Calculation of errors in direct and indirect measurements
Measurement is understood as a comparison of the measured value with another value, taken as a unit of measurement. Measurements are carried out empirically using special technical means.
Direct measurements are called measurements, the result of which is obtained directly from experimental data (for example, measuring length with a ruler, time with a stopwatch, temperature with a thermometer). Indirect measurements are measurements in which the desired value of a quantity is found on the basis of a known relationship between this quantity and the quantities whose values are obtained in the process of direct measurements (for example, determining the speed along the distance traveled and time https://pandia.ru/text/78/ 464/images/image002_23.png" width="65" height="21 src=">).
Any measurement, no matter how carefully it is performed, is necessarily accompanied by an error (error) - a deviation of the measurement result from the true value of the measured quantity.
Systematic errors are errors, the magnitude of which is the same in all measurements carried out by the same method using the same measuring instruments, under the same conditions. Systematic errors occur:
As a result of the imperfection of the instruments used in measurements (for example, the ammeter needle may deviate from zero division in the absence of current; the balance beam may have unequal arms, etc.);
As a result of insufficient development of the theory of the measurement method, i.e. the measurement method contains a source of errors (for example, an error occurs when heat loss to the environment is not taken into account in calorimetric works or when weighing on an analytical balance is performed without taking into account the buoyancy force of air) ;
As a result of the fact that the change in the conditions of the experiment is not taken into account (for example, during the long-term passage of current through the circuit, as a result of the thermal effect of the current, the electrical parameters of the circuit change).
Systematic errors can be eliminated if the features of the instruments are studied, the theory of experiment is developed more fully, and on the basis of this, corrections are made to the measurement results.
Random errors are errors whose magnitude is different even for measurements made in the same way. Their reasons lie both in the imperfection of our senses, and in many other circumstances that accompany measurements, and which cannot be taken into account in advance (random errors occur, for example, if the equality of illumination fields of the photometer is set by eye; if the moment of maximum deviation of the mathematical pendulum is determined by eye ; when finding the moment of sound resonance by ear; when weighing on an analytical balance, if the vibrations of the floor and walls are transmitted to the balance, etc.).
Random errors cannot be avoided. Their occurrence is manifested in the fact that when repeating measurements of the same quantity with the same care, numerical results are obtained that differ from each other. Therefore, if the same values were obtained when repeating the measurements, then this indicates not the absence of random errors, but the insufficient sensitivity of the measurement method.
Random errors change the result both in one direction and in the other direction from the true value, therefore, in order to reduce the influence of random errors on the measurement result, measurements are usually repeated many times and the arithmetic mean of all measurement results is taken.
Knowingly incorrect results - misses occur due to violation of the basic conditions of measurement, as a result of inattention or negligence of the experimenter. For example, in poor lighting, instead of “3”, write “8”; due to the fact that the experimenter is distracted, he can go astray when counting the number of swings of the pendulum; due to negligence or inattention, he can confuse the masses of the loads when determining the stiffness of the spring, etc. An external sign of a miss is a sharp difference in magnitude from the results of other measurements. If a miss is detected, the measurement result should be discarded immediately, and the measurement itself should be repeated. The identification of blunders is also aided by a comparison of the measurement results obtained by different experimenters.
To measure a physical quantity means to find the confidence interval in which its true value lies https://pandia.ru/text/78/464/images/image005_14.png" width="16 height=21" height="21">. .png" width="21" height="17 src=">.png" width="31" height="21 src="> cases, the true value of the measured value falls within the confidence interval. The value is expressed either in fractions of a unit, or in percent Most measurements are limited to a confidence level of 0.9 or 0.95 Sometimes, when an extremely high degree of reliability is required, a confidence level of 0.999 is given Along with the confidence level, a significance level is often used, which specifies the probability that the true value does not fall within confidence interval The measurement result is presented as
where https://pandia.ru/text/78/464/images/image012_8.png" width="23" height="19"> is the absolute error. Thus, the interval limits, https://pandia.ru/ text/78/464/images/image005_14.png" width="16" height="21"> lies within this range.
To find and , perform a series of single measurements. Consider a specific example..png" width="71" height="23 src=">; ; https://pandia.ru/text/78/464/images/image019_5.png" width="72" height=" 23">.png" width="72" height="24">. Values can be repeated, like values and https://pandia.ru/text/78/464/images/image024_4.png" width="48 height=15" height="15">.png" width="52" height="21">. Accordingly, the significance level .
Mean value of measured value
The measuring device also contributes to the measurement error. This error is due to the design of the device (friction in the axis of the pointer device, rounding produced by a digital or discrete pointer device, etc.). By its nature, this is a systematic error, but neither the magnitude nor the sign of it for this particular instrument is known. The instrumental error is evaluated in the process of testing a large series of the same type of instruments.
The normalized range of accuracy classes of measuring instruments includes the following values: 0.05; 0.1; 0.2; 0.5; 1.0; 1.5; 2.5; 4.0. The accuracy class of the device is equal to the relative error of the device, expressed as a percentage, in relation to the full range of the scale. Passport error of the device
Any measurements are always made with some errors associated with the limited accuracy of measuring instruments, the wrong choice, and the error of the measurement method, the physiology of the experimenter, the characteristics of the measured objects, changes in measurement conditions, etc. Therefore, the measurement task includes finding not only the quantity itself, but also the measurement error, i.e. the interval in which the true value of the measured quantity is most likely to be found. For example, when measuring a time interval t with a stopwatch with a division value of 0.2 s, we can say that its true value is in the interval from s to 
with. Thus, the measured value always contains some error 
, where 
and X are, respectively, the true and measured values of the quantity under study. Value 
called absolute error(error) measurements, and the expression 
characterizing the measurement accuracy is called relative error.
It is quite natural for the experimenter to make every measurement with the greatest attainable accuracy, but such an approach is not always expedient. The more accurately we want to measure this or that quantity, the more complex the instruments we must use, the more time these measurements will require. Therefore, the accuracy of the final result should correspond to the purpose of the experiment. The theory of errors gives recommendations on how measurements should be taken and how results should be processed so that the margin of error is as small as possible.
All errors arising during measurements are usually divided into three types - systematic, random and misses, or gross errors.
Systematic errors due to the limited accuracy of the manufacture of devices (instrument errors), the shortcomings of the chosen measurement method, the inaccuracy of the calculation formula, improper installation of the device, etc. Thus, systematic errors are caused by factors that act in the same way when the same measurements are repeated many times. The value of this error is systematically repeated or changed according to a certain law. Some systematic errors can be eliminated (in practice, this is always easy to achieve) by changing the measurement method, introducing corrections to instrument readings, and taking into account the constant influence of external factors.
Although the systematic (instrumental) error during repeated measurements gives a deviation of the measured value from the true value in one direction, we never know in which direction. Therefore, the instrumental error is written with a double sign
Random errors are caused by a large number of random causes (changes in temperature, pressure, shaking of the building, etc.), the effect of which on each measurement is different and cannot be taken into account in advance. Random errors also occur due to the imperfection of the experimenter's sense organs. Random errors also include errors due to the properties of the measured object.
It is impossible to exclude random errors of individual measurements, but it is possible to reduce the influence of these errors on the final result by carrying out multiple measurements. If the random error turns out to be significantly less than the instrumental (systematic) error, then there is no point in further reducing the random error by increasing the number of measurements. If the random error is greater than the instrumental one, then the number of measurements should be increased in order to reduce the value of the random error and make it less than or one order of magnitude with the instrumental error.
Mistakes or blunders- these are incorrect readings on the device, incorrect recording of the reading, etc. As a rule, misses due to the indicated reasons are clearly visible, since the readings corresponding to them differ sharply from other readings. Misses must be eliminated by control measurements. Thus, the width of the interval in which the true values of the measured quantities lie will be determined only by random and systematic errors.
2 . Estimation of systematic (instrumental) error
For direct measurements the value of the measured quantity is read directly on the scale of the measuring instrument. The reading error can reach several tenths of a scale division. Usually, in such measurements, the magnitude of the systematic error is considered equal to half the scale division of the measuring instrument. For example, when measuring with a caliper with a division value of 0.05 mm, the value of the instrumental measurement error is taken equal to 0.025 mm.
Digital measuring instruments give the value of the quantities they measure with an error equal to the value of one unit of the last digit on the scale of the instrument. So, if a digital voltmeter shows a value of 20.45 mV, then the absolute error in the measurement is 
mV.
Systematic errors also arise when using constant values determined from tables. In such cases, the error is taken equal to half of the last significant digit. For example, if in the table the value of steel density is given by a value equal to 7.9∙10 3 kg / m 3, then the absolute error in this case is equal to 
kg / m 3.
Some features in the calculation of instrumental errors of electrical measuring instruments will be discussed below.
When determining the systematic (instrumental) error of indirect measurements functional value 
the formula is used
, (1)
where 
- instrument errors of direct measurements of quantity 
, 
- partial derivatives of the function with respect to the variable .
As an example, we will obtain a formula for calculating the systematic error when measuring the volume of a cylinder. The formula for calculating the volume of a cylinder is
.
Partial derivatives with respect to variables d and h will be equal
, 
.
Thus, the formula for determining the absolute systematic error in measuring the volume of a cylinder in accordance with (2. ..) has the following form
,
where 
and 
instrumental errors in measuring the diameter and height of the cylinder
3. Random error estimation.
Confidence Interval and Confidence Probability
For the vast majority of simple measurements, the so-called normal law of random errors is satisfied quite well ( Gauss law), derived from the following empirical provisions.
measurement errors can take a continuous series of values;
with a large number of measurements, errors of the same magnitude, but of a different sign, occur equally often,
The larger the random error, the less likely it is to occur.
The graph of the normal Gaussian distribution is shown in Fig.1. The curve equation has the form
, (2)
where 
- distribution function of random errors (errors), characterizing the probability of an error 
, σ is the root mean square error.
The value σ is not a random variable and characterizes the measurement process. If the measurement conditions do not change, then σ remains constant. The square of this quantity is called dispersion of measurements. The smaller the dispersion, the smaller the spread of individual values and the higher the measurement accuracy.
The exact value of the root-mean-square error σ, as well as the true value of the measured quantity, is unknown. There is a so-called statistical estimate of this parameter, according to which the mean square error is equal to the mean square error of the arithmetic mean 
. The value of which is determined by the formula
, (3)
where 
- result i-th dimension; 
- arithmetic mean of the obtained values; n
is the number of measurements.
The larger the number of measurements, the smaller and the more it approaches σ. If the true value of the measured value μ, its arithmetic mean value obtained as a result of measurements , and the random absolute error , then the measurement result will be written as 
.
Value interval from 
before 
, in which the true value of the measured quantity μ falls, is called confidence interval. Since it is a random variable, the true value falls into the confidence interval with a probability α, which is called confidence probability, or reliability measurements. This value is numerically equal to the area of the shaded curvilinear trapezoid. (see pic.)
All this is true for a sufficiently large number of measurements, when is close to σ. To find the confidence interval and confidence level for a small number of measurements, which we deal with in the course of laboratory work, we use Student's probability distribution. This is the probability distribution of the random variable 
called Student's coefficient, gives the value of the confidence interval in fractions of the root mean square error of the arithmetic mean .
. (4)
The probability distribution of this quantity does not depend on σ 2 , but essentially depends on the number of experiments n. With an increase in the number of experiments n Student's distribution tends to a Gaussian distribution.
The distribution function is tabulated (Table 1). The value of the Student's coefficient is at the intersection of the line corresponding to the number of measurements n, and the column corresponding to the confidence level α
Table 1.
Using the data in the table, you can:
determine the confidence interval, given a certain probability;
choose a confidence interval and determine the confidence level.
For indirect measurements, the root mean square error of the arithmetic mean of the function is calculated by the formula
. (5)
Confidence interval and confidence probability are determined in the same way as in the case of direct measurements.
Estimation of the total measurement error. Recording the final result.
The total error of the measurement result of X will be defined as the mean square value of the systematic and random errors
, (6)
where δx - instrumental error, Δ X is a random error.
X can be either a directly or indirectly measured quantity.
, α=…, Е=… (7)
It should be borne in mind that the formulas of the theory of errors themselves are valid for a large number of measurements. Therefore, the value of the random, and consequently, the total error is determined for a small n with a big mistake. When calculating Δ X with the number of measurements 
it is recommended to be limited to one significant figure if it is greater than 3 and two if the first significant figure is less than 3. For example, if Δ X= 0.042, then discard 2 and write Δ X=0.04, and if Δ X=0.123, then we write Δ X=0,12.
The number of digits of the result and the total error must be the same. Therefore, the arithmetic mean of the error should be the same. Therefore, the arithmetic mean is first calculated by one digit more than the measurement, and when recording the result, its value is refined to the number of digits of the total error.
4. Methodology for calculating measurement errors.
Errors of direct measurements
When processing the results of direct measurements, it is recommended to adopt the following order of operations.
. (8)
.
.
The total error is determined
The relative error of the measurement result is estimated
.
The final result is written as
, with α=… E=…%.
5. Error of indirect measurements
When evaluating the true value of an indirectly measured quantity , which is a function of other independent quantities 
, two methods can be used.
First way is used if the value y determined under various experimental conditions. In this case, for each of the values, 
, and then the arithmetic mean of all values is determined y i
. (9)
The systematic (instrumental) error is found on the basis of the known instrumental errors of all measurements according to the formula. The random error in this case is defined as the direct measurement error.
Second way applies if the function y determined several times with the same measurements. In this case, the value is calculated from the average values. In our laboratory practice, the second method of determining the indirectly measured quantity is more often used y. The systematic (instrumental) error, as in the first method, is found on the basis of the known instrumental errors of all measurements according to the formula
To find the random error of an indirect measurement, the root mean square errors of the arithmetic mean of individual measurements are first calculated. Then the root mean square error is found y. Setting the confidence probability α, finding the Student's coefficient , determining random and total errors are carried out in the same way as in the case of direct measurements. Similarly, the result of all calculations is presented in the form
, with α=… E=…%.
6. An example of designing a laboratory work
Lab #1
CYLINDER VOLUME DETERMINATION
Accessories: vernier caliper with a division value of 0.05 mm, a micrometer with a division value of 0.01 mm, a cylindrical body.
Objective: familiarization with the simplest physical measurements, determining the volume of a cylinder, calculating the errors of direct and indirect measurements.
Work order
Take at least 5 measurements of the cylinder diameter with a caliper, and its height with a micrometer.
Calculation formula for calculating the volume of a cylinder
where d is the diameter of the cylinder; h is the height.
Measurement results
Table 2.
Measurement No. | ||||||
| ; |
; 
.
; E = 0.5%.
, E = 0.5%.
In most cases, the ultimate goal of laboratory work is to calculate the desired value using some formula, which includes quantities that are measured in a direct way. Such measurements are called indirect. As an example, we give the formula for the density of a solid cylindrical body
where r is the density of the body, m- body mass, d- diameter of the cylinder, h- his high.
Dependence (A.5) in general form can be represented as follows:
where Y is an indirectly measured quantity, in formula (A.5) it is the density r; X 1 , X 2 ,... ,X n are directly measured quantities, in formula (A.5) these are m, d, and h.
The result of an indirect measurement cannot be accurate, since the results of direct measurements of quantities X 1 , x2, ... ,X n always contain errors. Therefore, for indirect measurements, as well as for direct measurements, it is necessary to estimate the confidence interval (absolute error) of the obtained value DY and relative error e.
When calculating errors in the case of indirect measurements, it is convenient to follow the following sequence of actions:
1) get the average values of each directly measured quantity á x1ñ, á x2ñ, …, á X nñ;
2) get the average value of the indirectly measured quantity á Yñ by substituting into formula (A.6) the average values of directly measured quantities;
3) to evaluate the absolute errors of directly measured quantities DX 1 , DX 2 , ..., DXn, using formulas (A.2) and (A.3);
4) based on the explicit form of the function (A.6), obtain a formula for calculating the absolute error of the indirectly measured value DY and calculate it;
6) write down the measurement result, taking into account the error.
Below, without derivation, a formula is given that allows one to obtain formulas for calculating the absolute error, if the explicit form of the function (A.6) is known:
where ¶Y¤¶ x1 etc. - partial derivatives of Y with respect to all directly measured quantities X 1 , X 2 , …, X n (when a partial derivative is taken, for example X 1 , then all other quantities X i are considered constant in the formula), D X i– absolute errors of directly measured quantities, calculated according to (A.3).
Having calculated DY, they find the relative error.
However, if the function (A.6) is a monomial, then it is much easier to first calculate the relative error, and then the absolute one.
Indeed, dividing both sides of equality (A.7) by Y, we get
But since , we can write
Now, knowing the relative error, determine the absolute.
As an example, we obtain a formula for calculating the error in the density of a substance, determined by formula (A.5). Since (A.5) is a monomial, then, as mentioned above, it is easier to first calculate the relative measurement error according to (A.8). In (A.8), under the root we have the sum of squares of partial derivatives of logarithm measured quantity, so first we find the natural logarithm r:
ln r = ln 4 + ln m– ln p –2 ln d–ln h,
and then we use formula (A.8) and obtain that
As can be seen, in (A.9) the average values of directly measured quantities and their absolute errors, calculated by the method of direct measurements according to (A.3), are used. The error introduced by the number p is not taken into account, since its value can always be taken with an accuracy exceeding the measurement accuracy of all other quantities. Calculating e, we find .
If indirect measurements are independent (the conditions of each subsequent experiment differ from the conditions of the previous one), then the values of quantity Y calculated for each individual experiment. Having produced n experiences, get n values Y i. Further, taking each of the values Y i(where i- number of experience) for the result of direct measurement, calculate á Yñ and D Y according to formulas (A.1) and (A.2), respectively.
The final result of both direct and indirect measurements should look like this:
where m- exponent, u– units of measure Y.
ERRORS OF MEASUREMENTS OF PHYSICAL QUANTITIES AND
MEASUREMENT RESULTS PROCESSING
by measurement called finding the values of physical quantities empirically with the help of special technical means. Measurements are either direct or indirect. At direct measurement, the desired value of a physical quantity is found directly with the help of measuring instruments (for example, measuring the dimensions of bodies using a caliper). Indirect called a measurement in which the desired value of a physical quantity is found on the basis of a known functional relationship between the measured quantity and the quantities subjected to direct measurements. For example, when determining the volume V of a cylinder, its diameter D and height H are measured, and then according to the formula p D 2 /4 calculate its volume.
Due to the inaccuracy of measuring instruments and the difficulty of taking into account all side effects in measurements, measurement errors inevitably arise. error or mistake measurement refers to the deviation of the measurement result from the true value of the measured physical quantity. The measurement error is usually unknown, as is the true value of the measured quantity. Therefore, the task of elementary processing of measurement results is to establish the interval within which the true value of the measured physical quantity is located with a given probability.
Classification of measurement errors
Errors are divided into three types:
1) gross or misses,
2) systematic,
3) random.
gross errors- these are erroneous measurements resulting from careless reading on the device, illegible recording of readings. For example, writing a result of 26.5 instead of 2.65; reading on a scale of 18 instead of 13, etc. If a gross error is detected, the result of this measurement should be immediately discarded, and the measurement itself should be repeated.
Systematic errors- errors that remain constant during repeated measurements or change according to a certain law. These errors may be due to the wrong choice of measurement method, imperfection or malfunction of instruments (for example, measurements using an instrument that has a zero offset). In order to eliminate systematic errors as much as possible, one should always carefully analyze the measurement method, compare instruments with standards. In the future, we will assume that all systematic errors have been eliminated, except for those caused by inaccuracies in the manufacture of devices and reading errors. We will call this error hardware.
Random errors - These are errors, the cause of which cannot be taken into account in advance. Random errors depend on the imperfection of our sense organs, on the continuous action of changing external conditions (changes in temperature, pressure, humidity, air vibration, etc.). Random errors are unavoidable, they are inevitably present in all measurements, but they can be estimated using the methods of probability theory.
Processing the results of direct measurements
Let, as a result of direct measurements of a physical quantity, a series of its values be obtained:
x 1 , x 2 , ... x n .
Knowing this series of numbers, you need to indicate the value closest to the true value of the measured value, and find the value of the random error. This problem is solved on the basis of probability theory, a detailed presentation of which is beyond the scope of our course.
The most probable value of the measured physical quantity (close to the true value) is the arithmetic mean
. (1)
Here x i is the result of the i-th measurement; n is the number of measurements. Random measurement error can be estimated by the absolute error D x, which is calculated by the formula
,
(2)
where t(a ,n) - Student's coefficient, depending on the number of measurements n and the confidence level a . Confidence value a set by the experimenter.
Probability random event is the ratio of the number of cases favorable for this event to the total number of equally likely cases. The probability of a sure event is 1, and an impossible one is 0.
The value of the Student's coefficient corresponding to a given confidence level a and a certain number of measurements n, find according to the table. one.
Table 1
|
Number measurements n |
Confidence probability a |
|||
|
0,95 |
0,98 |
|||
|
1,38 |
12,7 |
31,8 |
||
|
1,06 |
||||
|
0,98 |
||||
|
0,94 |
||||
|
0,92 |
||||
|
0,90 |
||||
|
0,90 |
||||
|
0,90 |
||||
|
0,88 |
||||
|
0,84 |
||||
From Table. 1 it can be seen that the value of the Student's coefficient and the random measurement error are the smaller, the larger n and the smaller a . Practically choose a =0.95. However, a simple increase in the number of measurements cannot reduce the total error to zero, since any measuring device gives an error.
Let us explain the meaning of the terms absolute error D x and confidence level a using the number line. Let the average value of the measured quantity
Fig.1
If the measurements of the same quantity are repeated by the same instruments under the same conditions, then the true value of the measured quantity x ist will fall into the same confidence interval, but the hit will not be reliable, but with a probability a.
Calculating the magnitude of the absolute error D x by formula (2), the true value x of the measured physical quantity can be written as x=
To assess the accuracy of measuring a physical quantity, calculate relative error which is usually expressed as a percentage
. (3)
Thus, when processing the results of direct measurements, it is necessary to do the following:
1. Take measurements n times.
2. Calculate the arithmetic mean using formula (1).
3. Set a confidence level a (usually take a = 0.95).
4. According to Table 1, find the Student's coefficient corresponding to the given confidence level a and the number of dimensions n.
5. Calculate the absolute error using formula (2) and compare it with the instrumental one. For further calculations, take the one that is larger.
6. Using formula (3), calculate the relative error e.
7. Write down the final result
x=
Processing the results of indirect measurements
Let the desired physical quantity y be associated with other quantities x 1 , x 2 , ... x k by some functional dependence
Y=f(x 1 , x 2 , ... x k) (4)
Among the values x 1 , x 2 , ... x k there are values obtained from direct measurements and tabular data. It is required to determine the absolute D y and relative e errors in the value of y.
In most cases, it is easier to calculate the relative error first, and then the absolute error. From the theory of probability, the relative error of indirect measurement
.
(5)
Here 
, where is the partial derivative of the function with respect to the variable x i, in the calculation of which all values, except for x i , are considered constant; D x i is the absolute error of x i . If x i is obtained as a result of direct measurements, then its average value
Let's write the final result:
y=
Here
Usually, both random and systematic (instrumental) errors are present in real measurements. If the calculated random error of direct measurements is equal to zero or less than the hardware error by two or more times, then when calculating the error of indirect measurements, the hardware error should be taken into account. If these errors differ by less than two times, then the absolute error is calculated by the formula
.
Consider an example. Let it be necessary to calculate the volume of the cylinder:
. (6)
Here D is the diameter of the cylinder, H is its height, measured with a vernier caliper with a division value of 0.1 mm. As a result of repeated measurements, we find the average values
,
(7)
where D D and D H are absolute errors of direct measurements of diameter and height. Their values are calculated by formula (2): D D=0.01 mm; D H=0.13 mm. Let's compare the calculated errors with the hardware one, equal to the division value of the caliper. D D<0.1, поэтому в формуле (7) подставим вместо D D is not 0.01 mm, but 0.1 mm.
p value must be chosen such that the relative error Dp/p in formula (7) could be neglected. From analysis of measured values and calculated absolute errors D D and D H, it can be seen that the height measurement error makes the greatest contribution to the relative volume measurement error. Calculating the relative height error gives e H =0.01. Therefore, the value p you need to take 3.14. In this case Dp / p » 0.001 (Dp =3.142-3.14=0.002).
One significant figure is left in the absolute error.
Notes.
1. If the measurements are made once or the results of multiple measurements are the same, then the absolute measurement error should be taken as the instrumental error, which for most of the instruments used is equal to the division value of the instrument (for more details on the instrumental error, see the “Measuring instruments” section).
2. If tabular or experimental data are given without specifying the error, then the absolute error of such numbers is taken equal to half the order of the last significant digit.
Actions with approximate numbers
The issue of different calculation accuracy is very important, since overestimation of the calculation accuracy leads to a large amount of unnecessary work. Students often calculate the value they are looking for with an accuracy of five or more significant figures. It should be understood that this precision is excessive. It makes no sense to conduct calculations beyond the limit of accuracy, which is provided by the accuracy of determining directly measured quantities. After processing the measurements, they often do not calculate the errors of individual results and judge the error of the approximate value of the quantity, indicating the number of correct significant digits in this number.
Significant figures An approximate number is called all digits except zero, as well as zero in two cases:
1) when it stands between significant figures (for example, in the number 1071 - four significant figures);
2) when it stands at the end of the number and when it is known that the unit of the corresponding digit is not available in the given number. Example. There are three significant figures in the number 5.20, and this means that when measuring we took into account not only units, but also tenths and hundredths, and in the number 5.2 - only two significant figures, which means that we took into account only integers and tenths.
Approximate calculations should be made in compliance with the following rules.
1. When adding and subtracting as a result, retain as many decimal places as there are in the number with the least number of decimal places. For example: 0.8934+3.24+1.188=5.3214» 5.32. The amount should be rounded to hundredths, i.e. take equal to 5.32.
2. When multiplying and dividing as a result, as many significant digits are retained as the approximate number with the fewest significant digits has. For example, you need to multiply 8.632´ 2.8´ 3.53. Instead, expressions should be evaluated
8.6 ´ 2.8 ´ 3.5 » 81.
When calculating intermediate results, they save one digit more than the rules recommend (the so-called spare digit). In the final result, the spare digit is discarded. To clarify the value of the last significant digit of the result, you need to calculate the digit behind it. If it turns out to be less than five, it should simply be discarded, and if five or more than five, then, having discarded it, the previous figure should be increased by one. Usually, one significant digit is left in the absolute error, and the measured value is rounded up to the digit in which the significant digit of the absolute error is located.
3. The result of calculating the values of the functions x n , , lg( x) some approximate number x must contain as many significant digits as there are in the number x. For example: 
.
Plotting
The results obtained during the performance of laboratory work are often important and must be presented in a graphical relationship. In order to build a graph, it is necessary, on the basis of the measurements made, to compile a table in which each value of one of the quantities corresponds to a certain value of the other.
Graphs are made on graph paper. When constructing a graph, the values of the independent variable should be plotted on the abscissa, and the values of the function on the ordinate. Near each axis, you need to write the designation of the displayed value and indicate in what units it is measured (Fig. 2).
Fig.2
For the correct construction of the graph, the choice of scale is important: the curve occupies the entire sheet, and the dimensions of the graph in length and height are approximately the same. The scale should be simple. The easiest way is if the unit of the measured value (0.1; 10; 100, etc.) corresponds to 1, 2 or 5 cm. It should be borne in mind that the intersection of the coordinate axes does not have to coincide with the zero values of the values being plotted (Fig. 2).
Each experimental value obtained is plotted on the graph in a fairly noticeable way: a dot, a cross, etc.
Errors are indicated for the measured values in the form of segments with a length of a confidence interval, in the center of which the experimental points are located. Since the indication of errors clutters up the graph, this is done only when information about the errors is really needed: when constructing a curve from experimental points, when determining errors using a graph, when comparing experimental data with a theoretical curve (Figure 2). Often it is enough to specify the error for one or more points.
It is necessary to draw a smooth curve through the experimental points. Often, the experimental points are connected by a simple broken line. Thus, as it were, it is indicated that the quantities depend on each other in some jumpy way. And this is incredible. The curve must be smooth and may pass not through the marked points, but close to them so that these points are on both sides of the curve at the same distance from it. If any point strongly falls out of the graph, then this measurement should be repeated. Therefore, it is desirable to build a graph directly during the experiment. The graph can then serve to control and improve observations.
MEASURING INSTRUMENTS AND ACCOUNTING FOR THEIR ERRORS
Measuring instruments are used for direct measurements of physical quantities. Any measuring instruments do not give the true value of the measured value. This is due, firstly, to the fact that it is impossible to accurately read the measured value on the scale of the instrument, and secondly, to the inaccuracy in the manufacture of measuring instruments. To take into account the first factor, the reading error Δx o is introduced, for the second - the allowable errorΔ x d. The sum of these errors forms the instrumental or absolute error of the deviceΔ x:
.
The permissible error is normalized by state standards and indicated in the passport or description of the device.
The reading error is usually taken equal to half the division of the instrument, but for some instruments (stopwatch, aneroid barometer) - equal to the division of the instrument (since the position of the arrow of these instruments changes in jumps by one division) and even several divisions of the scale, if the conditions of the experiment do not allow confidently count up to one division (for example, with a thick pointer or poor lighting). Thus, the counting error is set by the experimenter himself, actually reflecting the conditions of a particular experiment.
If the allowable error is much less than the reading error, then it can be ignored. Usually, the absolute error of the instrument is taken equal to the scale division of the instrument.
Measuring rulers usually have millimeter divisions. For measurement, it is recommended to use steel or drawing rulers with a bevel. The permissible error of such rulers is 0.1 mm and it can be ignored, since it is much less than the reading error equal to ± 0.5 mm. Permissible error of wooden and plastic rulers± 1 mm.
The permissible measurement error of a micrometer depends on the upper limit of measurement and can be ± (3-4) µm (for micrometers with measuring range 0-25 mm). Half of the division value is taken as the reading error. Thus, the absolute error of the micrometer can be taken equal to the division value, i.e. 0.01 mm.
When weighing, the permissible error of technical scales depends on the load and amounts to 50 mg for a load of 20 to 200 g, and 25 mg for a load of less than 20 g.
The error of digital instruments is determined by the accuracy class.
The formulas for calculating the errors of indirect measurements are based on the representations of differential calculus.
Let the dependence of the quantity Y from the measured value Z has a simple form: .
Here and are constants whose values are known. If z is increased or decreased by some number , then it will change to :
If - the error of the measured value Z, then, respectively, will be the error of the calculated value Y.
We obtain the formula for the absolute error in the general case of a function of one variable. Let the graph of this function have the form shown in Fig.1. The exact value of the argument z 0 corresponds to the exact value of the function y 0 = f(z 0).
The measured value of the argument differs from the exact value of the argument by the value of Δz due to measurement errors. The value of the function will differ from the exact value by Δy.
From the geometric meaning of the derivative as the tangent of the slope of the tangent to the curve at a given point (Fig. 1), it follows:
. (10)
The formula for the relative error of indirect measurement in the case of a function of one variable will be: 
. (11)
Considering that the differential of the function is , we get
(12)
If the indirect measurement is a function m variables 
, then the error of indirect measurement will depend on the errors of direct measurements. We denote the partial error associated with the measurement error of the argument . It constitutes the increment of the function by the increment, provided that all other arguments are unchanged. Thus, we write the partial absolute error according to (10) in the following form:
(13)
Thus, in order to find the partial error of indirect measurement , it is necessary, according to (13), to multiply the partial derivative by the error of direct measurement . When calculating the partial derivative of a function with respect to the remaining arguments, they are considered constant.
The resulting absolute error of indirect measurement is determined by the formula, which includes the squares of partial errors
indirect measurement :
or taking into account (13)
(14)
The relative error of indirect measurement is determined by the formula:
Or taking into account (11) and (12)
. (15)
Using (14) and (15), one of the errors is found, absolute or relative, depending on the convenience of calculations. So, for example, if the working formula has the form of a product, the ratio of the measured quantities, it is easy to take a logarithm and use formula (15) to determine the relative error of indirect measurement. Then calculate the absolute error using formula (16):
To illustrate the above procedure for determining the error of indirect measurements, let's return to the virtual laboratory work "Determining the acceleration of free fall using a mathematical pendulum".
The working formula (1) has the form of the ratio of the measured values:
Therefore, we begin with the definition of the relative error. To do this, we take the logarithm of this expression, and then calculate the partial derivatives:
; ; 
.
Substitution into formula (15) leads to the formula for the relative error of indirect measurement:
(17)
After substituting the results of direct measurements
{ 
; 
) in (17) we get:
(18)
To calculate the absolute error, we use expression (16) and the previously calculated value (9) of the gravitational acceleration g:
The result of calculating the absolute error is rounded up to one significant figure. The calculated value of the absolute error determines the accuracy of recording the final result:
, α ≈ 1. (19)
In this case, the confidence probability is determined by the confidence probability of those of the direct measurements that made a decisive contribution to the error of the indirect measurement. In this case, these are period measurements.
Thus, with a probability close to 1, the value g lies between 8 and 12.
To obtain a more accurate value of the free fall acceleration g it is necessary to improve the measurement technique. To this end, it is necessary to reduce the relative error , which, as follows from formula (18), is mainly determined by the time measurement error.
To do this, it is necessary to measure the time of not one complete oscillation, but, for example, 10 complete oscillations. Then, as follows from (2), the relative error formula will take the form:
. (20)
Table 4 presents the results of measuring time for N = 10
For the quantity L take the measurement results from Table 2. Substituting the results of direct measurements into formula (20), we find the relative error of indirect measurements:
Using formula (2), we calculate the value of the indirectly measured quantity:
.
.
The final result is written as:
; ; .
This example shows the role of the relative error formula in the analysis of possible directions for improving the measurement technique.
