How to calculate the relative measurement error. The concept of relative error. Values ​​of quantiles of Student's distribution t(n) with confidence

X = X avg X

Absolute

error

Relative

error

a+ b

a+ b

In our age, man has invented and uses a huge variety of all kinds of measuring instruments. But no matter how perfect the technology for their manufacture is, they all have a greater or lesser error. This parameter, as a rule, is indicated on the instrument itself, and to assess the accuracy of the value being determined, you need to be able to understand what the numbers indicated on the marking mean. In addition, relative and absolute errors inevitably arise during complex mathematical calculations. It is widely used in statistics, industry (quality control) and in a number of other areas. How this value is calculated and how to interpret its value - this is exactly what will be discussed in this article.

Absolute error

Let us denote by x the approximate value of a quantity obtained, for example, through a single measurement, and by x 0 its exact value. Now let's calculate the magnitude of the difference between these two numbers. The absolute error is exactly the value that we got as a result of this simple operation. In the language of formulas, this definition can be written in this form: Δ x = | x - x 0 |.

Relative error

Absolute deviation has one important drawback - it does not allow assessing the degree of importance of the error. For example, we buy 5 kg of potatoes at the market, and dishonest seller When measuring weight, I made a mistake of 50 grams in my favor. That is, the absolute error was 50 grams. For us, such an oversight will be a mere trifle and we will not even pay attention to it. Imagine what will happen if a similar error occurs while preparing the medicine? Here everything will be much more serious. And when loading a freight car, deviations are likely to occur much larger than this value. Therefore, the absolute error itself is not very informative. In addition to it, the relative deviation is very often additionally calculated, equal to the ratio absolute error to the exact value of the number. This is written by the following formula: δ = Δ x / x 0 .

Error Properties

Suppose we have two independent quantities: 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 the pre-calculated absolute deviations of each of them. In some measurements, it may happen that errors in the determination of x and y values ​​cancel each other out. Or it may happen that as a result of addition, the deviations become maximally intensified. Therefore, when total absolute error is calculated, the worst-case scenario must be considered. The same is true for the difference between errors of several quantities. This property is characteristic only of absolute error, and cannot be applied to relative deviation, since this will inevitably lead to an incorrect result. Let's look at this situation using the following example.

Suppose measurements inside the cylinder showed that the inner radius (R 1) is 97 mm, and the outer radius (R 2) is 100 mm. It is necessary to determine the thickness of its wall. First, let's find the difference: h = R 2 - R 1 = 3 mm. If the problem does not indicate what the absolute error is, then it is taken as half the scale division of the measuring device. Thus, Δ(R 2) = Δ(R 1) = 0.5 mm. The total absolute error is: Δ(h) = Δ(R 2) + Δ(R 1) = 1 mm. Now let’s calculate the relative deviation of all values:

δ(R 1) = 0.5/100 = 0.005,

δ(R 1) = 0.5/97 ≈ 0.0052,

δ(h) = Δ(h)/h = 1/3 ≈ 0.3333>> δ(R 1).

As you can see, the error in measuring both radii does not exceed 5.2%, and the error in calculating their difference - the thickness of the cylinder wall - was as much as 33.(3)%!

The following property states: the relative deviation of the product of several numbers is approximately equal to the sum of the relative deviations of the individual factors:

δ(xy) ≈ δ(x) + δ(y).

Moreover this rule is true regardless of the number of values ​​being evaluated. The third and final property of relative error is that relative score kth numbers degree approximately in | k | times the relative error of the original number.

Physical quantities are characterized by the concept of “error accuracy”. There is a saying that by taking measurements you can come to knowledge. This way you can find out the height of the house or the length of the street, like many others.

Introduction

Let us understand the meaning of the concept of “measure a quantity”. The measurement process is to compare it with homogeneous quantities, which are taken as a unit.

Liters are used to determine volume, grams are used to calculate mass. To make calculations more convenient, the SI system of international classification of units was introduced.

For measuring the length of the stick in meters, mass - kilograms, volume - cubic liters, time - seconds, speed - meters per second.

When calculating physical quantities it is not always necessary to use traditional way, it is enough to apply the calculation using the formula. For example, to calculate indicators such as average speed, you need to divide the distance traveled by the time spent on the road. This is how the average speed is calculated.

When using units of measurement that are ten, one hundred, thousand times higher than the accepted measurement units, they are called multiples.

The name of each prefix corresponds to its multiplier number:

1. Deca.
2. Hecto.
3. Kilo.
4. Mega.
5. Giga.
6. Tera.

In physical science, powers of 10 are used to write such factors. For example, a million is written as 10 6 .

In a simple ruler, length has a unit of measurement - centimeters. She's 100 times less than a meter. A 15 cm ruler is 0.15 m long.

A ruler is the simplest type of measuring instrument for measuring lengths. More complex devices are represented by a thermometer - to a hygrometer - to determine humidity, an ammeter - to measure the level of force with which electric current propagates.

How accurate will the measurements be?

Take a ruler and a simple pencil. Our task is to measure the length of this stationery.

First you need to determine what the division price indicated on the scale is measuring instrument. On the two divisions, which are the closest strokes of the scale, numbers are written, for example, “1” and “2”.

It is necessary to count how many divisions are between these numbers. If counted correctly it will be "10". Let us subtract from the number that is larger the number that will be smaller and divide by the number that is the division between the digits:

(2-1)/10 = 0.1 (cm)

So we determine that the price that determines the division of stationery is the number 0.1 cm or 1 mm. It is clearly shown how the price indicator for division is determined using any measuring device.

When measuring a pencil with a length that is slightly less than 10 cm, we will use the knowledge gained. If there were no fine divisions on the ruler, it would be concluded that the object has a length of 10 cm. This approximate value is called the measurement error. It indicates the level of inaccuracy that can be tolerated when making measurements.

Determining the length parameters of a pencil with more high level accuracy, greater dividing costs achieve greater measuring accuracy, which provides less error.

In this case, absolutely accurate measurements cannot be taken. And the indicators should not exceed the size of the division price.

It has been established that the measurement error is ½ of the price, which is indicated on the graduations of the device used to determine the dimensions.

After taking measurements of a pencil of 9.7 cm, we will determine its error indicators. This is the interval 9.65 - 9.85 cm.

The formula that measures this error is the calculation:

A = a ± D (a)

A - in the form of a quantity for measuring processes;

a is the value of the measurement result;

D - designation of absolute error.

When subtracting or adding values ​​with an error, the result will be equal to the sum of the error indicators, which is each individual value.

Introduction to the concept

If we consider depending on the method of its expression, we can distinguish the following varieties:

• Absolute.
• Relative.
• Given.

The absolute measurement error is indicated by the letter “Delta” in capital. This concept is defined as the difference between the measured and actual values ​​of the physical quantity that is being measured.

The expression of absolute measurement error is the units of the quantity that needs to be measured.

When measuring mass, it will be expressed, for example, in kilograms. This is not a measurement accuracy standard.

How to calculate the error of direct measurements?

There are ways to depict measurement errors and calculate them. To do this, it is important to be able to determine a physical quantity with the required accuracy, to know what the absolute measurement error is, that no one will ever be able to find it. Only its boundary value can be calculated.

Even if this term is used conventionally, it indicates precisely the boundary data. Absolute and relative measurement errors are indicated by the same letters, the difference is in their spelling.

When measuring length, the absolute error will be measured in the units in which the length is calculated. And the relative error is calculated without dimensions, since it is the ratio of the absolute error to the measurement result. This value is often expressed as a percentage or fraction.

Absolute and relative measurement errors have several different methods of calculation, depending on what physical quantity.

Concept of direct measurement

The absolute and relative errors of direct measurements depend on the accuracy class of the device and the ability to determine the weighing error.

Before we talk about how the error is calculated, it is necessary to clarify the definitions. Direct measurement is a measurement in which the result is directly read from the instrument scale.

When we use a thermometer, ruler, voltmeter or ammeter, we always carry out direct measurements, since we directly use a device with a scale.

There are two factors that influence the effectiveness of the readings:

• Instrument error.
• The error of the reference system.

The absolute error limit for direct measurements will be equal to the sum of the error that the device shows and the error that occurs during the counting process.

D = D (flat) + D (zero)

Example with a medical thermometer

The error indicators are indicated on the device itself. A medical thermometer has an error of 0.1 degrees Celsius. The counting error is half the division value.

D ots. = C/2

If the division value is 0.1 degrees, then for medical thermometer you can make calculations:

D = 0.1 o C + 0.1 o C / 2 = 0.15 o C

On the back of the scale of another thermometer there is a specification and it is indicated that for correct measurements it is necessary to immerse the entire back of the thermometer. not specified. All that remains is the counting error.

If the scale division value of this thermometer is 2 o C, then it is possible to measure temperature with an accuracy of 1 o C. These are the limits of the permissible absolute measurement error and the calculation of the absolute measurement error.

A special system for calculating accuracy is used in electrical measuring instruments.

Accuracy of electrical measuring instruments

To specify the accuracy of such devices, a value called accuracy class is used. The letter “Gamma” is used to designate it. To accurately determine the absolute and relative measurement error, you need to know the accuracy class of the device, which is indicated on the scale.

Let's take an ammeter for example. Its scale indicates the accuracy class, which shows the number 0.5. It is suitable for measurements at constant and alternating current, refers to devices of the electromagnetic system.

This is enough precision instrument. If you compare it with a school voltmeter, you can see that it has an accuracy class of 4. You must know this value for further calculations.

Application of knowledge

Thus, D c = c (max) X γ /100

We will use this formula for specific examples. Let's use a voltmeter and find the error in measuring the voltage provided by the battery.

Let's connect the battery directly to the voltmeter, first checking whether the needle is at zero. When connecting the device, the needle deviated by 4.2 divisions. This state can be characterized as follows:

1. It can be seen that the maximum value of U for of this subject equals 6.
2. Accuracy class -(γ) = 4.
3. U(o) = 4.2 V.
4. C=0.2 V

Using these formula data, the absolute and relative measurement error is calculated as follows:

D U = DU (ex.) + C/2

D U (ex.) = U (max) X γ /100

D U (ex.) = 6 V X 4/100 = 0.24 V

This is the error of the device.

The calculation of the absolute measurement error in this case will be performed as follows:

D U = 0.24 V + 0.1 V = 0.34 V

Using the formula discussed above, you can easily find out how to calculate the absolute measurement error.

There is a rule for rounding errors. It allows you to find the average between the absolute and relative error limits.

Learning to determine weighing error

This is one example of direct measurements. Weighing has a special place. After all, lever scales do not have a scale. Let's learn how to determine the error of such a process. The accuracy of mass measurement is influenced by the accuracy of the weights and the perfection of the scales themselves.

We use lever scales with a set of weights that must be placed on the right pan of the scale. To weigh, take a ruler.

Before starting the experiment, you need to balance the scales. Place the ruler on the left bowl.

The mass will be equal to the sum of the installed weights. Let us determine the error in measuring this quantity.

D m = D m (scales) + D m (weights)

The error in mass measurement consists of two terms associated with scales and weights. To find out each of these values, factories producing scales and weights provide products with special documents that allow the accuracy to be calculated.

Using tables

Let's use a standard table. The error of the scale depends on what mass is put on the scale. The larger it is, the correspondingly larger the error.

Even if you put a very light body, there will be an error. This is due to the friction process occurring in the axes.

The second table is for a set of weights. It indicates that each of them has its own mass error. The 10 gram has an error of 1 mg, the same as the 20 gram. Let's calculate the sum of the errors of each of these weights taken from the table.

It is convenient to write the mass and mass error in two lines, which are located one below the other. The smaller the weights, the more accurate the measurement.

Results

In the course of the material reviewed, it was established that it is impossible to determine the absolute error. You can only set its boundary indicators. To do this, use the formulas described above in the calculations. This material proposed for study at school for students in grades 8-9. Based on the knowledge gained, you can solve problems to determine the absolute and relative errors.

Due to the errors inherent in the measuring instrument, the chosen method and measurement procedure, differences external conditions, in which the measurement is performed, for established and other reasons, the result of almost every measurement is burdened with error. This error is calculated or estimated and assigned to the result obtained.

Measurement result error(in short - measurement error) - the deviation of the measurement result from the true value of the measured value.

The true value of the quantity remains unknown due to the presence of errors. It is used in solving theoretical problems of metrology. In practice, the actual value of the quantity is used, which replaces the true value.

The measurement error (Δx) is found by the formula:

x = x meas. - x valid (1.3)

where x meas. - the value of the quantity obtained on the basis of measurements; x valid — the value of the quantity taken as real.

For single measurements, the actual value is often taken to be the value obtained using a standard measuring instrument; for multiple measurements, the arithmetic mean of the values ​​of individual measurements included in a given series.

Measurement errors can be classified according to the following criteria:

By the nature of the manifestations - systematic and random;

According to the method of expression - absolute and relative;

According to the conditions of change in the measured value - static and dynamic;

According to the method of processing a number of measurements - arithmetic averages and root mean squares;

According to the completeness of coverage of the measurement task - partial and complete;

In relation to a unit of physical quantity - errors in reproducing the unit, storing the unit and transmitting the size of the unit.

Systematic measurement error(in short - systematic error) - a component of the error of a measurement result that remains constant for a given series of measurements or changes naturally with repeated measurements of the same physical quantity.

According to the nature of their manifestation, systematic errors are divided into permanent, progressive and periodic. Constant systematic errors(in short - constant errors) - errors, long time retaining their value (for example, throughout the entire series of measurements). This is the most common type of error.

Progressive systematic errors(in short - progressive errors) - continuously increasing or decreasing errors (for example, errors from wear of measuring tips that come into contact with the part during the grinding process when monitoring it with an active control device).

Periodic systematic error(briefly - periodic error) - an error, the value of which is a function of time or a function of the movement of the pointer of a measuring device (for example, the presence of eccentricity in goniometer devices with a circular scale causes a systematic error that varies according to a periodic law).

Based on the reasons for the appearance of systematic errors, a distinction is made between instrumental errors, method errors, subjective errors and errors due to deviations of external measurement conditions from those established by the methods.

Instrumental measurement error(in short - instrumental error) is a consequence of a number of reasons: wear of device parts, excessive friction in the device mechanism, inaccurate marking of strokes on the scale, discrepancy between the actual and nominal values ​​of the measure, etc.

Measurement method error(in short - method error) may arise due to the imperfection of the measurement method or its simplifications established by the measurement methodology. For example, such an error may be due to insufficient performance of the measuring instruments used when measuring the parameters of fast processes or unaccounted for impurities when determining the density of a substance based on the results of measuring its mass and volume.

Subjective measurement error(in short - subjective error) is due to the individual errors of the operator. This error is sometimes called personal difference. It is caused, for example, by a delay or advance in the operator's acceptance of a signal.

Error due to deviation(in one direction) the external measurement conditions from those established by the measurement technique leads to the emergence of a systematic component of the measurement error.

Systematic errors distort the measurement result, so they must be eliminated as far as possible by introducing corrections or adjusting the device to bring systematic errors to an acceptable minimum.

Unexcluded systematic error(in short - non-excluded error) is the error of the measurement result, due to the error in calculation and introduction of a correction for the action of a systematic error, or a small systematic error, the correction for which is not introduced due to its smallness.

Sometimes this type of error is called non-excluded residuals of systematic error(in short - non-excluded balances). For example, when measuring the length of a line meter in wavelengths of reference radiation, several non-excluded systematic errors were identified (i): due to inaccurate temperature measurement - 1; due to inaccurate determination of the refractive index of air - 2, due to inaccurate wavelength - 3.

Usually the sum of non-excluded systematic errors is taken into account (their boundaries are set). When the number of terms is N ≤ 3, the limits of non-excluded systematic errors are calculated using the formula

When the number of terms is N ≥ 4, the formula is used for calculations

(1.5)

where k is the coefficient of dependence of non-excluded systematic errors on the selected confidence probability P when they are uniformly distributed. At P = 0.99, k = 1.4, at P = 0.95, k = 1.1.

Random measurement error(in short - random error) - a component of the error of a measurement result that changes randomly (in sign and value) in a series of measurements of the same size of a physical quantity. Reasons for random errors: rounding errors when taking readings, variation in readings, changes in random measurement conditions, etc.

Random errors cause scattering of measurement results in a series.

The theory of errors is based on two principles, confirmed by practice:

1. With a large number of measurements, random errors of the same numerical value, but different sign, occur equally often;

2. Large (in absolute value) errors are less common than small ones.

From the first position follows an important conclusion for practice: as the number of measurements increases, the random error of the result obtained from a series of measurements decreases, since the sum of the errors of individual measurements of a given series tends to zero, i.e.

(1.6)

For example, as a result of measurements, a number of values ​​were obtained electrical resistance(corrected for systematic errors): R 1 = 15.5 Ohm, R 2 = 15.6 Ohm, R 3 = 15.4 Ohm, R 4 = 15.6 Ohm and R 5 = 15.4 Ohm . Hence R = 15.5 Ohm. Deviations from R (R 1 = 0.0; R 2 = +0.1 Ohm, R 3 = -0.1 Ohm, R 4 = +0.1 Ohm and R 5 = -0.1 Ohm) are random errors of individual measurements in this series. It is easy to verify that the sum R i = 0.0. This indicates that the errors in individual measurements of this series were calculated correctly.

Despite the fact that as the number of measurements increases, the sum of random errors tends to zero (in this example it accidentally turned out to be zero), the random error of the measurement result must be assessed. In the theory of random variables, the dispersion o2 serves as a characteristic of the dispersion of the values ​​of a random variable. "|/o2 = a is called the mean square deviation of the population or standard deviation.

It is more convenient than dispersion, since its dimension coincides with the dimension of the measured quantity (for example, the value of the quantity is obtained in volts, the standard deviation will also be in volts). Since in measurement practice we deal with the term “error,” the derivative term “mean square error” should be used to characterize a number of measurements. A characteristic of a series of measurements can be the arithmetic mean error or the range of measurement results.

The range of measurement results (span for short) is the algebraic difference between the largest and smallest results of individual measurements, forming a series (or sample) of n measurements:

R n = X max - X min (1.7)

where R n is the range; X max and X min - the greatest and smallest value values ​​in a given series of measurements.

For example, out of five measurements of the hole diameter d, the values ​​R 5 = 25.56 mm and R 1 = 25.51 mm turned out to be its maximum and minimum values. In this case, R n = d 5 - d 1 = 25.56 mm - 25.51 mm = 0.05 mm. This means that the remaining errors in this series are less than 0.05 mm.

Arithmetic mean error of an individual measurement in a series(briefly - arithmetic mean error) - a generalized characteristic of the scattering (due to random reasons) of individual measurement results (of the same quantity) included in a series of n equal-precision independent measurements, calculated by the formula

(1.8)

where X i is the result of the i-th measurement included in the series; x is the arithmetic mean of n values: |Х і - X| — absolute value errors of the i-th measurement; r is the arithmetic mean error.

The true value of the average arithmetic error p is determined from the relation

p = lim r, (1.9)

With the number of measurements n > 30 between the arithmetic mean (r) and the root mean square (s) there are correlations between errors

s = 1.25 r; r and= 0.80 s. (1.10)

The advantage of the arithmetic mean error is the simplicity of its calculation. But still, the mean square error is more often determined.

Mean square error individual measurement in a series (in short - mean square error) - a generalized characteristic of the scattering (due to random reasons) of individual measurement results (of the same value) included in a series of P equal-precision independent measurements, calculated by the formula

(1.11)

The mean square error for the general sample o, which is the statistical limit S, can be calculated at /i-mx > using the formula:

Σ = lim S (1.12)

In reality, the number of measurements is always limited, so it is not σ , and its approximate value (or estimate), which is s. The more P, the closer s is to its limit σ .

With a normal distribution law, the probability that the error of an individual measurement in a series will not exceed the calculated mean square error is small: 0.68. Therefore, in 32 cases out of 100 or 3 cases out of 10, the actual error may be greater than the calculated one.

Figure 1.2 Decrease in the value of the random error of the result of multiple measurements with an increase in the number of measurements in a series

In a series of measurements, there is a relationship between the root mean square error of an individual measurement s and the root mean square error of the arithmetic mean S x:

which is often called the “U n rule”. From this rule it follows that the measurement error due to random causes can be reduced by n times if n measurements of the same size of any quantity are performed, and the arithmetic mean is taken as the final result (Fig. 1.2).

Performing at least 5 measurements in a series makes it possible to reduce the influence of random errors by more than 2 times. With 10 measurements, the influence of random error is reduced by 3 times. A further increase in the number of measurements is not always economically feasible and, as a rule, is carried out only for critical measurements that require high accuracy.

The root mean square error of a single measurement from a number of homogeneous double measurements S α is calculated by the formula

(1.14)

where x" i and x"" i are the i-th results of measurements of the same size quantity in the forward and reverse directions with one measuring instrument.

In case of unequal measurements, the root mean square error of the arithmetic average in the series is determined by the formula

(1.15)

where p i is the weight of the i-th measurement in a series of unequal measurements.

The root mean square error of the result of indirect measurements of the value Y, which is a function of Y = F (X 1, X 2, X n), is calculated using the formula

(1.16)

where S 1, S 2, S n are the root mean square errors of the measurement results of the quantities X 1, X 2, X n.

If, for greater reliability in obtaining a satisfactory result, several series of measurements are carried out, the root mean square error of an individual measurement from m series (S m) is found by the formula

(1.17)

Where n is the number of measurements in the series; N— total number measurements in all series; m is the number of series.

With a limited number of measurements, it is often necessary to know the root mean square error. To determine the error S, calculated by formula (2.7), and the error S m, calculated by formula (2.12), you can use the following expressions

(1.18)

(1.19)

where S and S m are the mean square errors of S and S m , respectively.

For example, when processing the results of a number of measurements of length x, we obtained

= 86 mm 2 at n = 10,

= 3.1 mm

= 0.7 mm or S = ±0.7 mm

The value S = ±0.7 mm means that due to the calculation error, s is in the range from 2.4 to 3.8 mm, therefore tenths of a millimeter are unreliable here. In the case considered, we must write: S = ±3 mm.

To have greater confidence in assessing the error of a measurement result, calculate the confidence error or confidence limits of the error. Under the normal distribution law, the confidence limits of the error are calculated as ±t-s or ±t-s x, where s and s x are the mean square errors, respectively, of an individual measurement in the series and the arithmetic mean; t is a number depending on the confidence probability P and the number of measurements n.

An important concept is the reliability of the measurement result (α), i.e. the probability that the desired value of the measured quantity will fall within a given confidence interval.

For example, when processing parts on machine tools in a stable technological mode, the distribution of errors obeys the normal law. Let's assume that the part length tolerance is set to 2a. In this case, the confidence interval in which the desired value of the length of the part a is located will be (a - a, a + a).

If 2a = ±3s, then the reliability of the result is a = 0.68, i.e. in 32 cases out of 100 one should expect the part size to exceed tolerance 2a. When assessing the quality of a part according to a tolerance of 2a = ±3s, the reliability of the result will be 0.997. In this case, we can expect only three parts out of 1000 to exceed the established tolerance. However, an increase in reliability is possible only by reducing the error in the length of the part. Thus, to increase reliability from a = 0.68 to a = 0.997, the error in the length of the part must be reduced by three times.

Recently, the term “measurement reliability” has become widespread. In some cases, it is unreasonably used instead of the term “measurement accuracy.” For example, in some sources you can find the expression “establishing the unity and reliability of measurements in the country.” Whereas it would be more correct to say “establishing the unity and required accuracy of measurements.” We consider reliability as a qualitative characteristic that reflects the proximity to zero of random errors. It can be quantitatively determined through the unreliability of measurements.

Unreliability of measurements(in short - unreliability) - an assessment of the discrepancy between the results in a series of measurements due to the influence of the total influence of random errors (determined by statistical and non-statistical methods), characterized by the range of values ​​in which the true value of the measured value is located.

In accordance with the recommendations of the International Bureau of Weights and Measures, unreliability is expressed in the form of a total mean square measurement error - Su, including the mean square error S (determined by statistical methods) and the mean square error u (determined by non-statistical methods), i.e.

(1.20)

Maximum measurement error(briefly - maximum error) - the maximum measurement error (plus, minus), the probability of which does not exceed the value P, while the difference 1 - P is insignificant.

For example, with a normal distribution law, the probability of a random error equal to ±3s is 0.997, and the difference 1-P = 0.003 is insignificant. Therefore, in many cases, the confidence error of ±3s is taken as the maximum, i.e. pr = ±3s. If necessary, pr may have other relationships with s at a sufficiently large P (2s, 2.5s, 4s, etc.).

Due to the fact that in the GSI standards, instead of the term “mean square error,” the term “mean square deviation” is used, in further discussions we will adhere to this very term.

Absolute measurement error(in short - absolute error) - measurement error expressed in units of the measured value. Thus, the error X in measuring the length of a part X, expressed in micrometers, represents an absolute error.

The terms “absolute error” and “absolute value of error” should not be confused, which is understood as the value of the error without taking into account the sign. So, if the absolute measurement error is ±2 μV, then the absolute value of the error will be 0.2 μV.

Relative measurement error(in short - relative error) - measurement error, expressed in fractions of the value of the measured value or as a percentage. The relative error δ is found from the relations:

(1.21)

For example, there is a real value of the part length x = 10.00 mm and an absolute value of the error x = 0.01 mm. The relative error will be

Static error— error of the measurement result due to the conditions of static measurement.

Dynamic error— error of the measurement result due to the conditions of dynamic measurement.

Unit reproduction error— error in the result of measurements performed when reproducing a unit of physical quantity. Thus, the error in reproducing a unit using a state standard is indicated in the form of its components: the non-excluded systematic error, characterized by its boundary; random error characterized by standard deviation s and instability over the year ν.

Unit size transmission error— error in the result of measurements performed when transmitting the size of a unit. The error in transmitting the unit size includes non-excluded systematic errors and random errors of the method and means of transmitting the unit size (for example, a comparator).

Essay

Absolute and relative error

Introduction

Absolute error - is an estimate of the absolute measurement error. Calculated different ways. The calculation method is determined by the distribution of the random variable. Accordingly, the magnitude of the absolute error depending on the distribution of the random variable may be different. If is the measured value, and is the true value, then the inequality must be fulfilled with some probability close to 1. If random value is distributed according to a normal law, then its standard deviation is usually taken as the absolute error. Absolute error is measured in the same units as the quantity itself.

There are several ways to write a quantity along with its absolute error.

· Signed notation is usually used ± . For example, the 100 meter record, set in 1983, is 9.930±0.005 s.

· To record quantities measured with very high accuracy, a different notation is used: the numbers corresponding to the error of the last digits of the mantissa are added in brackets. For example, the measured value of Boltzmann's constant is 1,380 6488 (13)×10?23 J/C, which can also be written much longer as 1,380 6488×10?23 ± 0.000 0013×10?23 J/C.

Relative error- measurement error, expressed as the ratio of the absolute measurement error to the actual or average value of the measured value (RMG 29-99):.

The relative error is a dimensionless quantity or measured as a percentage.

1. What is an approximate value?

With excess and insufficient? In the process of calculations, one often has to deal with approximate numbers. Let A- the exact value of a certain quantity, hereinafter called exact number A.Under the approximate value A,or approximate numberscalled number A, replacing the exact value of the quantity A.If A< A,That Acalled the approximate value of the number And for lack.If A> A,- That by excess.For example, 3.14 is an approximation of the number ? by deficiency, and 3.15 - by excess. To characterize the degree of accuracy of this approximation, the concept is used errors or errors.

Error ?Aapproximate number Acalled a difference of the form

?a = A - a,

Where A- the corresponding exact number.

From the figure it can be seen that the length of segment AB is between 6 cm and 7 cm.

This means that 6 is an approximate value of the length of segment AB (in centimeters) > with a deficiency, and 7 with an excess.

Denoting the length of the segment by the letter y, we get: 6< у < 1. Если a < х < b, то а называют приближенным значением числа х с недостатком, a b - приближенным значением х с избытком. Длина segmentAB (see Fig. 149) is closer to 6 cm than to 7 cm. It is approximately equal to 6 cm. They say that the number 6 was obtained by rounding the length of the segment to whole numbers.

. What is approximation error?

A) Absolute?

B) Relative?

A) The absolute error of the approximation is the magnitude of the difference between the true value of a quantity and its approximate value. |x - x_n|, where x is the true value, x_n is the approximate value. For example: The length of a sheet of A4 paper is (29.7 ± 0.1) cm. And the distance from St. Petersburg to Moscow is (650 ± 1) km. The absolute error in the first case does not exceed one millimeter, and in the second - one kilometer. The question is to compare the accuracy of these measurements.

If you think that the length of the sheet is measured more accurately because the absolute error does not exceed 1 mm. Then you are wrong. These values ​​cannot be directly compared. Let's do some reasoning.

When measuring the length of a sheet, the absolute error does not exceed 0.1 cm per 29.7 cm, that is, in percentage this is 0.1/29.7 *100% = 0.33% of the measured value.

When we measure the distance from St. Petersburg to Moscow, the absolute error does not exceed 1 km per 650 km, which as a percentage is 1/650 * 100% = 0.15% of the measured value. We see that the distance between cities is measured more accurately than the length of an A4 sheet.

B) The relative approximation error is the ratio of the absolute error to the absolute value of the approximate value of a quantity.

mathematical error fraction

where x is the true value, x_n is the approximate value.

Relative error is usually expressed as a percentage.

Example. Rounding the number 24.3 to units gives the number 24.

The relative error is equal. They say that the relative error in this case is 12.5%.

) What kind of rounding is called rounding?

A) With a disadvantage?

B) In excess?

A) Rounding down

When rounding a number expressed as a decimal fraction to the nearest 10^(-n), the first n decimal places are retained and the subsequent ones are discarded.

For example, rounding 12.4587 to the nearest thousandth, we get 12.458.

B) Rounding up

When rounding a number expressed as a decimal fraction to the nearest 10^(-n), the first n decimal places are retained in excess, and the subsequent ones are discarded.

For example, rounding 12.4587 to the nearest thousandth, we get 12.459.

) Rule for rounding decimals.

Rule. To round decimal to a certain digit of the integer or fractional part, all smaller digits are replaced by zeros or discarded, and the digit preceding the digit discarded during rounding does not change its value if it is followed by the numbers 0, 1, 2, 3, 4, and is increased by 1 (one) , if the numbers are 5, 6, 7, 8, 9.

Example. Round the fraction 93.70584 to:

ten thousandths: 93.7058

thousandths: 93.706

hundredths: 93.71

tenths: 93.7

whole number: 94

tens: 90

Despite equality absolute errors, because the measured quantities are different. The larger the measured size, the smaller the relative error while the absolute error remains constant.

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