Why is absolute error needed? Measurement of physical quantities

The result of measuring a physical quantity always differs from the true value by a certain amount, which is called error

CLASSIFICATION:

1. By way of expression: absolute, reduced and relative

2. By source of origin: methodological and instrumental.

3. According to the conditions and causes of occurrence: main and additional

4. By the nature of the changes: systematic and random.

5. Depending on the input measured value: additive and multiplicative

6. Depending on inertia: static and dynamic.

13. Absolute, relative and reduced errors.

Absolute error is the difference between the measured and actual values ​​of the measured quantity:

where A is measured, A is the measured and actual values; ΔA - absolute error.

The absolute error is expressed in units of the measured value. The absolute error taken with the opposite sign is called the correction.

Relativeerror p is equal to the ratio absolute errorΔA to the actual value of the measured value and is expressed as a percentage:

Givenerror of a measuring instrument is the ratio of the absolute error to the nominal value. The nominal value for a device with a one-sided scale is equal to the upper limit of measurement, for a device with a double-sided scale (with a zero in the middle) - the arithmetic sum of the upper limits of measurement:

pr. no.

14. Methodological, instrumental, systematic and random errors.

Method error is due to the imperfection of the measurement method used, the inaccuracy of the formulas and mathematical dependencies that describe this measurement method, as well as the influence of the measuring instrument on the object whose properties change.

Instrumental error(instrument error) is due to the design features of the measuring device, inaccuracy of the calibration and scale, as well as incorrect installation of the measuring device.

The instrumental error, as a rule, is indicated in the passport for the measuring instrument and can be assessed in numerical terms.

Systematic error- a constant or naturally varying error during repeated measurements of the same quantity under the same measurement conditions. For example, the error that occurs when measuring resistance with an ampere-voltmeter is caused by a low battery.

Random error- measurement error, the nature of which changes during repeated measurements of the same quantity under the same conditions is random. For example, the counting error during several repeated measurements.

The cause of random error is the simultaneous action of many random factors, each of which has little effect individually.

Random error can be assessed and partially reduced through proper processing methods mathematical statistics, as well as probability methods.

15. Basic and additional, static and dynamic errors.

Basic error- error that occurs under normal conditions of use of a measuring instrument (temperature, humidity, supply voltage, etc.), which are standardized and specified in standards or technical specifications.

Additional error is caused by the deviation of one or more influencing quantities from the normal value. For example, temperature change environment, changes in humidity, fluctuations in supply voltage. The value of the additional error is standardized and indicated in the technical documentation for the measuring instruments.

Static error- error when measuring a time-constant value. For example, the measurement error of a constant current voltage during measurement.

Dynamic error- measurement error of a time-varying quantity. For example, the error in measuring the switched DC voltage due to transient processes during switching, as well as limited speed measuring instrument.

MEASUREMENT OF PHYSICAL QUANTITIES.

INTRODUCTION

The K-402.1 complex is necessary list laboratory work provided for by the educational standard and work program under the section "Dynamics" solid"discipline "Physics". It includes a description of laboratory installations, the procedure for measurements and an algorithm for calculating certain physical quantities.

If a student begins getting acquainted with a specific work in the classroom during a lesson, then the two hours allotted for completing one laboratory work, he will not have enough and will begin to lag behind the semester work schedule. To eliminate this, the second generation educational standard requires 50% of the hours allocated to studying the discipline to be spent on independent work, which is a necessary component of the learning process. Purpose independent work is to consolidate and deepen knowledge and skills, prepare for lectures, practical and laboratory classes, as well as develop students’ independence in acquiring new knowledge and skills.

Curricula for various specialties are provided for self-study discipline "Physics" during the semester from 60 to 120 hours. Of these, laboratory classes account for 20–40 hours, or 2–4 hours per work. During this time, the student must: read the relevant paragraphs in textbooks; learn basic formulas and laws; become familiar with the installation and measurement procedure. To be allowed to perform work on the installation, a student must know the device of the installation, be able to determine the division value of the measuring instrument, know the sequence of measurements, be able to process measurement results, and evaluate the error.

After all the calculations and preparation of the report, the student must draw a conclusion, specifically indicating those physical laws that were tested during the work.

There are two types of measurements: direct and indirect.

Direct measurements are those in which a comparison of a measure and an object is made. For example, measure the height and diameter of a cylinder using a caliper.

In indirect measurements, a physical quantity is determined on the basis of a formula establishing its relationship with quantities found by direct measurements.

The measurement cannot be made absolutely accurately. Its result always contains some error.

Measurement errors are usually divided into systematic and random.

Systematic errors are caused by factors that act in the same way when the same measurements are repeated many times.

Contribution to systematic errors comes from instrumental or instrument error, which is determined by the sensitivity of the device. In the absence of such data on the instrument, the instrument error is taken to be the price or half the price of the smallest scale division of the instrument.

Random errors caused by the simultaneous action of many factors that cannot be taken into account. Most measurements are accompanied by random errors, characterized in that with each repeated measurement they take on a different, unpredictable value.

Absolute error will include systematic and random errors:

. (1.1)

The true value of the measured value will be in the range:

which is called the confidence interval.

To determine the random error, first calculate the average of all values ​​obtained during the measurement:

, (1.2)

where is the result i-th dimension, – number of dimensions.

Then, the errors of individual measurements are found

, , …, .

. (1.3)

When processing measurement results, the Student distribution is used. Taking into account the Student coefficient, random error

.

Table 1.1

Student's coefficient table

n
0,6	0,7	0,9	0,95	0,99
	1,36	2,0	6,3	12,7	636,6
	1,06	1,3	2,9	4,3	31,6
	0,98	1,3	2,4	3,2	12,9
	0,94	1,2	2,1	2,8	8,7
…	…	…	…	…	…
	0,85	1,0	1,7	2,0	3,5
	0,84	1,0	1,7	2,0	3,4

The Student coefficient shows the deviation of the arithmetic mean from the true value, expressed as a fraction of the mean square error. Student's coefficient depends on the number of measurements n and on reliability and is indicated in table. 1.1.

The absolute error is calculated using the formula

.

In most cases, it is not the absolute, but the relative error that plays a more significant role

Or . (1.4)

All calculation results are entered into the table. 1.2.

Table 1.2

The result of calculating the measurement error

No.
	mm	mm	mm	mm 2	mm 2	mm	mm	mm	mm	mm	%

Calculation of errors of indirect measurements

Terms measurement error And measurement error are used interchangeably.) It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. In this case, the average statistical value obtained by statistical processing of the results of a series of measurements is taken as the true value. This obtained value is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, the measurement error is indicated along with the result obtained. For example, record T=2.8±0.1 c. means that the true value of the quantity T lies in the range from 2.7 s. to 2.9 s. some specified probability (see confidence interval, confidence probability, standard error).

In 2006 at international level was accepted new document, dictating the conditions for carrying out measurements and establishing new rules for comparing state standards. The concept of “error” became obsolete, and the concept of “measurement uncertainty” was introduced instead.

Determination of error

Depending on the characteristics of the measured quantity, various methods are used to determine the measurement error.

• The Kornfeld method consists in choosing 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:

• Mean square error:

• Root mean square error of the arithmetic mean:

Error classification

According to presentation form

• Absolute error - Δ X is an estimate of the absolute measurement error. The magnitude of this error depends on the method of its calculation, which, in turn, is determined by the distribution of the random variable X meas . In this case the equality:

Δ X = | X true − X meas | ,

Where X true is the true value, and X meas - measured value must be fulfilled with some probability close to 1. If random variable X meas is distributed according to the normal law, then, usually, its standard deviation is taken as the absolute error. Absolute error is measured in the same units as the quantity itself.

• Relative error- the ratio of the absolute error to the value that is accepted as true:

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

• Reduced error- relative error, expressed as the ratio of the absolute error of the measuring instrument to the conventional accepted value a value that is constant over the entire measurement range or 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 in normal conditions operation for this 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. For normal temperature ambient air is taken to be 20°C as normal atmospheric pressure 01.325 kPa.

A generalized characteristic of measuring instruments is the accuracy class, determined limit values 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 quantity 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).

The absolute calculation error is found by the formula:

The modulus sign shows that we do not care which value is greater and which is less. Important, how far the approximate result deviated from the exact value in one direction or another.

The relative error of calculations is found by the formula:
, or the same thing:

The relative error shows by what percentage the approximate result deviated from the exact value. There is a version of the formula without multiplying by 100%, but in practice I almost always see the above version with percentages.

After a short reference, let's return to our problem, in which we calculated the approximate value of the function using a differential.

Let's calculate the exact value of the function using a microcalculator:
, strictly speaking, the value is still approximate, but we will consider it accurate. Such problems do occur.

Let's calculate the absolute error:

Let's calculate relative error:
, thousandths of a percent were obtained, so the differential provided just an excellent approximation.

Answer: , absolute calculation error, relative calculation error

The following example for an independent solution:

Example 4

at point . Calculate a more accurate value of the function at a given point, estimate the absolute and relative error of calculations.

An approximate sample of the final design and the answer at the end of the lesson.

Many people have noticed that roots appear in all the examples considered. This is not accidental; in most cases, the problem under consideration actually offers functions with roots.

But for suffering readers, I dug up a small example with arcsine:

Example 5

Calculate approximately the value of a function using a differential at the point

This short but informative example is also for you to solve on your own. And I rested a little so that with renewed vigor I could consider the special task:

Example 6

Calculate approximately using differential, round the result to two decimal places.

Solution: What's new in the task? The condition requires rounding the result to two decimal places. But that’s not the point; I think the school rounding problem is not difficult for you. The fact is that we are given a tangent with an argument, which is expressed in degrees. What should you do when you are asked to solve a trigonometric function with degrees? For example , etc.

The solution algorithm is fundamentally the same, that is, it is necessary, as in previous examples, to apply the formula

Let's write an obvious function

The value must be presented in the form . Will provide serious assistance table of values ​​of trigonometric functions . By the way, for those who have not printed it out, I recommend doing so, since you will have to look there throughout the entire course of studying higher mathematics.

Analyzing the table, we notice a “good” tangent value, which is close to 47 degrees:

Thus:

After preliminary analysis degrees must be converted to radians. Yes, and only this way!

IN in this example directly from the trigonometric table you can find out that. Using the formula for converting degrees to radians: (formulas can be found in the same table).

What follows is formulaic:

Thus: (we use the value for calculations). The result, as required by condition, is rounded to two decimal places.

Answer:

Example 7

Calculate approximately using a differential, round the result to three decimal places.

This is an example for you to solve on your own. Full solution and answer at the end of the lesson.

As you can see, there is nothing complicated, we convert degrees to radians and adhere to the usual solution algorithm.

Approximate calculations using the total differential of a function of two variables

Everything will be very, very similar, so if you came to this page specifically for this task, then first I recommend looking at at least a couple of examples of the previous paragraph.

To study a paragraph you must be able to find second order partial derivatives , where would we be without them? In the above lesson, I denoted a function of two variables using the letter . In relation to the task under consideration, it is more convenient to use the equivalent notation.

As in the case of a function of one variable, the condition of the problem can be formulated in different ways, and I will try to consider all the formulations encountered.

Example 8

Solution: No matter how the condition is written, in the solution itself to denote the function, I repeat, it is better to use not the letter “zet”, but .

And here is the working formula:

In fact, before us older sister formulas of the previous paragraph. The variable has only increased. What can I say, myself the solution algorithm will be fundamentally the same!

According to the condition, it is required to find the approximate value of the function at the point.

Let's represent the number 3.04 as . The bun itself asks to be eaten:
,

Let's represent the number 3.95 as . The turn has come to the second half of Kolobok:
,

And don’t look at all the fox’s tricks, there is a Kolobok - you have to eat it.

Let's calculate the value of the function at the point:

We find the differential of a function at a point using the formula:

From the formula it follows that we need to find partial derivatives first order and calculate their values ​​at point .

Let's calculate the first order partial derivatives at the point:

Total differential at point:

Thus, according to the formula, the approximate value of the function at the point:

Let's calculate the exact value of the function at the point:

This value is absolutely accurate.

Errors are calculated using standard formulas, which have already been discussed in this article.

Absolute error:

Relative error:

Answer: , absolute error: , relative error:

Example 9

Calculate the approximate value of a function at a point using a total differential, estimate the absolute and relative error.

This is an example for you to solve on your own. Whoever dwells in more detail on this example will notice that the calculation errors turned out to be very, very noticeable. This happened for the following reason: in the proposed problem the increments of arguments are quite large: .

General pattern that's how it is a - the larger these increments in absolute value, the lower the accuracy of the calculations. So, for example, for a similar point the increments will be small: , and the accuracy of the approximate calculations will be very high.

This feature is also true for the case of a function of one variable (the first part of the lesson).

Example 10

Solution: Let's calculate this expression approximately using the total differential of a function of two variables:

The difference from Examples 8-9 is that we first need to construct a function of two variables: . I think everyone understands intuitively how the function is composed.

The value 4.9973 is close to “five”, therefore: , .
The value 0.9919 is close to “one”, therefore, we assume: , .

Let's calculate the value of the function at the point:

We find the differential at a point using the formula:

To do this, we calculate the first order partial derivatives at the point.

The derivatives here are not the simplest, and you should be careful:

;

.

Total differential at point:

Thus, the approximate value of this expression is:

Let's calculate a more accurate value using a microcalculator: 2.998899527

Let's find the relative calculation error:

Answer: ,

Just an illustration of the above, in the problem considered, the increments of arguments are very small, and the error turned out to be fantastically tiny.

Example 11

Using the complete differential of a function of two variables, calculate approximately the value of this expression. Calculate the same expression using a microcalculator. Estimate the relative calculation error as a percentage.

This is an example for you to solve on your own. An approximate sample of the final design at the end of the lesson.

As already noted, the most common guest in this type of task is some kind of roots. But from time to time there are other functions. And a final simple example for relaxation:

Example 12

Using the total differential of a function of two variables, calculate approximately the value of the function if

The solution is closer to the bottom of the page. Once again, pay attention to the wording of the lesson tasks, in various examples in practice, the formulations may be different, but this does not fundamentally change the essence and algorithm of the solution.

To be honest, I was a little tired because the material was a bit boring. It was not pedagogical to say this at the beginning of the article, but now it’s already possible =) Indeed, the tasks computational mathematics usually not very complex, not very interesting, the most important thing, perhaps, is not to make a mistake in ordinary calculations.

May the keys of your calculator not be erased!

Solutions and answers:

Example 2:

Solution: We use the formula:
In this case: , ,

Thus:

Answer:

Example 4:

Solution: We use the formula:
In this case: , ,

Thus:

Let's calculate a more accurate value of the function using a microcalculator:

Absolute error:

Relative error:

Answer: , absolute calculation error, relative calculation error

Example 5:

Solution: We use the formula:

In this case: , ,

Thus:

Answer:

Example 7:

Solution: We use the formula:
In this case: , ,

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 may be, 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 an unscrupulous seller, when measuring the weight, made a mistake of 50 grams in his 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. Can you 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, very often they additionally calculate the relative deviation, equal to the ratio of the absolute error to exact value numbers. 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 the relative estimate of the number kth degree approximately in | k | times the relative error of the original number.