The relative error can be greater than 100. What is an approximate value? How to record a measurement result

Errors in measurements of physical quantities

1.Introduction(measurement and measurement error)

2.Random and systematic errors

3.Absolute and relative errors

4. Errors of measuring instruments

5. Accuracy class of electrical measuring instruments

6.Reading error

7.Total absolute error of direct measurements

8.Recording the final result of direct measurement

9. Errors of indirect measurements

10.Example

1. Introduction(measurement and measurement error)

Physics as a science was born more than 300 years ago, when Galileo essentially created the scientific study of physical phenomena: physical laws are established and tested experimentally by accumulating and comparing experimental data, represented by a set of numbers, laws are formulated in the language of mathematics, i.e. using formulas that connect numerical values of physical quantities by functional dependence. Therefore, physics is an experimental science, physics is a quantitative science.

Let's get acquainted with some characteristic features of any measurements.

Measurement is finding a numerical value physical quantity experimentally using measuring instruments (ruler, voltmeter, watch, etc.).

Measurements can be direct or indirect.

Direct measurement is the determination of the numerical value of a physical quantity directly by means of measurement. For example, length - with a ruler, atmospheric pressure - with a barometer.

Indirect measurement is finding the numerical value of a physical quantity using a formula that connects the desired quantity with other quantities determined by direct measurements. For example, the resistance of a conductor is determined by the formula R=U/I, where U and I are measured by electrical measuring instruments.

Let's look at an example of measurement.

Measure the length of the bar with a ruler (division value is 1 mm). We can only say that the length of the bar is between 22 and 23 mm. The width of the interval of “unknown” is 1 mm, that is, equal to the division price. Replacing the ruler with a more sensitive device, such as a caliper, will reduce this interval, which will lead to increased measurement accuracy. In our example, the measurement accuracy does not exceed 1mm.

Therefore, measurements can never be made absolutely accurately. The result of any measurement is approximate. Uncertainty in measurement is characterized by error - the deviation of the measured value of a physical quantity from its true value.

Let us list some of the reasons leading to errors.

1. Limited manufacturing accuracy of measuring instruments.

2. Influence on the measurement of external conditions (temperature changes, voltage fluctuations...).

3. Actions of the experimenter (delay in starting the stopwatch, different eye positions...).

4. The approximate nature of the laws used to find measured quantities.

The listed causes of errors cannot be eliminated, although they can be minimized. To establish the reliability of conclusions obtained as a result of scientific research, there are methods for assessing these errors.

2. Random and systematic errors

Errors arising during measurements are divided into systematic and random.

Systematic errors are errors corresponding to the deviation of the measured value from the true value of a physical quantity, always in one direction (increase or decrease). With repeated measurements, the error remains the same.

Reasons for systematic errors:

1) non-compliance of measuring instruments with the standard;

2) incorrect installation of measuring instruments (tilt, imbalance);

3) discrepancy between the initial indicators of the instruments and zero and ignoring the corrections that arise in connection with this;

4) discrepancy between the measured object and the assumption about its properties (presence of voids, etc.).

Random errors are errors that change their numerical value in an unpredictable way. Such errors are caused by a large number of uncontrollable reasons that affect the measurement process (irregularities on the surface of the object, wind blowing, power surges, etc.). The influence of random errors can be reduced by repeating the experiment many times.

3. Absolute and relative errors

To quantify the quality of measurements, the concepts of absolute and relative measurement errors are introduced.

As already mentioned, any measurement gives only an approximate value of a physical quantity, but you can specify an interval that contains its true value:

A pr - D A< А ист < А пр + D А

Value D A is called the absolute error in measuring the quantity A. The absolute error is expressed in units of the measured quantity. The absolute error is equal to the modulus of the maximum possible deviation of the value of a physical quantity from the measured value. And pr is the value of a physical quantity obtained experimentally; if the measurement was carried out repeatedly, then the arithmetic mean of these measurements.

But to assess the quality of measurement it is necessary to determine the relative error e. e = D A/A pr or e= (D A/A pr)*100%.

If a relative error of more than 10% is obtained during a measurement, then they say that only an estimate of the measured value has been made. In physics workshop laboratories, it is recommended to carry out measurements with a relative error of up to 10%. In scientific laboratories, some precise measurements (for example, determining the wavelength of light) are performed with an accuracy of millionths of a percent.

4. Errors of measuring instruments

These errors are also called instrumental or instrumental. They are determined by the design of the measuring device, the accuracy of its manufacture and calibration. Usually they are content with the permissible instrumental errors reported by the manufacturer in the passport for this device. These permissible errors are regulated by GOSTs. This also applies to standards. Usually the absolute instrumental error is denoted D and A.

If there is no information about the permissible error (for example, with a ruler), then half the division value can be taken as this error.

When weighing, the absolute instrumental error consists of the instrumental errors of the scales and weights. The table shows the most common permissible errors

measuring instruments encountered in school experiments.

Measuring	Measurement limit	Value of division	Allowable error
student ruler
demonstration ruler
measuring tape
beaker
weights 10,20, 50 mg
weights 100,200 mg
weights 500 mg







calipers
micrometer
dynamometer
training scales
Stopwatch			1s in 30 min
aneroid barometer	720-780 mm Hg.	1 mmHg	3 mmHg
laboratory thermometer	0-100 degrees C
school ammeter
school voltmeter

5. Accuracy class of electrical measuring instruments

Pointer electrical measuring instruments according to acceptable values errors are divided into accuracy classes, which are indicated on instrument scales by numbers 0.1; 0.2; 0.5; 1.0; 1.5; 2.5; 4.0. Accuracy class g pr The device shows what percentage the absolute error is from the entire scale of the device.

g pr = (D and A/A max)*100% .

For example, the absolute instrumental error of a class 2.5 device is 2.5% of its scale.

If the accuracy class of the device and its scale are known, then the absolute instrumental measurement error can be determined

D and A = (g pr * A max)/100.

To increase the accuracy of measurements with a pointer electrical measuring instrument, it is necessary to select a device with such a scale that during the measurement process it is located in the second half of the instrument scale.

6. Reading error

The reading error results from insufficiently accurate readings of the measuring instruments.

In most cases, the absolute reading error is taken equal to half the division value. Exceptions are made when measuring with a clock (the hands move jerkily).

The absolute error of reading is usually denoted D oA

7. Total absolute error of direct measurements

When performing direct measurements of physical quantity A, the following errors must be assessed: D and A, D oA and D сА (random). Of course, other sources of errors associated with incorrect installation of instruments, misalignment of the initial position of the instrument arrow with 0, etc. should be excluded.

The total absolute error of direct measurement must include all three types of errors.

If the random error is small compared to the smallest value that can be measured by a given measuring instrument (compared to the division value), then it can be neglected and then one measurement is sufficient to determine the value of a physical quantity. Otherwise, probability theory recommends finding the measurement result as the average arithmetic value results of the entire series of repeated measurements, the error of the result is calculated by the method of mathematical statistics. Knowledge of these methods goes beyond the school curriculum.

8. Recording the final result of direct measurement

The final result of measuring the physical quantity A should be written in this form;

A=A pr + D A, e= (D A/A pr)*100%.

And pr is the value of a physical quantity obtained experimentally; if the measurement was carried out repeatedly, then the arithmetic mean of these measurements. D A is the total absolute error of direct measurement.

Absolute error is usually expressed in one significant figure.

Example: L=(7.9 + 0.1) mm, e=13%.

9. Errors of indirect measurements

When processing the results of indirect measurements of a physical quantity that is functionally related to physical quantities A, B and C, which are measured directly, the relative error of the indirect measurement is first determined e=D X/X pr, using the formulas given in the table (without evidence).

The absolute error is determined by the formula D X=X pr *e,

where e expressed as a decimal fraction rather than a percentage.

The final result is recorded in the same way as in the case of direct measurements.

Function type	Formula
X=A+B+C
X=A-B
X=ABC
X=A n
X=A/B

Example: Let's calculate the error in measuring the friction coefficient using a dynamometer. The experiment consists of pulling a block evenly over a horizontal surface and measuring the applied force: it is equal to the sliding friction force.

Using a dynamometer, weigh the block with weights: 1.8 N. F tr =0.6 N

μ = 0.33. The instrumental error of the dynamometer (we find it from the table) is Δ and = 0.05 N, Reading error (half the division value)

Δ o =0.05 N. The absolute error in measuring weight and friction force is 0.1 N.

Relative measurement error (5th line in the table)

, therefore the absolute error of indirect measurement μ is 0.22*0.33=0.074

Absolute and relative errors

Absolute approximation error

When dealing with infinite decimal fractions in calculations, you have to approximate these numbers for convenience, that is, round them. Approximate numbers are also obtained from various measurements.

It can be useful to know how much the approximate value of a number differs from its exact value. It is clear that the smaller this difference, the better, the more accurately the measurement or calculation is performed.

To determine the accuracy of measurements (calculations), such a concept as approximation error is introduced. In another way it is called absolute error.

Absolute error approaching The modulus of the difference between the exact value of a number and its approximate value is called.

Where X - this is the exact value of the number, A - its approximate value.

For example, as a result of measurements a number was obtained. However, as a result of the calculation using the formula, the exact value of this number is. Then the absolute error of the approximation

In the case of infinite fractions, the approximation error is determined by the same formula. In place of the exact number, the infinite fraction itself is written. For example, . Here it turns out that the absolute error of the approximation is expressed by an irrational number.

The approximation can be done as by lack , so by excess .

The same number π when approximating by the deficiency with an accuracy of 0.01 is equal to 3.14, and when approximating by the excess with an accuracy of 0.01 it is equal to 3.15.

Rounding rule: if the first digit to be discarded is five or greater than five, then the excess approximation is performed; if less than five, then due to deficiency.

For example, because the third digit after the decimal point of the number π is 1, then when approximating with an accuracy of 0.01 it is carried out by deficiency.

Let us calculate the absolute errors of approximation up to 0.01 of the number π by deficiency and excess:

As we can see, the absolute error of approximation for deficiency is less than for excess. This means that the approximation by disadvantage in this case has higher accuracy.

Relative approximation error

The absolute error has one important drawback - it does not allow one to assess 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 g in his favor. Those. the absolute error was 50 g. For us, such an oversight will be a mere trifle and we will not even pay attention to it. What 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 this, the relative deviation is often additionally calculated.

Relative approximation error is called the ratio of the absolute error to the exact value of the number.

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

Let's give a few examples.

Example 1. The company has 1,284 workers and employees. Round the number of employees to the nearest whole number with excess and deficiency. Find their absolute and relative errors (in percent). Draw a conclusion.

So, .

Absolute error:

Relative error:

This means that the accuracy of an approximation with a deficiency is higher than the accuracy of an approximation with an excess.

Example 2. The school has 197 students. Round the number of students to the nearest whole number with excess and deficiency. Find their absolute and relative errors (in percent). Draw a conclusion.

So, .

Absolute error:

Relative error:

This means that the accuracy of approximation with an excess is higher than the accuracy of approximation with a deficiency.

Find absolute error proximity:

Approximate value of the numberX equalsA . Find the absolute error of the approximation if:

Write it as a double inequality:

Find the approximate value of a numberX , equal to the arithmetic mean of the approximations with deficiency and excess, if:

Prove that the arithmetic mean of the numbersA Andb is an approximate value of each of these numbers, accurate to.

Round the numbers:

up to units

up to tenths

to thousandths

up to thousands

up to hundred thousandths

up to units

up to tens

up to tenths

to thousandths

up to hundreds

up to ten thousandths

Imagine common fraction as a decimal and round it to thousandths and find the absolute error:

Prove that each of the numbers 0.368 and 0.369 is an approximation of the number to within 0.001. Which of them is an approximate value of the number accurate to 0.0005?

Prove that each of the numbers 0.38 and 0.39 is an approximate value of the number to within 0.01. Which one is the approximate value of the number to within 0.005?

Round the number to units and find the relative rounding error:

5,12

9,736

49,54

1,7

9,85

5,314

99,83

Present each number as a decimal fraction. Having rounded the resulting fractions to tenths, find the absolute and relative errors of the approximations.

The radius of the Earth is 6380 km with an accuracy of 10 km. Estimate the relative error of the approximate value.

The shortest distance from the Earth to the Moon is 356,400 km with an accuracy of 100 km. Estimate the relative error of the approximation.

Compare mass measurement qualitiesM electric locomotive and massT medicine tablets, if t (to the nearest 0.5 t), and g (to the nearest 0.01 g).

Compare the quality of measuring the length of the Volga River and the diameter of a table tennis ball, if km (with an accuracy of 5 km) and mm (with an accuracy of 1 mm).

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 it is necessary from more subtract the lesser.

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 more important has a definition of relative measurement error.

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 occurs as a result of influence on the measurement process external factors and can change its meaning.

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 decimals. 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.

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

The concept of absolute error

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

How to calculate absolute error

Let's show what this might look like 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 equal to 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 let's 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 cannot 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 value of the angle between two straight lines using a protractor, then we will get approximate values. But the exact value is impossible to calculate. In this case, we can specify a number such that the absolute error value cannot be greater.

In the example with a 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 because the protractor scale is graduated at every degree. Thus, the absolute error values in the first case are 0.1, and in the second case 1.

For direct measurements

1. Let two voltages be measured once on a voltmeter U 1 = 10 V, U 2 = 200 V. The voltmeter has the following characteristics: accuracy class d class t = 0.2, U max = 300 V.

Let us determine the absolute and relative errors of these measurements.

Since both measurements were made on the same device, then D U 1 = D U 2 and are calculated using formula (B.4)

According to the definition, relative errors U 1 and U 2 are respectively equal

ε 1 = 0.6 ∙ V / 10 V = 0.06 = 6%,

ε 2 = 0.6 ∙ V / 200 V = 0.003 = 0.3%.

From the given results of calculations ε 1 and ε 2 it is clear that ε 1 is significantly larger than ε 2.

This leads to the rule: you should choose a device with such a measurement limit that the readings are in the last third of the scale.

2. Let some quantity be measured many times, that is, produced n individual measurements of this quantity A x 1 , A x 2 ,...,A x 3 .

Then, to calculate the absolute error, the following operations are performed:

1) using formula (B.5) determine the arithmetic mean value A 0 measured value;

2) calculate the sum of squared deviations of individual measurements from the found arithmetic mean and, using formula (B.6), determine the root mean square error, which characterizes the absolute error of a single measurement for multiple direct measurements of a certain value;

3) relative error ε is calculated using formula (B.2).

Calculation of absolute and relative error

With indirect measurement

Calculation of errors in indirect measurements – more difficult task, since in this case the desired quantity is a function of other auxiliary quantities, the measurement of which is accompanied by the appearance of errors. Usually in measurements, apart from mistakes, random errors turn out to be very small compared to the measured value. They are so small that the second or more high degrees errors lie beyond the measurement accuracy and can be neglected. Due to the smallness of the errors to obtain the error formula
methods of differential calculus are used to measure an indirectly measured quantity. When measuring a quantity indirectly, when quantities associated with some desired mathematical relationship are directly measured, it is more convenient to first determine the relative error and then
Using the found relative error, calculate the absolute measurement error.

Differential calculus provides the simplest way to determine the relative error in indirect measurement.

Let the required quantity A is connected by a functional dependence with several independent directly measurable quantities x 1 ,
x 2 , ..., x k, i.e.

A= f(x 1 , x 2 , ..., x k).

To determine the relative error of the value A take the natural logarithm of both sides of the equality

ln A= log f(x 1 , x 2 , ..., x k).

Then the differential of the natural logarithm of the function is calculated
A= f(x 1 ,x 2 , ..., x k),

dln A=dln f(x 1 , x 2 , ..., x k)

All possible algebraic transformations and simplifications are performed in the resulting expression. After this, all differential symbols d are replaced by error symbols D, and the negative signs in front of the differentials of the independent variables are replaced by positive ones, i.e., the most unfavorable case is taken, when all the errors are added up. In this case, the maximum error of the result is calculated.

With that said

but ε = D A / A

This expression is the formula for the relative error of the quantity A in indirect measurements, it determines the relative error of the desired value, through the relative errors of the measured values. Having calculated the relative error using formula (B.11),
determine the absolute error of the value A as the product of the relative error and the calculated value A i.e.

D A = ε A, (AT 12)

where ε is expressed as a dimensionless number.

So, the relative and absolute errors of the indirectly measured quantity should be calculated in the following sequence:

1) take a formula by which the desired value is calculated ( calculation formula);

2) take the natural logarithm of both sides of the calculation formula;

3) the total differential of the natural logarithm of the desired quantity is calculated;

4) all possible algebraic transformations and simplifications are performed in the resulting expression;

5) the symbol of differentials d is replaced by the symbol of error D, while all negative signs in front of the differentials of independent variables are replaced by positive ones (the value of the relative error will be maximum) and the relative error formula is obtained;

6) the relative error of the measured value is calculated;

7) based on the calculated relative error, the absolute error of indirect measurement is calculated using formula (B.12).

Let's look at several examples of calculating relative and absolute errors in indirect measurements.

1. Required quantity A related to directly measurable quantities X, at, z ratio

Where a And b– constant values.

2. Take the natural logarithm of expression (B.13)

3. Calculate the total differential of the natural logarithm of the desired quantity A, that is, we differentiate (B.13)

4. We make transformations. Considering that d A= 0, since A= const,cos at/sin y=ctg y, we get:

5. Replace the differential symbols with error symbols and the minus sign in front of the differential with a plus sign.

6. We calculate the relative error of the measured value.

7. Based on the calculated relative error, the absolute error of indirect measurement is calculated according to formula (B.12), i.e.

The wavelength is determined yellow color spectral line of mercury using a diffraction grating (using the accepted sequence for calculating the relative and absolute errors for the yellow wavelength).

1. The wavelength of yellow color in this case is determined by the formula:

Where WITH– constant of the diffraction grating (indirectly measured value); φ f – diffraction angle of the yellow line in a given spectral order (directly measured value); K g – order of the spectrum in which the observation was made.

The diffraction grating constant is calculated by the formula

Where K h – order of the spectrum of the green line; λ з – known wavelength of green color (λ з – constant); φз – diffraction angle of the green line in a given spectral order (directly measured value).

Then, taking into account expression (B.15)

(B.16)

Where K h, K g – observables, which are considered constant; φ h, φ w – are
directly measurable quantities.

Expression (B.16) is the calculation formula for the yellow wavelength determined using a diffraction grating.

4. d K z = 0; d K w = 0; dλ з = 0, since K h, K g and λ h – constant values;

Then

5. (B.17)

where Dφ w, Dφ h – absolute errors in measuring the diffraction angle of yellow
and green lines of the spectrum.

6. Calculate the relative error of the yellow wavelength.

7. Calculate the absolute error of the yellow wavelength:

Dλ f = ελ f.