Specified by the language of the formula, this definition can be written in this form: Δ x \u003d | x - x 0 |.

Relative error

Absolute deviation has one important disadvantage - it does not allow to assess the degree of importance of the error. For example, we buy 5 kg of potatoes on the market, and the unscrupulous seller when measuring weight was mistaken by 50 grams in its favor. That is, an absolute error was 50 grams. For us, such an oversight will be trifle and we will not even pay attention to it. And imagine what happens if a similar error occurs when cooking medicine? Here everything will be much more serious. And when loading a commercial car, probably deviations occur much more than this value. Therefore, the absolute error in itself is uninformative. In addition to her, relative deviations are very often calculated. equal to relation absolute error to the exact value of the number. This is written by the following formula: Δ \u003d Δ x / x 0.

Properties of error

Suppose we have two independent values: x and y. We need to calculate the deviation of the approximate value of their sum. In this case, we can calculate the absolute error as the sum of pre-calculated absolute deviations of each of them. In some measurements, it may occur so that errors in determining x and y values \u200b\u200bwill compensate each other. And it may happen that as a result of the addition, the deviation will increase as much as possible. Therefore, when the total absolute error is calculated, the worst of all options should be taken into account. The same is true for the difference of errors of several quantities. This property is characteristic only for absolute error, and it cannot be applied to relative deviation, since it will inevitably lead to an incorrect result. Consider this situation in the following example.

Suppose, measurements inside the cylinder showed that the inner radius (R 1) is 97 mm, and the outer (R 2) is 100 mm. It is required to determine the thickness of its wall. First we find the difference: H \u003d R 2 - R 1 \u003d 3 mm. If the problem does not indicate that the absolute error is equal, then it is taken over half of the division of the scale of the measuring instrument. Thus, δ (R 2) \u003d δ (R 1) \u003d 0.5 mm. The total absolute error is: Δ (H) \u003d δ (R 2) + δ (R 1) \u003d 1 mm. Now we calculate the relatively deviation of all values:

δ (R 1) \u003d 0.5 / 100 \u003d 0.005,

δ (R 1) \u003d 0.5 / 97 ≈ 0.0052,

δ (h) \u003d δ (H) / H \u003d 1/3 ≈ 0.3333 \u003e\u003e δ (R 1).

As we can see, the measurement error of both radii does not exceed 5.2%, but an error in calculating their difference - the thickness of the cylinder wall - amounted to as many as 33, (3)%!

The following property reads: the relative deviation of the work of several number is about equal to the sum of the relative deviations of individual factors:

δ (Hu) ≈ Δ (x) + δ (y).

Moreover, this rule is valid regardless of the amount of valued values. The third and last property of the relative error is that the relative estimate of the number k-th degree approximately in | K | Once exceeds the relative error in the original number.

The main qualitative characteristic of any kip sensor is the measurement error of the controlled parameter. The error in the measurement of the device is the magnitude of the discrepancy between what shown (measured) the kip sensor and what is really. Measurement error for each specific type of sensor is indicated in the accompanying documentation (passport, instruction manual, calibration technique), which comes with this sensor.

In the form of the presentation of the error are divided into absolute, relative and led Error.

Absolute error - This is the difference between the sensor measured by the value of hism and the valid value of the HD of this value.

The actual value of the value of the measured value is the experimentally found value of the measured value as close as possible to its true value. Speaking simple language The actual value of the HD is a value measured by the reference device, or a high-end accuracy generated by a calibrator or a valiant. The absolute error is expressed in the same measurement units as the measured value (for example, in M3 / h, MA, MPa, etc.). Since the measured value may turn out to be both more and less than its actual value, the measurement error can be both with a plus sign (the instrument readings are overestimated) and with a minus sign (the device underson).

Relative error - This is the ratio of the absolute measurement error Δ to the actual value of the HD of the measured value.

The relative error is expressed in percent, or is a dimensionless value, and can also take both positive and negative values.

Limited error - This is the ratio of the absolute error of measurement Δ to the normalizing value of the Xn, constant in the entire range of measurement or its part.

The rational value of XN depends on the type of scale of the Kip sensor:

If the sensor scale is unilateral and the lower measurement limit is zero (for example, a sensor scale from 0 to 150 m3 / h), Xn is accepted with an equal to the upper measurement limit (in our case xn \u003d 150 m3 / h).
If the sensor scale is unilateral, but the lower measurement limit is not zero (for example, a sensor scale is from 30 to 150 m3 / h), xn is taken equal to the difference in the upper and lower measurement limits (in our case xn \u003d 150-30 \u003d 120 m3 / h ).
If the scale of the sensor is double-sided (for example, from -50 to +150 ˚С), xn is equal to the width of the measurement range of the sensor (in our case xn \u003d 50 + 150 \u003d 200 ˚С).

The above error is expressed as a percentage, or is a dimensionless value, and can also take both positive and negative values.

Quite often, in the description on one or another sensor, not only the measurement range is indicated, for example, from 0 to 50 mg / m3, but also the test range, for example, from 0 to 100 mg / m3. The reduced error in this case is normalized by the end of the measurement range, that is, to 50 mg / m3, and in the indication range from 50 to 100 mg / m3, the sensor measurement error is not determined at all - in fact, the sensor can show anything and have any measurement error. The measurement range of the sensor can be divided into several measuring subbands, for each of which the error can be defined both in size and in the form of the presentation. At the same time, during the calibration of such sensors, their sample measurement means can be used for each subband, the list of which is specified in the calibration technique to this device.

Some instruments in passports instead of the measurement error indicate the accuracy class. Such devices include mechanical pressure gauges showing bimetallic thermometers, thermostats, flow pointers, arrogant ammeters and voltmeters for shield mounting, etc. The accuracy class is a generalized characteristics of measuring instruments, which are determined by the limits of allowable and additional errors, as well as a number of other properties that affect the accuracy of measurements with their help. In this case, the accuracy class is not the direct characteristic of the accuracy of measurements performed by this device, it only indicates a possible tool constituent measurement error. The instrument accuracy class is applied on its scale or body according to GOST 8.401-80.

When assigning the quality of the accuracy class, it is selected from the row 1 · 10 n; 1.5 · 10 n; (1.6 · 10 n); 2 · 10 n; 2.5 · 10 n; (3 · 10 n); 4 · 10 n; 5 · 10 n; 6 · 10 n; (where n \u003d 1, 0, -1, -2, etc.). The values \u200b\u200bof the accuracy classes specified in brackets are not installed for the newly developed measurement tools.

The definition of the measurement error of the sensors is performed, for example, when they are periodic calibration and calibration. With the help of various accuracy and calibrators with high accuracy generate certain values \u200b\u200bfor one or another physical quantity And the readings of the indicated sensor with the readings of the measuring measurement to which the same value of the physical value is supplied. Moreover, the error of measuring the sensor is controlled both at the direct course (an increase in the measured physical value from a minimum to the maximum scale) and during the reverse course (decrease in the measured value from a maximum to a minimum scale). This is due to the fact that due to the elastic properties of the sensitive element of the sensor (the membrane of the pressure sensor), various intensity of the flow chemical reactions (electrochemical sensor), thermal inertia, etc. The sensor readings will be different depending on how the physical value affecting the sensor changes: decreases or increases.

Quite often, according to the calibration method, the countdown of the sensor readings when calibration should be performed not by its display or scale, but by the output signal value, for example, by the value of the current output of the current output 4 ... 20 mA.

At the test pressure sensor with a measurement scale from 0 to 250 MBAR, the main relative measurement error in the entire measurement range is 5%. The sensor has a current output of 4 ... 20 mA. A pressure of 125 MBAR is fed to the sensor by the calibrator, and its output signal is 12.62 mA. It is necessary to determine whether the sensor readings are stacked in the permissible limits.

First, it is necessary to calculate what should be the output current of the sensor I ,.t at a pressure of the RT \u003d 125 MBAR.

Ivy.t \u003d Ish.Vy.M. + ((Ish.Vykh.maks - Ish.Vykh.min) / (RS. Max - Rsh.min)) * RT

where I take the output current of the sensor at a given pressure of 125 MBAR, MA.

Ш.Vy.Min is the minimum sensor output current, Ma. For a sensor with a yield of 4 ... 20 mA Ш.Vy.Min \u003d 4 mA, for a sensor with a yield of 0 ... 5 or 0 ... 20 mA Ш.Vy.Min \u003d 0.

Ish.Vy.Max is the maximum output current of the sensor, ma. For a sensor with a yield of 0 ... 20 or 4 ... 20 mA Ish. Max \u003d 20 mA, for a sensor with an output of 0 ... 5 mA Ish.mak. Max \u003d 5 mA.

RS. Max is the maximum of the pressure sensor scale, MBAR. RSh.Max \u200b\u200b\u003d 250 MBAR.

RSh.min - Minimum Pressure Sensor Scale, MBAR. RSh.min \u003d 0 MBAR.

RT - submitted from the calibrator to the pressure sensor, MBAR. RT \u003d 125 MBAR.

Substituting the known values \u200b\u200bwe get:

Ivy.t \u003d 4 + ((20-4) / (250-0)) * 125 \u003d 12 mA

That is, with a 125 MBAR pressure submitted to a pressure sensor, 12 mA should be on its current output. We believe in what limits the calculated value of the output current may vary, given that the main relative measurement error is ± 5%.

ΔIV.t \u003d 12 ± (12 * 5%) / 100% \u003d (12 ± 0.6) ma

That is, with a pressure of 125 MBAR submitted to a pressure sensor at its current output, the output signal must be in the range from 11.40 to 12.60 mA. Under the task condition, we have a 12.62 mA output signal, which means our sensor did not meet the measurement error defined by the manufacturer and requires configuration.

The main relative error of measuring our sensor is equal to:

Δ \u003d ((12.62 - 12.00) / 12.00) * 100% \u003d 5.17%

Verification and calibration of instruments of instruments should be performed under normal conditions. ambient According to atmospheric pressure, humidity and temperature and at rated sensor power voltage, since higher or low temperature and supply voltage can be brought to the appearance of an additional measurement error. Conditions for calibration are indicated in the verification method. Devices, the measurement error of which did not specify the frame-mounted procedures, or adjusted and adjust, after which they re-pass calibration, or if the setup did not bring results, for example, due to aging or excessive sensor deformation, repaired. If the repair is not possible, the devices are brave and are output.

If nevertheless, the devices managed to repair, then they are no longer periodic, but the primary verification with the execution of all the checks outlined in the technique of verification of items for this type of calibration. In some cases, the device is specifically exposed to minor repairs () as according to the verification technique, perform the primary calibration is significantly easier and cheaper than periodic, due to differences in the set of exemplary measurement tools that are used during periodic and primary verification.

To consolidate and check the knowledge gained, I recommend to perform.

The true meaning of physical quantity is absolutely accurately impossible, because Any measurement operation is associated with a number of errors or, otherwise, errors. The causes of errors can be the most different. Their occurrence can be associated with the inaccuracies of the manufacture and adjustment of the measuring device, due to the physical features of the object under study (for example, when measuring the diameter of the wire of the non-uniform thickness, the result randomly depends on the selection of the measurement section), reasons of random nature, etc.

The task of the experimenter is to reduce their effect on the result, as well as to indicate how close the result is close to true.

There are the concepts of absolute and relative error.

Under absolute error Measurements will understand the difference between the measurement result and the true meaning of the measured value:

Δx i \u003d x i -x and (2)

where Δx i is the absolute error of the i-th dimension, x i _- The result of the i-th measurement, x and is the true value of the measured value.

Result of any physical dimension It is customary to record as:

where - average arithmetic value The measured value that is closest to the true value (justice x and≈ will be shown below), is an absolute measurement error.

Equality (3) should be understood in such a way that the true value of the measured value is in the interval [-, +].

The absolute error is the size of the size, it has the same dimension as the measured value.

The absolute error does not fully characterize the accuracy of the measurements. In fact, if we measure with one and the same absolute error ± 1 mm segments of 1 m and 5 mm long, measurement accuracy will be incomparable. Therefore, along with the absolute measurement error, a relative error is calculated.

Relative error Measurements are called the ratio of absolute error to the most measured value:

Relative error - the magnitude is dimensionless. It is expressed as a percentage:

In the above example, the relative errors are 0.1% and 20%. They differ significantly among themselves, although absolute values \u200b\u200bare the same. Relative error gives information about accuracy

Measurement errors

By the nature of the manifestation and reasons, the appearance of errors can be divided into the following classes: instrument, systematic, random, and misses (coarse errors).

PR O M A X and due to either a malfunction of the device, or a violation of the technique or experimental conditions, or are subjective. Practically they are defined as the results sharply different from others. To eliminate their appearance, it is necessary to follow accuracy and care in operation with the instruments. Results containing misses must be excluded from consideration (discarded).

Instrument errors. If the measuring device is working and adjusted, then it can be measured with a limited accuracy defined by the type of instrument. Adopted instrumentary error of the arrow instrument to be considered equal to half the smallest division of its scale. In the instruments with a digital sample, the instrument error is equal to the value of one smallest discharge of the instrument scale.

Systematic errors are errors, the value and the sign are constant for the entire measurement series, carried out by the same method and using the same measuring instruments.

When performing measurements, not only accounting for systematic errors is important, but it is also necessary to achieve their exceptions.

Systematic errors are conditionally divided into four groups:

1) error, the nature of which is known and their value can be fairly defined. This error is, for example, a change in the measured mass in the air, which depends on temperature, humidity, air pressure, etc.;

2) errors whose nature is known, but the very magnitude of the error itself is unknown. Such errors include errors caused by the measuring instrument: a malfunction of the device itself, the discrepancy between the scale by zero value, the class of accuracy of this device;

3) errors, the existence of which can not be suspected, but their value will often be significant. Such errors occur most often with complex measurements. A simple example of such an error is to measure the density of a certain sample containing inside the cavity;

4) errors caused by the features of the measurement object itself. For example, when measuring the electrical conductivity of the metal from the last, the wire segment is taken. Error may occur if there is any defect in the material - a crack, the thickening of the wire or the heterogeneity, changing its resistance.

Random errors are errors that change randomly on the sign and value under identical conditions of repeated measurements of the same value.

Similar information.

Terms measurement error and measurement error Used as synonyms.) It is possible only to estimate the value of this deviation, for example, with the help of statistical methods. At the same time, the average value obtained during the statistical processing of the measurement series results is taken for its true meaning. This value obtained is not accurate, but only the most likely. Therefore, in measurements it is necessary to indicate what is their accuracy. To do this, together with the result, the measurement error is indicated. For example, writing T \u003d 2.8 ± 0.1c. means that the true value of the magnitude T. lies in the interval from 2.7 p. before 2.9 s. some agreed probability (see the confidence interval, trust probability, standard error).

In 2006, at the international level, a new document was adopted, dictating measurement conditions and established new rules for the comparison of state standards. The concept of "error" began to obstruct, instead, the concept of "measurement uncertainty" was introduced instead.

Definition of error

Depending on the characteristics of the measured value, various methods use various methods for determining measurement errors.

The Cornfeld method is to choose a confidence interval ranging from the minimum to the maximum measurement result, and the error as half the difference between the maximum and minimum measurement result:

Medium quadratic error:

The average quadratic error of the average arithmetic:

Classification of errors

In the form of representation

Absolute error - Δ X. is an assessment absolute error Measurements. The magnitude of this error depends on the method of its calculation, which, in turn, is determined by the distribution of random variable X. m.e.a.s. . At the same time, equality:

Δ X. = | X. t.r.u.e. − X. m.e.a.s. | ,

where X. t.r.u.e. - true meaning, and X. m.e.a.s. - The measured value must be performed with some probability close to 1. If a random value X. m.e.a.s. Distributed according to the normal law, then, usually, its standard deviation is taken for absolute error. The absolute error is measured in the same measurement units as the size itself.

Relative error - The ratio of absolute error towards the meaning that is taken for true:

The relative error is a dimensionless value, or is measured as a percentage.

Limited error - relative error, expressed by the ratio of the absolute error of measurements to the conditionally adopted value of the value, permanent in the entire range of measurements or in terms of the range. Calculated by formula

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

If the device is unilateral, i.e. The lower limit of measurements is zero, then X. n. It is determined equal to the upper limit of measurements;
- If the device is double-sided, then the rational value is equal to the width of the measurement range of the instrument.

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

Due to the occurrence

Tool / instrumentation errors - errors that are determined by the errors of the measured tools used and are caused by the imperfection of the principle of operation, the inaccuracy of the graduation of the scale, the belly of the device.
Methodical errors - errors caused by the imperfection of the method, as well as simplifications imposed on the basis of the methodology.
Subjective / Operator / Personal Error - errors caused by the degree of care, concentration, preparedness and other operator qualities.

The technique uses instruments for measuring only with a certain predetermined accuracy - the main error that is permissible normally in normal operating conditions for this instrument.

If the device works in conditions other than normal, an additional error occurs, which increases the overall error of the device. Additional errors include: temperature caused by a deviation of the ambient temperature from normal, installation, due to the deviation of the position of the device from the normal operating position, etc. For the normal ambient temperature take 20 ° C for normal atmospheric pressure 01,325 kPa.

The generalized characteristic of measuring instruments is the accuracy class, determined by the limit values \u200b\u200bof the allowed primary and additional errors, as well as other parameters affecting the accuracy of measuring instruments; The parameter value is established by standards for separate types of measurement. The accuracy class of measurement means characterizes their accuracy properties, but is not a direct indicator of measurement accuracy performed using these funds, since the accuracy also depends on the measurement method and the conditions of their execution. Measuring instruments, the limits of the allowed basic error of which are given in the form of the above basic (relative) errors, are assigned the accuracy classes selected from a number of the following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0 ; 5.0; 6.0) * 10n, where n \u003d 1; 0; -one; -2, etc.

By character of manifestation

Random error - Error changing (largest and on the sign) from measurement to the measurement. Random errors can be associated with imperfection of instruments (friction in mechanical devices, etc.), shaking in urban conditions, with the imperfection of the measurement object (for example, when measuring the diameter of a thin wire, which may not have a very round cross-section as a result of the imperfection of the manufacturing process ), with the peculiarities of the most measured value (for example, when measuring the number of elementary particles passing in a minute through the Geiger counter).
Systematic error - the error variable in time on a certain law (a special case is a constant error that does not change over time). Systematic errors can be associated with instrument errors (irregular scales, calibration, etc.), unrecorded experimenter.
Progressive (drift) error - Unpredictable error, slowly changing in time. It is a non-stationary random process.
Rough Error (Promach) - The error that arose as a result of the absence of an experimenter or malfunction of the equipment (for example, if the experimenter is incorrectly read the division number on the instrument scale, if a closure occurred in the electrical circuit).