Some people have a question where to look when talking with an interlocutor. In the process of communication, they simply do not know where to put their eyes and what to look at. The interlocutor diligently tells something and drills you with his eyes and probably expects interesting stories from you, but you cannot concentrate and have already searched everything around with your eyes, but thoughts continue to get confused. Others are tormented by the question of where to look in the subway, because they are nose to nose with strangers and their views intersect every now and then.

To overcome this disease, you need to work out your look.

To begin with, you will need a loved one, if one is not found nearby, you can try to get by with the help of a mirror. Sit opposite each other and try to reconsider each other or yourself, the longer you can look into each other's eyes without showing any emotion, the better. Periodically increase the power of your gaze - as if ordering your opponent to perform some action with your eyes, or suppress him with your pressure and try to subdue him. Gather all the strength and energy that you have and send it to your opponent.

This exercise should be repeated periodically and gradually increase its time. You need to reach the mark of at least 2 minutes so that you can seriously, without smiles and grins, gaze intently into the mirror of the soul of an opponent sitting opposite you.

When you are done with this exercise and you can easily withstand and resist someone else's gaze, proceed to the next step - absorb the energy and willpower of your interlocutor by translating it into information and looking at him. Study him, absorb his gaze, try to understand his mood and thoughts, what he does, why he talks to you on this topic, etc., and do it sincerely and kindly. After that, you can start studying passers-by on the street, in the subway, at work, in cafes and in other places - become a kind of researcher, but without excessive fanaticism - this is all just to overcome your phobia.

After some time and having worked out these skills to perfection, you will no longer have a question where to look when talking - you will look 70% of the time of communication into the eyes of your interlocutor and will not experience any discomfort and tightness, but will only think about the subject of the conversation, and finally, get rid of the extra thoughts that bothered you before.

In society, it is considered bad form when a person does not look into the eyes of his interlocutor when communicating. Such people are suspected of hiding something or not saying anything, they are unfriendly. However, psychologists say that this behavior has a variety of reasons.

Anger and excitement

Not so long ago, through a series of experiments, British scientists found that in just one second, when people meet eyes, they exchange an amount of information comparable to what is obtained in three hours of live communication. In psychology, it is said that because of this, some people find it difficult to look into the eyes of the interlocutor for a long time.

Practice not looking away while talking. This will help you make new friends faster and also build favorable business relationships.

Another reason is already in the person whose eyes are looked into. This can be very annoying, irritating, and nervous. It seems that the interlocutor is trying to "read" you, listening to every word and creating his own personal opinion. It is unlikely that such moments cause positive emotions, and a person tends to quickly look away.

It is very difficult for men or women who seem to deliberately drill with their heavy eyes to show, for example, their superiority over the interlocutor. Already from the first seconds of such communication it becomes uncomfortable, there is a strong desire to lower your eyes to the floor.

Uncertainty and boredom

Very often, looking away while talking can be a sign of shyness. With the help of a glance, you can express your attitude to the object, show interest, demonstrate a feeling of love. Also in the look it can be read that it is difficult for a person to find words for a conversation, his nervousness and so on. Therefore, the eyes are turned aside so as not to tell too much about themselves ahead of time and show themselves not in the best possible way.

Uncertainty and lack of concentration also often cause people not to look into the eyes of the interlocutor. Sometimes it can be difficult to find a common language with this or that person, because of which the interlocutor lowers his eyes, begins to nervously touch something in his hands, pulls his ears or hair, thereby giving out his excitement. Such people are simply not sure that they behave and speak correctly.

Let's say we run a series of n measurements of the same quantity X. Due to the presence of random errors, individual values X 1 ,X 2 ,X 3, X n are not the same, and the arithmetic mean is chosen as the best value of the desired value, equal to the arithmetic sum of all measured values ​​divided by the number of measurements:

. (P.1)

where å is the sign of the sum, i- measurement number, n- number of measurements.

So, - the value closest to the true. Nobody knows the true meaning. We can only calculate the interval D X near , in which the true value can be located with some degree of probability R. This interval is called confidence interval. The probability with which the true value falls into it is called confidence level, or reliability factor(because the knowledge of the confidence level allows us to estimate the degree of reliability of the result obtained). When calculating the confidence interval, the required degree of reliability is specified in advance. It is determined by practical needs (for example, more stringent requirements are imposed on aircraft engine parts than on a boat engine). Obviously, to obtain greater reliability, an increase in the number of measurements and their accuracy is required.

Due to the fact that the random errors of individual measurements are subject to probabilistic laws, the methods of mathematical statistics and probability theory make it possible to calculate the root mean square error of the arithmetic mean Dx sl. We write down without proof the formula for calculating Dx cl for a small number of measurements ( n < 30).

The formula is called Student's formula:

, (A.2)

where t n, p - Student's coefficient, depending on the number of measurements n and confidence level R.

The Student's coefficient is found in the table below, having previously determined, based on practical needs (as mentioned above), the values n and R.

When processing the results of laboratory work, it is enough to carry out 3-5 measurements, and take the confidence probability equal to 0.68.

But it happens that with repeated measurements, the same values ​​\u200b\u200bof the quantity are obtained X. For example, the wire diameter was measured 5 times and the same value was obtained 5 times. So, this does not mean at all that there is no error. It only means that the random error of each measurement is less accuracy device d, which is also called instrumentation,or instrumental, error. The instrumental error of the device d is determined by the accuracy class of the device indicated in its passport, or is indicated on the device itself. And sometimes it is taken equal to the division price of the device (the division price of the device is the value of its smallest division) or half the division price (if half the division price of the device can be approximately determined by eye).

Since each of the values X i obtained with error d, then the full confidence interval Dx, or absolute measurement error, is calculated by the formula:

. (P.3)

Note that if in formula (A.3) one of the quantities is at least 3 times larger than the other, then the smaller one is neglected.

The absolute error by itself does not reflect the quality of the measurements. For example, only according to the information, the absolute error is 0.002 m², it is impossible to judge how well this measurement was carried out. An idea of ​​the quality of the measurements taken is given by relative error e, equal to the ratio of the absolute error to the average value of the measured value. Relative error shows what proportion of the absolute error is from the measured value. As a rule, the relative error is expressed as a percentage:

Consider an example. Let the ball diameter be measured with a micrometer, the instrumental error of which is d = 0.01 mm. As a result of three measurements, the following diameter values ​​were obtained:

d 1 = 2.42 mm, d 2 = 2.44 mm, d 3 = 2.48 mm.

According to formula (A.1), the arithmetic mean value of the ball diameter is determined

Then, according to the table of Student's coefficients, it is found that for a confidence probability of 0.68 with three measurements t n, p = 1.3. After that, according to the formula (A.2), a random measurement error is calculated Dd sl

Since the resulting random error is only twice the instrumental error, when finding the absolute measurement error Dd according to (A.3), both the random error and the instrument error should be taken into account, i.e.

mm » ±0.03 mm.

The error was rounded to hundredths of a millimeter, since the accuracy of the result cannot exceed the accuracy of the measuring device, which in this case is 0.01 mm.

So the wire diameter is

mm.

This entry indicates that the true value of the ball diameter with a probability of 68% lies in the interval (2.42 ¸ 2.48) mm.

The relative error e of the obtained value according to (A.4) is

%.

Absolute and relative error

Elements of the theory of errors

Exact and approximate numbers

The accuracy of the number is generally beyond doubt when it comes to integer data values ​​(2 pencils, 100 trees). However, in most cases, when it is impossible to indicate the exact value of a number (for example, when measuring an object with a ruler, taking results from a device, etc.), we are dealing with approximate data.

An approximate value is a number that differs slightly from the exact value and replaces it in calculations. The degree of difference between the approximate value of a number and its exact value is characterized by error .

There are the following main sources of errors:

1. Errors in the formulation of the problem arising as a result of an approximate description of a real phenomenon in terms of mathematics.

2. Errors of the method associated with the difficulty or impossibility of solving the problem and replacing it with a similar one, so that you can apply a well-known and accessible solution method and get a result close to the desired one.

3. Fatal errors, associated with the approximate values ​​of the initial data and due to the performance of calculations on approximate numbers.

4. Rounding errors associated with the rounding of the values ​​of the initial data, intermediate and final results obtained with the use of computational tools.

Absolute and relative error

Accounting for errors is an important aspect of the application of numerical methods, since the error of the final result of solving the entire problem is the product of the interaction of all types of errors. Therefore, one of the main tasks of the theory of errors is to estimate the accuracy of the result based on the accuracy of the initial data.

If is an exact number and is its approximate value, then the error (error) of the approximate value is the degree of closeness of its value to its exact value .

The simplest quantitative measure of error is absolute error, which is defined as

(1.1.2-1)

As can be seen from formula 1.1.2-1, the absolute error has the same units of measurement as the value. Therefore, by the magnitude of the absolute error, it is far from always possible to draw a correct conclusion about the quality of the approximation. For example, if , and we are talking about a machine part, then the measurements are very rough, and if we are talking about the size of the vessel, then they are very accurate. In this regard, the concept of relative error is introduced, in which the value of the absolute error is related to the modulus of the approximate value ( ).

(1.1.2-2)

The use of relative errors is convenient, in particular, because they do not depend on the scale of values ​​and data units. Relative error is measured in fractions or percentages. So, for example, if

,a , then , what if and ,

so then .

To numerically evaluate the error of a function, you need to know the basic rules for calculating the error of actions:

· when adding and subtracting numbers absolute errors of numbers add up

· when multiplying and dividing numbers their relative errors are stacked on top of each other

· when raised to a power of an approximate number its relative error is multiplied by the exponent

Example 1.1.2-1. Given a function: . Find the absolute and relative errors of the value (the error of the result of performing arithmetic operations), if the values are known, and 1 is an exact number and its error is zero.

Having thus determined the value of the relative error, one can find the value of the absolute error as , where the value is calculated by the formula for approximate values

Since the exact value of the quantity is usually unknown, the calculation and according to the above formulas is impossible. Therefore, in practice, the marginal errors of the form are evaluated:

(1.1.2-3)

where and - known values, which are the upper limits of the absolute and relative errors, otherwise they are called - the limiting absolute and limiting relative errors. Thus, the exact value lies within:

If the value known, then , and if the value is known , then

Due to the errors inherent in the measuring instrument, the chosen method and measurement technique, the difference in the external conditions in which the measurement is performed from the established ones, and other reasons, the result of almost every measurement is burdened with an error. This error is calculated or estimated and attributed to the result obtained.

Measurement error(briefly - measurement error) - deviation of the measurement result from the true value of the measured quantity.

The true value of the quantity due to the presence of errors remains unknown. 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 actual (1.3)

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

The real value for single measurements is often taken as the value obtained with the help of an exemplary measuring instrument, for repeated measurements - the arithmetic mean of the values ​​of individual measurements included in this series.

Measurement errors can be classified according to the following criteria:

By the nature of the manifestation - systematic and random;

By way of expression - absolute and relative;

According to the conditions for changing the measured value - static and dynamic;

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

According to the completeness of the coverage of the measuring task - private and complete;

In relation to the unit of physical quantity - the error of reproduction of the unit, storage of the unit and transmission of the size of the unit.

Systematic measurement error(briefly - systematic error) - a component of the error of the measurement result, which remains constant for a given series of measurements or regularly changes during repeated measurements of the same physical quantity.

According to the nature of the manifestation, systematic errors are divided into constant, progressive and periodic. Permanent systematic errors(briefly - constant errors) - errors that retain their value for a long time (for example, during the entire series of measurements). This is the most common type of error.

Progressive systematic errors(briefly - progressive errors) - continuously increasing or decreasing errors (for example, errors due to wear of measuring tips that come into contact during grinding with a part when it is controlled by 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 the measuring device (for example, the presence of eccentricity in goniometers 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, there are instrumental errors, method errors, subjective errors and errors due to deviation of external measurement conditions from established methods.

Instrumental measurement error(briefly - instrumental error) is the result of a number of reasons: wear of instrument parts, excessive friction in the instrument mechanism, inaccurate streaks on the scale, discrepancy between the actual and nominal values ​​​​of the measure, etc.

Measurement method error(briefly - the error of the method) may arise due to the imperfection of the measurement method or its simplifications, established by the measurement procedure. For example, such an error may be due to the insufficient speed 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(briefly - subjective error) is due to the individual errors of the operator. Sometimes this error is called personal difference. It is caused, for example, by a delay or advance in the acceptance of a signal by the operator.

Deviation error(in one direction) external measurement conditions from those established by the measurement procedure leads to the occurrence 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 instrument to bring the systematic errors to an acceptable minimum.

Non-excluded systematic error(briefly - non-excluded error) - this is the error of the measurement result due to the error in calculating and introducing a correction for the effect of a systematic error, or a small systematic error, the correction for which is not introduced due to smallness.

This type of error is sometimes referred to as non-excluded bias residuals(briefly - non-excluded balances). For example, when measuring the length of a line meter in the wavelengths of the reference radiation, several non-excluded systematic errors were revealed (i): due to inaccurate temperature measurement - 1 ; due to the inaccurate determination of the refractive index of air - 2, due to the inaccurate value of the wavelength - 3.

Usually, the sum of non-excluded systematic errors is taken into account (their boundaries are set). With the number of terms N ≤ 3, the boundaries of non-excluded systematic errors are calculated by 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 chosen confidence probability P with their uniform distribution. At P = 0.99, k = 1.4, at P = 0.95, k = 1.1.

Random measurement error(briefly - random error) - a component of the error of the measurement result, changing randomly (in sign and value) in a series of measurements of the same size of a physical quantity. Causes of random errors: rounding errors when reading readings, variation in readings, changes in measurement conditions of a random nature, etc.

Random errors cause dispersion of measurement results in a series.

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

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

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

An important conclusion for practice follows from the first position: with an increase in the number of measurements, the random error of the result obtained from a series of measurements decreases, since the sum of the errors of individual measurements of this series tends to zero, i.e.

(1.6)

For example, as a result of measurements, a series of electrical resistance values ​​\u200b\u200bare obtained (which are corrected for the effects of systematic errors): R 1 \u003d 15.5 Ohm, R 2 \u003d 15.6 Ohm, R 3 \u003d 15.4 Ohm, R 4 \u003d 15, 6 ohms and R 5 = 15.4 ohms. Hence R = 15.5 ohms. Deviations from R (R 1 \u003d 0.0; R 2 \u003d +0.1 Ohm, R 3 \u003d -0.1 Ohm, R 4 \u003d +0.1 Ohm and R 5 \u003d -0.1 Ohm) are random errors of individual measurements in a given series. It is easy to see that the sum R i = 0.0. This indicates that the errors of individual measurements of this series are calculated correctly.

Despite the fact that with an increase in the number of measurements, 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 is necessarily estimated. In the theory of random variables, the dispersion of o2 serves as a characteristic of the dispersion of the values ​​of a random variable. "| / o2 \u003d a is called the standard deviation of the general 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 the practice of measurements one deals with the term “error”, the term “rms error” derived from it should be used to characterize a number of measurements. A number of measurements can be characterized by the arithmetic mean error or the range of measurement results.

The range of measurement results (briefly - range) is the algebraic difference between the largest and smallest results of individual measurements that form a series (or sample) of n measurements:

R n \u003d X max - X min (1.7)

where R n is the range; X max and X min - the largest and smallest values ​​​​of the quantity 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 \u003d d 5 - d 1 \u003d 25.56 mm - 25.51 mm \u003d 0.05 mm. This means that the remaining errors of this series are less than 0.05 mm.

Average arithmetic error of a single measurement in a series(briefly - the arithmetic mean error) - the generalized scattering characteristic (due to random reasons) of individual measurement results (of the same value), included in a series of n equally accurate independent measurements, is 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 ​​of the quantity: |X i - X| is the absolute value of the error of the i-th measurement; r is the arithmetic mean error.

The true value of the arithmetic mean error p is determined from the ratio

p = lim r, (1.9)

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

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

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

Root mean square error individual measurement in a series (briefly - root mean square error) - a generalized scattering characteristic (due to random reasons) of individual measurement results (of the same value) included in a series of P equally accurate independent measurements, calculated by the formula

(1.11)

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

Σ = limS (1.12)

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

With a normal distribution, the probability that the error of a single measurement in a series will not exceed the calculated root 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 rms error of a single measurement s and the rms error of the arithmetic mean S x:

which is often called the "rule of Y n". It follows from this rule that the measurement error due to the action of random causes can be reduced by n times if n measurements of the same size of any quantity are performed, and the arithmetic mean value is taken as the final result (Fig. 1.2).

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

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

(1.14)

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

With unequal measurements, the root mean square error of the arithmetic mean 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 quantity Y, which is a function of Y \u003d F (X 1, X 2, X n), is calculated by the formula

(1.16)

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

If, for greater reliability of 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 is the total number of measurements in all series; m is the number of series.

With a limited number of measurements, it is often necessary to know the RMS 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 series of measurements of the 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 considered case it is necessary to write down: S = ±3 mm.

In order to have greater confidence in the estimation of the error of the measurement result, the confidence error or confidence limits of the error are calculated. With a 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 root mean square errors, respectively, of a single measurement in a series and the arithmetic mean; t is a number depending on the confidence level 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 falls 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. 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, the part size should be expected to go beyond the tolerance of 2a. When evaluating the quality of the part according to the tolerance 2a = ±3s, the reliability of the result will be 0.997. In this case, only three parts out of 1000 can be expected to go beyond the established tolerance. However, an increase in reliability is possible only with a decrease in the error in the length of the part. So, to increase reliability from a = 0.68 to a = 0.997, the error in the length of the part must be reduced by a factor of three.

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 “establishment of unity and the required accuracy of measurements”. Reliability is considered by us as a qualitative characteristic, reflecting the proximity to zero of random errors. Quantitatively, it can be determined through the unreliability of measurements.

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

In accordance with the recommendations of the International Bureau of Weights and Measures, the uncertainty is expressed as the total rms measurement error - Su including the rms error S (determined by statistical methods) and the rms error u (determined by non-statistical methods), i.e.

(1.20)

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

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

In connection with the fact that in the CSI standards, instead of the term "root mean square error", the term "root mean square deviation" is used, in further reasoning we will adhere to this term.

Absolute measurement error(briefly - absolute error) - measurement error, expressed in units of the measured value. So, the error X of measuring the length of the part X, expressed in micrometers, is an absolute error.

The terms “absolute error” and “absolute error value” 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(briefly - relative error) - measurement error, expressed as a fraction of the value of the measured value or as a percentage. The relative error δ is found from the ratios:

(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 is the error of the measurement result due to the conditions of the static measurement.

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

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

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