Calculation of absolute and relative error examples. Absolute measurement error

X \u003d X cf X

Absolute

error

Relative

error

a+ b

a+ b

Often in life we ​​have to deal with various approximate values. Approximate calculations are always calculations with some error.

The concept of absolute error

The absolute error of the approximate value is the modulus of the difference between the exact value and the approximate value.
That is, from the exact value, you need to subtract the approximate value and take the resulting number modulo. Thus, the absolute error is always positive.

How to Calculate Absolute Error

We will show how this might look in practice. For example, we have a graph of a certain value, let it be a parabola: y=x^2.

From the graph, we can determine the approximate value at some points. For example, at x=1.5, the value of y is approximately 2.2 (y≈2.2).

Using the formula y=x^2, we can find the exact value at the point x=1.5 y= 2.25.

Now we calculate the absolute error of our measurements. |2.25-2.2|=|0.05| = 0.05.

The absolute error is 0.05. In such cases, they also say the value is calculated with an accuracy of 0.05.

It often happens that the exact value can not always be found, and, therefore, the absolute error is not always possible to find.

For example, if we calculate the distance between two points using a ruler, or the angle between two straight lines using a protractor, then we will get approximate values. But the exact value cannot be calculated. In this case, we can specify a number that cannot exceed the value of the absolute error.

In the example with the ruler, this will be 0.1 cm, since the division value on the ruler is 1 millimeter. In the example for the protractor, 1 degree is because the protractor scale is graduated every degree. Thus, the values ​​of the absolute error in the first case are 0.1, and in the second case 1.

As mentioned earlier, when we compare the measurement accuracy of some approximate value, we use the absolute error.

The concept of absolute error

The absolute error of an approximate value is the modulus of the difference between the exact value and the approximate value.
Absolute error can be used to compare the accuracy of approximations of the same quantities, and if we are going to compare the accuracy of approximations of different quantities, then absolute error alone is not enough.

For example: The length of a sheet of A4 paper is (29.7 ± 0.1) cm. And the distance from St. Petersburg to Moscow is (650 ± 1) km. The absolute error in the first case does not exceed one millimeter, and in the second - one kilometer. The question is to compare the accuracy of these measurements.

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

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

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

The concept of relative error

Here, to assess the quality of the approximation, a new concept of relative error is introduced. Relative error is the quotient of dividing the absolute error by the modulus of the approximate values ​​of the measured quantity. Usually, the relative error is expressed as a percentage. In our example, we got two relative errors equal to 0.33% and 0.15%.

As you may have guessed, the relative error value is always positive. This follows from the fact that the absolute error is always positive, and we divide it by the modulus, and the modulus is also always positive.

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) of the 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 Р 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 “root mean square 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 GSI standards, instead of the term "root mean square error", the term "root mean square deviation" is used, in further reasoning we will stick 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).

In practice, usually the numbers on which calculations are made are approximate values ​​of certain quantities. For brevity, the approximate value of a quantity is called an approximate number. The true value of a quantity is called the exact number. An approximate number is of practical value only when we can determine with what degree of accuracy it is given, i.e. evaluate its error. Recall the basic concepts from the general course of mathematics.

Denote: x- exact number (true value of the quantity), a- approximate number (approximate value of a quantity).

Definition 1. The error (or true error) of an approximate number is the difference between the number x and its approximate value a. Approximate error a we will denote . So,

Exact number x most often it is unknown, therefore it is not possible to find the true and absolute errors. On the other hand, it may be necessary to estimate the absolute error, i.e. indicate a number that the absolute error cannot exceed. For example, when measuring the length of an object with this tool, we must be sure that the error of the obtained numerical value will not exceed a certain number, for example, 0.1 mm. In other words, we must know the bound on the absolute error. This limit will be called the limiting absolute error.

Definition 3. The limiting absolute error of the approximate number a is called a positive number such that , i.e.

Means, X by deficiency, by excess. The following entry is also used:

It is clear that the limiting absolute error is determined ambiguously: if a certain number is the limiting absolute error, then any larger number is also the limiting absolute error. In practice, they try to choose the smallest possible and simple (with 1-2 significant digits) number that satisfies inequality (2.3).

Example.Determine the true, absolute and limiting absolute errors of the number a \u003d 0.17, taken as an approximate value of the number.

True error:

Absolute error:

For the limiting absolute error, you can take a number and any larger number. In decimal notation we will have: Replacing this number with a large and possibly simpler record, we will accept:

Comment. If a a is the approximate value of the number X, and the limiting absolute error is equal to h, then they say that a is the approximate value of the number X up to h.

Knowing the absolute error is not enough to characterize the quality of a measurement or calculation. Let, for example, such results are obtained when measuring length. Distance between two cities S1=500 1 km and the distance between two buildings in the city S2=10 1 km. Although the absolute errors of both results are the same, however, it is essential that in the first case the absolute error of 1 km falls on 500 km, in the second - on 10 km. The measurement quality in the first case is better than in the second. The quality of a measurement or calculation result is characterized by a relative error.

Definition 4. Relative error of approximate value a numbers X is the ratio of the absolute error of the number a to the absolute value of the number X:

Definition 5. The limiting relative error of the approximate number a is called a positive number such that .

Since , it follows from formula (2.7) that it can be calculated from the formula

For brevity, in cases where this does not cause misunderstandings, instead of “limiting relative error”, they simply say “relative error”.

The limiting relative error is often expressed as a percentage.

Example 1. . Assuming , we can accept = . By dividing and rounding (necessarily upwards), we get = 0.0008 = 0.08%.

Example 2When weighing the body, the result was obtained: p=23.4 0.2 g. We have = 0.2. . By dividing and rounding, we get = 0.9%.

Formula (2.8) determines the relationship between absolute and relative errors. From formula (2.8) it follows:

Using formulas (2.8) and (2.9), we can, if the number is known a, according to the given absolute error, find the relative error and vice versa.

Note that formulas (2.8) and (2.9) often have to be applied even when we do not yet know the approximate number a with the required accuracy, but we know the rough approximate value a. For example, it is required to measure the length of an object with a relative error of no more than 0.1%. The question is: is it possible to measure the length with the required accuracy using a caliper that allows you to measure the length with an absolute error of up to 0.1 mm? Although we have not yet measured an object with an accurate instrument, we know that a rough approximate value of the length is about 12 cm. By formula (1.9) we find the absolute error:

From this it can be seen that with the help of a caliper it is possible to perform a measurement with the required accuracy.

In the process of computational work, it is often necessary to switch from absolute to relative error, and vice versa, which is done using formulas (1.8) and (1.9).

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