Absolute and relative measurement errors. Test questions and exercises

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The absolute error of determination does not exceed 0 01 μg of phosphorus. We used this method 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 determining zinc in zinc-manganese ferrites using the proposed method does not exceed 0 2% rel.  

The absolute error in determining hydrocarbons C2 - C4, when their content in the gas is 0 2 - 5 0%, is 0 01 - 0 2%, respectively.  

Here Ау is the absolute error in determining r/, which results from the error Yes in determining 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 the cube root is simply one-third of the error in determining the number.  

More complex considerations are necessary when choosing a measure for comparisons of absolute errors in determining the time of the start of the accident TV - Ts, where Tv and Ts are the time of the reconstructed and real accident, respectively. By analogy, the average travel time of the pollution peak from the actual discharge to those monitoring points that recorded the accident during the passage of the pollution Tsm can be used here. 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 power, respectively. Finally, the relative error in determining the duration of an emergency release is characterized by the value rv - rs / re, where rv and rs are, respectively, the reconstructed and real duration of the accidents.  

More complex considerations are necessary when choosing a measure for comparisons of absolute errors in determining the time of the start of the accident TV - Ts, where Tv and Ts are the time of the reconstructed and real accident, respectively. By analogy, the average travel time of the pollution peak from the actual discharge to those monitoring points that recorded the accident during the passage of the pollution Tsm can be used here. 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 power, 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, respectively, the reconstructed and real duration of the accidents.  

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

Since the errors are based not on random, but on systematic errors, the final absolute error in determining the suction cups can reach 10% of the theoretically required amount of air. Only with unacceptably leaky fireboxes (A a0 25) does the generally accepted method give more or less satisfactory results. This is well known to service technicians who, when balancing the air balance of dense fireboxes, often receive negative suction values.  

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 sample capacity, weighing, and the relative error due to fluctuations in the sample mass around the equilibrium value.  

If all the rules for selecting, measuring volumes and analyzing gases using the GKhP-3 gas analyzer are observed, the total absolute error in determining the content of CO2 and O2 should not exceed 0 2 - 0 4% of their true value.  

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

Random errors can be absolute and relative. A random error having the dimension of the measured value is called the absolute error of determination. The arithmetic mean of the absolute errors of all individual measurements is called the absolute error of the analytical method.  

The value of the permissible deviation, or confidence interval, is not set arbitrarily, but is calculated from specific measurement data and the 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. Knowing the error of an individual measurement has no independent meaning, and in any seriously conducted experiment several parallel measurements must be carried out, from which the experimental error is calculated. Measurement errors, depending on the reasons for their occurrence, are divided into three types.  

It is almost impossible to determine the true value of a physical quantity absolutely accurately, because any measurement operation is associated with a number of errors or, in other words, inaccuracies. The reasons for errors can be very different. Their occurrence may be associated with inaccuracies in the manufacture and adjustment of the measuring device, due to the physical characteristics of the object under study (for example, when measuring the diameter of a wire of non-uniform thickness, the result randomly depends on the choice of the measurement site), random reasons, etc.

The experimenter’s task is to reduce their influence on the result, and also to indicate how close the result obtained is to the true one.

There are concepts of absolute and relative error.

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

∆x i =x i -x and (2)

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

The result of any physical measurement is usually written in the form:

where is the arithmetic mean value of the measured value, closest to the true value (the validity of x and≈ will be shown below), is the absolute measurement error.

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

Absolute error is a dimensional quantity; it has the same dimension as the measured quantity.

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

Relative error measurements is the ratio of the absolute error to the measured value itself:

Relative error is a dimensionless quantity. It is expressed as a percentage:

In the example above, the relative errors are 0.1% and 20%. They differ markedly from each other, although the absolute values ​​are the same. Relative error gives information about accuracy

Measurement errors

According to the nature of the manifestation and the reasons for the occurrence of errors, they can be divided into the following classes: instrumental, systematic, random, and misses (gross errors).

Errors are caused either by a malfunction of the device, or a violation of the methodology or experimental conditions, or are of a subjective nature. In practice, they are defined as results that differ sharply from others. To eliminate their occurrence, it is necessary to be careful and thorough when working with devices. Results containing errors must be excluded from consideration (discarded).

Instrument errors. If the measuring device is in good working order and adjusted, then it can be used to carry out measurements with limited accuracy determined by the type of device. It is customary to consider the instrument error of a pointer instrument to be equal to half the smallest division of its scale. In instruments with digital readout, the instrument error is equated to the value of one smallest digit of the instrument scale.

Systematic errors are errors whose magnitude and sign are constant for the entire series of measurements carried out by the same method and using the same measuring instruments.

When carrying out measurements, it is important not only to take into account systematic errors, but it is also necessary to ensure their elimination.

Systematic errors are conventionally divided into four groups:

1) errors, the nature of which is known and their magnitude can be determined quite accurately. Such an error is, for example, a change in the measured mass in the air, which depends on temperature, humidity, air pressure, etc.;

2) errors, the nature of which is known, but the magnitude of the error itself is unknown. Such errors include errors caused by the measuring device: a malfunction of the device itself, a scale that does not correspond to the zero value, or the accuracy class of the device;

3) errors, the existence of which may not be suspected, but their magnitude can often be significant. Such errors occur most often in complex measurements. A simple example of such an error is the measurement of the density of some sample containing a cavity inside;

4) errors caused by the characteristics of the measurement object itself. For example, when measuring the electrical conductivity of a metal, a piece of wire is taken from the latter. Errors can occur if there is any defect in the material - a crack, thickening of the wire or inhomogeneity that changes its resistance.

Random errors are errors that change randomly in sign and magnitude under identical conditions of repeated measurements of the same quantity.

Related information.

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 correct signs. Approximate numbers should be written down, keeping only the correct 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 for searching for the roots of 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 applies to any other physical parameters. This also includes the inaccuracy of the 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.

Instructions

First of all, take several measurements with an instrument of the same value in order to be able to obtain the actual value. The more measurements are taken, 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 real value of the quantity (real, since the true value cannot be found). To do this, add up the results obtained 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. It represents a qualitative characteristic of the accuracy of the research. According to the form of presentation, it can be absolute and relative.

You will need

• - calculator.

Instructions

The second arise from the influence of causes and are random in nature. These include incorrect rounding when calculating readings and influence. If such errors are significantly less than the scale divisions of this measuring device, then it is advisable to take half the division as the absolute error.

Miss or Rough error represents an observational result that differs sharply from all others.

Absolute error approximate numerical value is the difference between the result during measurement and the true value of the measured value. The true or actual value reflects the physical quantity being studied. This error is the simplest quantitative measure of error. It can be calculated using the following formula: ∆Х = Hisl - Hist. It can take on positive and negative meanings. For a better understanding, let's look at . The school has 1205 students, rounded to 1200 absolute error equals: ∆ = 1200 - 1205 = 5.

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

Sources:

• how to determine 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 value.

You will need

• -meter:
• -calculator.

Instructions

Errors may result from the influence of various factors. Among them are the imperfection of measurement tools or methods, inaccuracies in their manufacture, and failure to comply with special conditions when conducting research.

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

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

According to the conditions of occurrence, they distinguish between basic and additional. If the measurements were carried out under normal conditions, then the first type appears. Deviations caused by values ​​going beyond normal limits are additional. To evaluate it, the documentation usually establishes standards within which the value can change if the measurement conditions are violated.

Also, errors in physical measurements are divided into systematic, random and gross. The former are caused by factors that act when measurements are repeated many times. The second arise from the influence of reasons and character. A miss is an observation that differs sharply from all others.

Depending on the nature of the measured value, various methods of measuring error can be used. The first of them is the Kornfeld method. It is based on calculating a confidence interval ranging from the minimum to maximum result. The error in this case will be half the difference between these results: ∆x = (xmax-xmin)/2. Another method is to calculate the mean square error.

Measurements can be taken with varying degrees of 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 larger and smaller.

You will need

• - measurement data;
• - calculator.

Instructions

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 may be a transfer of the original measurement point.

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

Random errors affect the results of even highly accurate measurements. Therefore, any result will be more or less close to the absolute, but there will always be discrepancies. Determine this interval. It can be expressed by the formula (Xizm- ΔХ)≤Xizm ≤ (Xizm+ΔХ).

Determine the value that is closest to the value. In measurements, the 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 the reference instrument is accepted 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 specific measurement. Determine the difference ΔХ by subtracting the smaller from the larger. When determining the error, only the modulus of this difference is taken into account.

Please note

As a rule, in practice it is not possible to carry out absolutely accurate measurements. Therefore, the maximum error is taken as the reference value. It represents the maximum value of the absolute error module.

Useful advice

In practical measurements, half of the smallest division value is usually taken as the absolute error. When working with numbers, the absolute error is taken to be half the value of the digit, which is in the digit next to the exact digits.

To determine the accuracy class of an instrument, 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 imperfection of instruments, tools, and techniques. Accuracy also depends on the attentiveness and state of the experimenter. Errors are divided into absolute, relative and reduced.

Instructions

Let a single measurement of a quantity give the result x. The true value is denoted by x0. Then absolute errorΔx=|x-x0|. She evaluates 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 quantity 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, absolute error gives the boundaries of the interval in which the true value is located with some probability.

Direct and indirect measurements. In direct measurements, the desired value is immediately measured with the appropriate device. For example, bodies with a ruler, voltage with a voltmeter. In indirect measurements, a value is found using the formula for the relationship between it and the measured values.

If the result is a dependence on three directly measured quantities having 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 for each of the directly measured quantities.

Useful advice

Errors are gross inaccuracies in measurements that occur due to malfunction of instruments, inattentiveness of the experimenter, or violation of the experimental methodology. To reduce the likelihood of such errors, be careful when taking measurements and write down the results obtained in detail.

Sources:

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

The result of any measurement is inevitably accompanied by a deviation from the true value. The measurement error can be calculated in several ways depending on its type, for example, by statistical methods of determining the confidence interval, standard deviation, etc.

Measurement error

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

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

Where X n- normalizing value, which depends on the type of scale of the measuring device and is determined by its calibration:

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

The given error is a dimensionless quantity (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 imperfections in the operating principle, inaccuracy of scale calibration, and 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 technology, instruments are used to measure only with a certain predetermined accuracy - the main error allowed by the normal under normal operating conditions for a given device.

If the device operates under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by a deviation of the ambient temperature from normal, installation, caused by a deviation of the device’s position from the normal operating position, etc. The normal ambient temperature is 20°C, and the normal atmospheric pressure is 01.325 kPa.

A generalized characteristic of measuring instruments is the accuracy class, determined by the maximum permissible main and additional errors, as well as other parameters affecting the accuracy of measuring instruments; the meaning of the parameters is established by standards for certain types of measuring instruments. The accuracy class of measuring instruments characterizes their precision 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 permissible basic error of which are specified in the form of the given basic (relative) errors, are assigned 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 = 1; 0; -1; -2, etc.

By nature of manifestation

• Random error- error that varies (in magnitude and sign) from measurement to measurement. Random errors may be associated with imperfection of instruments (friction in mechanical devices, etc.), shaking in urban conditions, with imperfection of the measurement object (for example, when measuring the diameter of a thin wire, which may not have a completely round cross-section as a result of imperfections in the manufacturing process ), with the characteristics of the measured quantity itself (for example, when measuring the number of 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 may be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
• Progressive (drift) error- an unpredictable error that changes slowly over time. It is a non-stationary random process.
• Gross error (miss)- an error resulting from an oversight by the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the division number on the instrument scale, if a short circuit occurred in the electrical circuit).

By measurement method

• Direct measurement error
• Error of indirect measurements- error of the calculated (not directly measured) quantity:

If F = F(x 1 ,x 2 ...x n) , Where x i- directly measured independent quantities with an error Δ x i, Then:

See also

• Measurement of physical quantities
• System for automated data collection from meters via radio channel

Literature

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

• Laboratory classes in physics. Textbook/Goldin L.L., Igoshin F.F., Kozel S.M. et al.; edited by Goldina L.L. - M.: Science. Main editorial office of physical and mathematical literature, 1983. - 704 p.

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