X \u003d X cf X

Absolute

error

Relative

error

a+ b

a+ b

a+ b

Physical quantities are characterized by the concept of "error accuracy". There is a saying that by taking measurements one can come to knowledge. So it will be possible to find out what is the height of the house or the length of the street, like many others.

Introduction

Let's understand the meaning of the concept of "measure the value." 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 it more convenient to make calculations, we introduced the SI system of the international classification of units.

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

When calculating physical quantities, it is not always necessary to use the traditional method; it is enough to apply the calculation using a 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.

Using units of measurement that are ten, one hundred, one thousand times higher than the indicators of the accepted measuring 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, a power of 10 is used to write such factors. For example, a million is denoted as 10 6 .

In a simple ruler, the length has a unit of measure - a centimeter. It is 100 times smaller than a meter. A 15 cm ruler is 0.15 m long.

A ruler is the simplest type of measuring instrument for measuring length. More complex devices are represented by a thermometer - so that a hygrometer - to determine humidity, an ammeter - to measure the level of force with which an 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 is the division value indicated on the scale of the measuring device. On the two divisions, which are the nearest strokes of the scale, numbers are written, for example, "1" and "2".

It is necessary to calculate how many divisions are enclosed in the interval of these numbers. If you count correctly, you get "10". Subtract from the number that is greater, the number that will be less, and divide by the number that makes up the divisions 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.

By measuring a pencil with a length that is slightly less than 10 cm, we will use the knowledge gained. If there were no small divisions on the ruler, the conclusion would follow 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 in the measurement.

By specifying the length of a pencil with a higher level of accuracy, a larger division value achieves a greater measurement accuracy, which provides a smaller error.

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

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

After measuring the pencil at 9.7 cm, we determine the indicators of its error. This is a gap of 9.65 - 9.85 cm.

The formula that measures such an error is the calculation:

A = a ± D (a)

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

a - the value of the measurement result;

D - the designation of the 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 way it is expressed, we can distinguish the following varieties:

• Absolute.
• Relative.
• Given.

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

The expression of the 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 represent measurement errors and calculate them. To do this, it is important to be able to determine the physical quantity with the required accuracy, to know what the absolute measurement error is, that no one will ever be able to find it. You can only calculate its boundary value.

Even if this term is conditionally used, 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 those 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 fractions.

The absolute and relative measurement errors have several different ways of calculating, depending on what physical quantities.

The concept of direct measurement

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

Before talking about how the error is calculated, it is necessary to clarify the definitions. A 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 use a device with a scale directly.

There are two factors that affect performance:

• 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 reading process.

D = D (pr.) + D (absent)

Medical thermometer example

Accuracy values ​​are indicated on the instrument itself. An error of 0.1 degrees Celsius is registered on a medical thermometer. The reading error is half the division value.

D = C/2

If the division value is 0.1 degrees, then for a medical thermometer, calculations can be made:

D \u003d 0.1 o C + 0.1 o C / 2 \u003d 0.15 o C

On the back side of the scale of another thermometer there is a technical specification and it is indicated that for the correct measurements it is necessary to immerse the thermometer with the entire back part. not specified. The only remaining error is the counting error.

If the division value of the scale of this thermometer is 2 o C, then you can measure the 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 the accuracy class is used. For its designation, the letter "Gamma" is used. To accurately determine the absolute and relative measurement errors, you need to know the accuracy class of the device, which is indicated on the scale.

Take, for example, an ammeter. Its scale indicates the accuracy class, which shows the number 0.5. It is suitable for measurements on direct and alternating current, refers to the devices of the electromagnetic system.

This is a fairly accurate device. If you compare it with a school voltmeter, you can see that it has an accuracy class of 4. This value must be known for further calculations.

Application of knowledge

Thus, D c \u003d c (max) X γ / 100

This formula will be used for specific examples. Let's use a voltmeter and find the error in measuring the voltage that the battery gives.

Let's connect the battery directly to the voltmeter, having previously checked whether the arrow is at zero. When the device was connected, the arrow deviated by 4.2 divisions. This state can be described as follows:

1. It can be seen that the maximum value of U for this item is 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 errors are calculated as follows:

D U \u003d DU (ex.) + C / 2

D U (pr.) \u003d U (max) X γ / 100

D U (pr.) \u003d 6 V X 4/100 \u003d 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 considered formula, 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 error limit and the relative one.

Learning to determine the weighing error

This is one example of direct measurements. In a special place is weighing. 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 affected by the accuracy of the weights and the perfection of the scales themselves.

We use a balance scale with a set of weights that must be placed exactly on the right side of the scale. Take a ruler for weighing.

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

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

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

The mass measurement error consists of two terms associated with scales and weights. To find out each of these values, at the factories for the production of scales and weights, products are supplied with special documents that allow you to calculate the accuracy.

Application of tables

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

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

The second table refers to a set of weights. It indicates that each of them has its own mass error. The 10-gram has an error of 1 mg, as well as the 20-gram. We calculate the sum of the errors of each of these weights, taken from the table.

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

Results

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

Measurements of many quantities occurring in nature cannot be accurate. The measurement gives a number expressing a value with varying degrees of accuracy (length measurement with an accuracy of 0.01 cm, calculation of the value of a function at a point with an accuracy of up to, etc.), that is, approximately, with some error. The error can be set in advance, or, conversely, it needs to be found.

The theory of errors has the object of its study mainly of approximate numbers. When calculating instead of usually use approximate numbers: (if accuracy is not particularly important), (if accuracy is important). How to carry out calculations with approximate numbers, determine their errors - this is the theory of approximate calculations (error theory).

In the future, exact numbers will be denoted by capital letters, and the corresponding approximate numbers will be denoted by lowercase letters.

Errors arising at one or another stage of solving the problem can be divided into three types:

1) Problem error. This type of error occurs when constructing a mathematical model of the phenomenon. It is far from always possible to take into account all the factors and the degree of their influence on the final result. That is, the mathematical model of an object is not its exact image, its description is not accurate. Such an error is unavoidable.

2) Method error. This error arises as a result of replacing the original mathematical model with a more simplified one, for example, in some problems of correlation analysis, a linear model is acceptable. Such an error is removable, since at the stages of calculation it can be reduced to an arbitrarily small value.

3) Computational ("machine") error. Occurs when a computer performs arithmetic operations.

Definition 1.1. Let be the exact value of the quantity (number), be the approximate value of the same quantity (). True absolute error approximate number is the modulus of the difference between the exact and approximate values:

. (1.1)

Let, for example, =1/3. When calculating on the MK, they gave the result of dividing 1 by 3 as an approximate number = 0.33. Then .

However, in reality, in most cases, the exact value of the quantity is not known, which means that (1.1) cannot be applied, that is, the true absolute error cannot be found. Therefore, another value is introduced that serves as some estimate (upper bound for ).

Definition 1.2. Limit absolute error approximate number, representing an unknown exact number, is called such a possibly smaller number, which does not exceed the true absolute error, that is . (1.2)

For an approximate number of quantities satisfying inequality (1.2), there are infinitely many, but the most valuable of them will be the smallest of all those found. From (1.2), based on the definition of the modulus, we have , or abbreviated as the equality

. (1.3)

Equality (1.3) determines the boundaries within which an unknown exact number is located (they say that an approximate number expresses an exact number with a limiting absolute error). It is easy to see that the smaller , the more precisely these boundaries are determined.

For example, if measurements of a certain value gave the result cm, while the accuracy of these measurements did not exceed 1 cm, then the true (exact) length cm.

Example 1.1. Given a number. Find the limiting absolute error of the number by the number .

Decision: From equality (1.3) for the number ( =1.243; =0.0005) we have a double inequality , i.e.

Then the problem is posed as follows: to find for the number the limiting absolute error satisfying the inequality . Taking into account the condition (*), we obtain (in (*) we subtract from each part of the inequality)

Since in our case , then , whence =0.0035.

Answer: =0,0035.

The limiting absolute error often gives a poor idea of ​​the accuracy of measurements or calculations. For example, =1 m when measuring the length of a building will indicate that they were not carried out accurately, and the same error =1 m when measuring the distance between cities gives a very qualitative estimate. Therefore, another value is introduced.

Definition 1.3. True relative error number, which is an approximate value of the exact number, is the ratio of the true absolute error of the number to the modulus of the number itself:

. (1.4)

For example, if, respectively, the exact and approximate values, then

However, formula (1.4) is not applicable if the exact value of the number is not known. Therefore, by analogy with the limiting absolute error, the limiting relative error is introduced.

Definition 1.4. Limiting relative error a number that is an approximation of an unknown exact number is called the smallest possible number , which is not exceeded by the true relative error , i.e

. (1.5)

From inequality (1.2) we have ; whence, taking into account (1.5)

Formula (1.6) has a greater practical applicability compared to (1.5), since the exact value does not participate in it. Taking into account (1.6) and (1.3), one can find the boundaries that contain the exact value of the unknown quantity.

It is practically impossible to determine the true value of a physical quantity absolutely exactly, because any measurement operation is associated with a number of errors or, otherwise, errors. The reasons for the errors can be very different. Their occurrence may be due to inaccuracies in 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 a wire of inhomogeneous thickness, the result randomly depends on the choice of the measurement area), random reasons, etc.

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

There are concepts of absolute and relative error.

Under absolute error measurement 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 i is the true value of the measured value.

The result of any physical measurement is usually written as:

where is the arithmetic mean value of the measured quantity 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 value lies in the interval [ - , + ].

Absolute error is a dimensional value, it has the same dimension as the measured value.

The absolute error does not fully characterize the accuracy of the measurements made. Indeed, if we measure with the same absolute error of ± 1 mm segments 1 m and 5 mm long, the measurement accuracy 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 appearance of the error, it can be conditionally divided into the following classes: instrumental, systematic, random, and misses (gross errors).

Misses are due either to a malfunction of the device, or to 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 appearance, it is necessary to observe accuracy and thoroughness in working with devices. Results containing misses must be excluded from consideration (discarded).

instrumental errors. If the measuring device is serviceable and adjusted, then measurements can be taken on it with limited accuracy, determined by the type of device. It is accepted that the instrumental error of the pointer instrument is considered equal to half of the smallest division of its scale. In devices with a digital readout, the instrument error is equated to the value of one smallest digit on 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 achieve their elimination.

Systematic errors are conditionally 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 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: malfunction of the device itself, non-compliance of the scale with the zero value, the accuracy class of this device;

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

4) errors due to 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 for repeated measurements of the same quantity.

Similar information.