Absolute and relative errors are used to assess the inaccuracy in highly complex calculations. They are also used in various measurements and for rounding calculation results. Let's look at how to determine absolute and relative error.

Absolute error

Absolute error of the number call the difference between this number and its exact value.
Let's look at an example : There are 374 students in the school. If we round this number to 400, then the absolute measurement error is 400-374=26.

To calculate the absolute error, you need to subtract the smaller number from the larger number.

There is a formula for absolute error. Let us denote the exact number by the letter A, and the letter a - the approximation to the exact number. An approximate number is a number that differs slightly from the exact one and usually replaces it in calculations. Then the formula will look like this:

Δa=A-a. We discussed above how to find the absolute error using the formula.

In practice, absolute error is not sufficient to accurately evaluate a measurement. It is rarely possible to know the exact value of the measured quantity in order to calculate the absolute error. Measuring a book 20 cm long and allowing an error of 1 cm, one can consider the measurement to be with a large error. But if an error of 1 cm was made when measuring a wall of 20 meters, this measurement can be considered as accurate as possible. Therefore, in practice, determining the relative measurement error is more important.

Record the absolute error of the number using the ± sign. For example , the length of a roll of wallpaper is 30 m ± 3 cm. The absolute error limit is called the maximum absolute error.

Relative error

Relative error They call the ratio of the absolute error of a number to the number itself. To calculate the relative error in the example with students, we divide 26 by 374. We get the number 0.0695, convert it to a percentage and get 6%. The relative error is denoted as a percentage because it is a dimensionless quantity. Relative error is an accurate estimate of measurement error. If we take an absolute error of 1 cm when measuring the length of segments of 10 cm and 10 m, then the relative errors will be equal to 10% and 0.1%, respectively. For a segment 10 cm long, an error of 1 cm is very large, this is an error of 10%. But for a ten-meter segment, 1 cm does not matter, only 0.1%.

There are systematic and random errors. Systematic is the error that remains unchanged during repeated measurements. Random error arises as a result of the influence of external factors on the measurement process and can change its value.

Rules for calculating errors

There are several rules for the nominal estimation of errors:

when adding and subtracting numbers, it is necessary to add up their absolute errors;
when dividing and multiplying numbers, it is necessary to add relative errors;
When raised to a power, the relative error is multiplied by the exponent.

Approximate and exact numbers are written using decimal fractions. Only the average value is taken, since the exact value can be infinitely long. To understand how to write these numbers, you need to learn about true and dubious numbers.

True numbers are those numbers whose rank exceeds the absolute error of the number. If the digit of a figure is less than the absolute error, it is called doubtful. For example , for the fraction 3.6714 with an error of 0.002, the correct numbers will be 3,6,7, and the doubtful ones will be 1 and 4. Only the correct numbers are left in the recording of the approximate number. The fraction in this case will look like this - 3.67.

What have we learned?

Absolute and relative errors are used to assess the accuracy of measurements. Absolute error is the difference between an exact and an approximate number. Relative error is the ratio of the absolute error of a number to the number itself. In practice, relative error is used since it is more accurate.

Test on the topic

Article rating

Average rating: 4.2. Total ratings received: 603.

Hello, forum users! I would like to ask everyone about the formula for determining the maximum permissible error in determining the storage area. Much has been written on the issue of point error, but very, very little has been written on area error.
At the moment, due to the fact that there are no approved formulas, in all programs in which cadastral engineers work, two formulas are used... - one of the “methodological recommendations for conducting land surveying” (approved by Roszemkadastr dated 02/17/2003) , looks like - ΔР= 3.5 Mt √Р
second of “Instructions for land surveying” (approved by Roskomzem 04/08/1996), it’s impossible to write it correctly, but you understand...
I want to discuss the use of formula No. 1 from the method.recommendations.. ΔР= 3.5 Mt √Р
To be honest, to my shame, I have never looked closely and analyzed these formulas thoroughly, leaving it to the conscience of the software developers, i.e. considers the error to be the program..... but now, after moving to another city, circumstances forced....
You know very well that there are cases (and often) when in an order, decree, etc. costs one area, but in fact (due to circumstances) it is slightly different, please do not confuse it with 10% and similar increases when clarifying.
I always used the first formula by default, and I was surprised by the remark of the local control center - “why do you have the actual area under the root sign?” At first, naturally, I wanted to be indignant, but then I decided to read the theoretical part anyway, I found out where the legs grow from.... and it seems like KP is right... In the source code, i.e. The method recommendations provide a completely understandable explanation of the permissible error. And the main thing is that the document area from the permits is used under the sign of the root...
I wrote to the software developers asking for comments on this point, and so - their position in brief - “under the root there should be an actual area, because this follows from the 921 order...
"The formulas used to calculate the maximum permissible error in determining the area of land plots (parts of land plots) () are indicated in the boundary plan with the values substituted into these formulas and calculation results"And it seems logical too...
But it is not entirely logical that the other formula from the instructions uses the actual area. Well, it can’t be like that... I’m certainly not a mathematician, but if you want to get the result of calculations, the formulas may be different, but the source codes won’t be...
So, gentlemen and ladies, I know very well that while there is no regulatory legal act, there cannot be a consensus, but still! Who has this formula in their software??? I don’t even stutter anymore about how correct it is... to use the actual or permissive area under the root?
I already asked my colleagues who work in other software, and it turned out that they calculate the formula exactly according to the methodological recommendations, i.e. based on their permitting area, it means who goes to the forest - who wants firewood...
Otherwise, I now have a small fork - the cadastral agency is waving its finger and threatening “we won’t accept”, I can’t change anything in the program, the developers are defending their position.. but I’m a bit confused with the argumentation..
Of course, I’ll try to make a boundary using the second formula, but I’m just afraid that the KP, by analogy, will not begin to require the area from the permits there too..

An integral part of any measurement is measurement error. With the development of instrumentation and measurement techniques, humanity strives to reduce the influence of this phenomenon on the final measurement result. I propose to understand in more detail the question of what measurement error is.

Measurement error is the deviation of the measurement result from the true value of the measured value. The measurement error is the sum of errors, each of which has its own cause.

According to the form of numerical expression, measurement errors are divided into absolute And relative

– this is the error expressed in units of the measured value. It is defined by the expression.

(1.2), where X is the measurement result; X 0 is the true value of this quantity.

Since the true value of the measured quantity remains unknown, in practice only an approximate estimate of the absolute measurement error is used, determined by the expression

(1.3), where X d is the actual value of this measured quantity, which, with an error in its determination, is taken as the true value.

is the ratio of the absolute measurement error to the actual value of the measured quantity:

According to the pattern of occurrence of measurement errors, they are divided into systematic, progressive, And random.

Systematic error– this is a measurement error that remains constant or changes naturally with repeated measurements of the same quantity.

Progressive error– This is an unpredictable error that changes slowly over time.

Systematic And progressive errors in measuring instruments are caused by:

the first - by the scale calibration error or its slight shift;
the second - aging of the elements of the measuring instrument.

The systematic error remains constant or changes naturally with repeated measurements of the same quantity. The peculiarity of the systematic error is that it can be completely eliminated by introducing corrections. The peculiarity of progressive errors is that they can only be corrected at a given point in time. They require continuous correction.

Random error– this measurement error varies randomly. When taking repeated measurements of the same quantity. Random errors can only be detected through repeated measurements. Unlike systematic errors, random ones cannot be eliminated from measurement results.

By origin they distinguish instrumental And methodological errors of measuring instruments.

Instrumental errors- these are errors caused by the properties of measuring instruments. They arise due to insufficiently high quality of measuring instrument elements. These errors include the manufacture and assembly of measuring instrument elements; errors due to friction in the mechanism of the device, insufficient rigidity of its elements and parts, etc. We emphasize that the instrumental error is individual for each measuring instrument.

Methodological error- this is the error of a measuring instrument that arises due to the imperfection of the measurement method, the inaccuracy of the ratio used to estimate the measured value.

Errors of measuring instruments.

is the difference between its nominal value and the true (real) value of the quantity reproduced by it:

(1.5), where X n is the nominal value of the measure; X d – actual value of the measure

is the difference between the instrument reading and the true (actual) value of the measured quantity:

(1.6), where X p – instrument readings; X d – actual value of the measured quantity.

is the ratio of the absolute error of a measure or measuring device to the true one

(real) value of the reproduced or measured quantity. The relative error of a measure or measuring device can be expressed in (%).

(1.7)

– the ratio of the error of the measuring device to the standard value. The normalizing value XN is a conventionally accepted value equal to either the upper measurement limit, or the measurement range, or the scale length. The given error is usually expressed in (%).

(1.8)

Limit of permissible error of measuring instruments– the largest error of a measuring instrument, without taking into account the sign, at which it can be recognized and approved for use. This definition applies to the main and additional errors, as well as to the variation of indications. Since the properties of measuring instruments depend on external conditions, their errors also depend on these conditions, therefore the errors of measuring instruments are usually divided into basic And additional.

Main– this is the error of a measuring instrument used under normal conditions, which are usually defined in the regulatory and technical documents for this measuring instrument.

Additional– this is a change in the error of a measuring instrument due to the deviation of influencing quantities from normal values.

The errors of measuring instruments are also divided into static And dynamic.

Static is the error of the measuring instrument used to measure a constant value. If the measured quantity is a function of time, then due to the inertia of the measuring instruments, a component of the total error arises, called dynamic error of measuring instruments.

There are also systematic And random the errors of measuring instruments are similar with the same measurement errors.

Factors influencing measurement error.

Errors arise for various reasons: these may be errors of the experimenter or errors due to the use of the device for other purposes, etc. There are a number of concepts that define factors influencing measurement error

Variation of instrument readings– this is the largest difference in the readings obtained during the forward and reverse strokes with the same actual value of the measured quantity and constant external conditions.

Instrument accuracy class– this is a generalized characteristic of a measuring instrument (device), determined by the limits of permissible main and additional errors, as well as other properties of measuring instruments that affect the accuracy, the value of which is established for certain types of measuring instruments.

The accuracy classes of a device are established upon release, calibrating it against a standard device under normal conditions.

Precision- shows how accurately or clearly a reading can be made. It is determined by how close the results of two identical measurements are to each other.

Device resolution is the smallest change in the measured value to which the device will respond.

Instrument range— determined by the minimum and maximum value of the input signal for which it is intended.

Device bandwidth is the difference between the minimum and maximum frequencies for which it is intended.

Device sensitivity- defined as the ratio of the output signal or reading of the device to the input signal or measured value.

Noises- any signal that does not carry useful information.

Accuracy is one of the most important metrological characteristics of a measuring instrument (a technical instrument intended for measurements). It corresponds to the difference between the readings of the measuring instrument and the true value of the measured value. The smaller the error, the more accurate the measuring instrument is considered, the higher its quality. The greatest possible error value for a certain type of measuring instrument under certain conditions (for example, in a given range of values of the measured value) is called the permissible error limit. Usually set the limits of permissible error, i.e. the lower and upper limits of the interval beyond which the error should not go.

Both the errors themselves and their limits are usually expressed in the form of absolute, relative or reduced errors. The specific form is selected depending on the nature of the change in errors within the measurement range, as well as on the conditions of use and purpose of the measuring instruments. The absolute error is indicated in units of the measured value, and the relative and reduced error is usually expressed as a percentage. The relative error can characterize the quality of a measuring instrument much more accurately than the given one, which will be discussed in more detail below.

The relationship between absolute (Δ), relative (δ) and reduced (γ) errors is determined by the formulas:

where X is the value of the measured quantity, X N is the normalizing value, expressed in the same units as Δ. The criteria for choosing the standard value X N are established by GOST 8.401-80 depending on the properties of the measuring instrument, and usually it should be equal to the measurement limit (X K), i.e.

It is recommended to express the limits of permissible errors in the form given in the case where the error limits can be assumed to be practically unchanged within the measurement range (for example, for dial analog voltmeters, when the error limits are determined depending on the scale division, regardless of the value of the measured voltage). Otherwise, it is recommended to express the limits of permissible errors in relative form in accordance with GOST 8.401-80.
However, in practice, the expression of the limits of permissible errors in the form of reduced errors is erroneously used in cases where the error limits cannot be assumed to be constant within the measurement range. This either misleads users (when they do not understand that the error specified in this way as a percentage is not calculated at all from the measured value), or significantly limits the scope of application of the measuring instrument, because Formally, in this case, the error in relation to the measured value increases, for example, tenfold, if the measured value is 0.1 of the measurement limit.
Expressing the limits of permissible errors in the form of relative errors makes it possible to quite accurately take into account the real dependence of the error limits on the value of the measured quantity when using a formula of the form

δ = ±

where c and d are coefficients, d

In this case, at point X=X k the limits of the permissible relative error, calculated according to formula (4), will coincide with the limits of the permissible reduced error

At points X

Δ 1 =δ·X=·X

Δ 2 =γ X K = c X k

Those. in a large range of values of the measured quantity, much higher measurement accuracy can be ensured if we normalize not the limits of the permissible reduced error according to formula (5), but the limits of the permissible relative error according to formula (4).

This means, for example, that for a measuring converter based on an ADC with a large bit width and a large dynamic range of the signal, the expression of the error limits in the relative form more adequately describes the real limits of the converter error, compared to the reduced form.

Use of terminology

This terminology is widely used in describing the metrological characteristics of various measuring instruments, for example, those listed below produced by L Card LLC:

ADC/DAC module
16/32 channels, 16 bits, 2 MHz, USB, Ethernet

Selection of measuring instruments according to acceptable

When choosing measuring instruments and methods for monitoring products, a set of metrological, operational and economic indicators is taken into account. Metrological indicators include: permissible error of the measuring instrument; scale division price; sensitivity threshold; measurement limits, etc. Operational and economic indicators include: cost and reliability of measuring instruments; duration of work (before repair); time spent on setup and measurement process; weight, overall dimensions and working load.

3.6.3.1. Selection of measuring instruments for dimensional control

In Fig. Figure 3.3 shows distribution curves of part sizes (for those) and measurement errors (for mets) with centers coinciding with the tolerance limits. As a result of the overlapping of the curves for met and those, the distribution curve y(s those, s met) is distorted, and probability regions appear T And P, causing the size to go beyond the tolerance limit for the value With. Thus, the more accurate the technological process (lower the IT/D met ratio), the fewer incorrectly accepted parts compared to incorrectly rejected ones.

The decisive factor is the permissible error of the measuring instrument, which follows from the standardized definition of the actual size as well as the size obtained as a result of measurement with a permissible error.

Permissible measurement errors d measurements during acceptance control for linear dimensions up to 500 mm are established by GOST 8.051, which amount to 35-20% of the tolerance for the manufacture of IT parts. This standard provides for the largest permissible measurement errors, including errors from measuring instruments, installation standards, temperature deformations, measuring force, and part location. The permissible measurement error dmeas consists of random and unaccounted for systematic error components. In this case, the random component of the error is assumed to be equal to 2s and should not exceed 0.6 of the measurement error dmeas.

In GOST 8.051, the error is specified for a single observation. The random component of the error can be significantly reduced due to repeated observations, in which it decreases by a factor, where n is the number of observations. In this case, the arithmetic mean from a series of observations is taken as the actual size.

During arbitration rechecking of parts, the measurement error should not exceed 30% of the error limit allowed during acceptance.

Permissible measurement error values d meas. The angular dimensions are set according to GOST 8.050 - 73.

those

6s those

y meth

2D met

y(s those; s met)

can be assumed during measurement: they include random and unaccounted for systematic measurement errors, all components depending on measuring instruments, installation measures, temperature deformations, basing, etc.

The random measurement error should not exceed 0.6 of the permissible measurement error and is taken equal to 2s, where s is the value of the standard deviation of the measurement error.

For tolerances that do not correspond to the values specified in GOST 8.051 - 81 and GOST 8.050 - 73, the permissible error is selected according to the nearest smaller tolerance value for the corresponding size.

The influence of measurement errors during acceptance inspection of linear dimensions is assessed by the following parameters:

T- some of the measured parts that have dimensions beyond the maximum dimensions are accepted as acceptable (incorrectly accepted);

P - some parts with dimensions not exceeding the maximum dimensions are rejected (incorrectly rejected);

With-probabilistic limiting value of the size exceeding the maximum dimensions for incorrectly accepted parts.

Parameter values t, p, s when the controlled sizes are distributed according to the normal law, they are shown in Fig. 3.4, 3.5 and 3.6.

Rice. 3.4. Graph for determining the parameter m

For determining T with another confidence probability, it is necessary to shift the origin of coordinates along the ordinate axis.

The graph curves (solid and dotted) correspond to a certain value of the relative measurement error equal to

where s is the standard deviation of the measurement error;

IT tolerance of controlled size.

When defining parameters t, p And With recommended to take

A met(s) = 16% for qualifications 2-7, A met(s) = 12% - for qualifications 8, 9,

And met(s) = 10% - for qualifications 10 and rougher.

Options t, p And With are shown on the graphs depending on the value of IT/s those, where s those is the standard deviation of the manufacturing error. Options m, n And With are given for a symmetrical location of the tolerance field relative to the center of grouping of the controlled parts. For determined m, n And With with the combined influence of systematic and random manufacturing errors, the same graphs are used, but instead of the IT/s value, it is taken

for one border,

and for the other - ,

Where a T - systematic manufacturing error.

When defining parameters m And n For each boundary, half of the resulting values are taken.

Possible limit values of parameters t, p And With/IT, corresponding to the extreme values of the curves (in Fig. 3.4 – 3.6), are given in Table 3.5.

Table 3.5

A meth(s)	m	n	c/IT	A meth(s)	m	n	c/IT
1,60	0,37-0,39	0,70-0,75	0,01	10,0	3,10-3,50	4,50-4,75	0,14
3,0	0,87-0,90	1,20-1,30	0,03	12,0	3,75-4,11	5,40-5,80	0,17
5,0	1,60-1,70	2,00-2,25	0,06	16,0	5,00-5,40	7,80-8,25	0,25
8,0	2,60-2,80	3,40-3,70	0,10

First values T And P correspond to the distribution of measurement errors according to the normal law, the second - according to the law of equal probability.

Parameter Limits t, p And With/IT take into account the influence of only the random component of the measurement error.

GOST 8.051-81 provides two ways to establish acceptance limits.

First way. Acceptance boundaries are set to coincide with the maximum dimensions (Fig. 3.7, A ).

Example. When designing a shaft with a diameter of 100 mm, it was estimated that the deviations in its dimensions for operating conditions should correspond to h6(100-0.022). In accordance with GOST 8.051 - 81, it is established that for a shaft size of 100 mm and a tolerance IT = 0.022 mm, the permissible measurement error dmeas = 0.006 mm.

In accordance with table. 3.5 establish that for A met (s) = 16% and unknown accuracy of the technological process m= 5.0 and With= 0.25IT, i.e. among suitable parts there may be up to 5.0% of incorrectly accepted parts with maximum deviations of +0.0055 and -0.0275 mm.

+d meas.

-d meas.

+d meas.

-d meas.

+d meas.

-d meas.

+d meas.

-d meas.

+d meas.

-d meas.

+d meas.

-d meas.

dmeas /2 With