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%.
If the desired physical quantity cannot be measured directly by the device, but is expressed through the measured quantities by means of a formula, then such measurements are called indirect.
As with direct measurements, you can calculate the mean absolute (arithmetic mean) error or the root mean square error of indirect measurements.
General rules for calculating errors for both cases are derived using differential calculus.
Let the physical quantity j( x, y, z, ...) is a function of a number of independent arguments x, y, z, ..., each of which can be determined experimentally. Quantities are determined by direct measurements and their mean absolute errors or root mean square errors are evaluated.
The average absolute error of indirect measurements of the physical quantity j is calculated by the formula
where are the partial derivatives of φ with respect to x, y, z calculated for the average values of the corresponding arguments.
Since the formula uses the absolute values of all terms of the sum, the expression for estimates the maximum error in measuring the function for given maximum errors of the independent variables.
Root mean square error of indirect measurements of the physical quantity j
Relative maximum error of indirect measurements of the physical quantity j
where, etc.
Similarly, we can write the relative root-mean-square error of indirect measurements j
If the formula represents an expression convenient for taking logarithms (that is, a product, a fraction, a power), then it is more convenient to first calculate the relative error. To do this (in the case of the average absolute error), the following should be done.
1. Take the logarithm of the expression for the indirect measurement of a physical quantity.
2. Differentiate it.
3. Combine all terms with the same differential and take it out of brackets.
4. Take the expression in front of various modulo differentials.
5. Formally replace the icons of the differentials with the icons of the absolute error D.
Then, knowing e, one can calculate the absolute error Dj by the formula
Example 1 Derivation of a formula for calculating the maximum relative error of indirect measurements of the volume of a cylinder.
Expression for indirect measurement of a physical quantity (initial formula)
Diameter value D and cylinder height h measured directly by instruments with direct measurement errors, respectivelyD D and D h.
We take the logarithm of the original formula and get
Differentiate the resulting equation
Replacing the icons of the differentials with the icons of the absolute error D, we finally obtain a formula for calculating the maximum relative error of indirect measurements of the cylinder volume
Now it is necessary to consider the question of how to find the error of the physical quantity U, which is determined by indirect measurements. General view of the measurement equation
Y=f(X 1 , X 2 , … , X n), (1.4)
where X j- various physical quantities that are obtained by the experimenter by direct measurements, or physical constants known with a given accuracy. In a formula, they are function arguments.
In measurement practice, two methods for calculating the error of indirect measurements are widely used. Both methods give almost the same result.
Method 1. Absolute D is found first, then relative d errors. This method is recommended for measurement equations that contain sums and differences of arguments.
General formula for calculating the absolute error in indirect measurements of a physical quantity Y for an arbitrary view f function looks like:
where the partial derivatives of the functions Y=f(X 1 , X 2 , … , X n) by argument X j,
The total error of direct measurements of the quantity X j.
To find the relative error, you must first find the average value of the quantity Y. To do this, it is necessary to substitute the arithmetic mean values of the quantities into the measurement equation (1.4) Xj.
That is, the average value of the value Y equals: . Now it is easy to find the relative error: .
Example: find the error in volume measurement V cylinder. Height h and diameter D of the cylinder are considered to be determined by direct measurements, and let the number of measurements n= 10.
The formula for calculating the volume of a cylinder, that is, the measurement equation is:
Let at P= 0,68;
At P= 0,68.
Then, substituting the average values into formula (1.5), we find:
Error DV in this example depends, as can be seen, mainly on the measurement error of the diameter.
The average volume is: , relative error d V is equal to:
Or d V = 19%.
V=(47±9) mm 3 , d V = 19%, P= 0,68.
Method 2. This method of determining the error of indirect measurements differs from the first method in less mathematical difficulties, so it is more often used.
First, find the relative error d, and only then absolute D. This method is especially convenient if the measurement equation contains only products and ratios of arguments.
The procedure can be considered using the same specific example - determining the error in measuring the volume of a cylinder
We will keep all numerical values of the quantities included in the formula the same as in the calculations for way 1.
Let be mm, ; at P= 0,68;
; at P=0.68.
Number rounding error p(see fig. 1.1)
Using way 2 should act like this:
1) take the logarithm of the measurement equation (we take the natural logarithm)
find the differentials of the left and right parts, considering independent variables,
2) replace the differential of each value with the absolute error of the same value, and the “minus” signs, if they are before the errors, with “plus”:
3) it would seem that with the help of this formula it is already possible to give an estimate for the relative error , but this is not so. It is required to estimate the error in such a way that the confidence probability of this estimate coincides with the confidence probabilities of estimating the errors of those terms that are on the right side of the formula. To do this, in order for this condition to be fulfilled, you need to square all the terms of the last formula, and then extract the square root from both sides of the equation:
Or in other notation, the relative error of the volume is:
moreover, the probability of this estimate of the volume error will coincide with the probability of estimating the errors of the terms included in the radical expression:
Having done the calculations, we will make sure that the result coincides with the estimate by method 1:
Now, knowing the relative error, we find the absolute:
D V=0.19 47=9.4 mm 3 , P=0,68.
Final result after rounding:
V\u003d (47 ± 9) mm 3, d V = 19%, P=0,68.
test questions
1. What is the task of physical measurements?
2. What types of measurements are distinguished?
3. How are measurement errors classified?
4. What are absolute and relative errors?
5. What are misses, systematic and random errors?
6. How to evaluate the systematic error?
7. What is the arithmetic mean of the measured value?
8. How to estimate the magnitude of the random error, how is it related to the standard deviation?
9. What is the probability of finding the true value of the measured value in the interval from X cf - s before X cf + s?
10. If, as an estimate for a random error, we choose the value 2s or 3s, then with what probability will the true value fall within the intervals determined by these estimates?
11. How to summarize errors and when should it be done?
12. How to round the absolute error and the average value of the measurement result?
13. What methods exist for estimating errors in indirect measurements? How to proceed with this?
14. What should be recorded as the measurement result? What values to indicate?
Lecture #8
Processing of measurement results
Direct single and multiple measurements.
1. Direct single measurements .
In the general case, the task of estimating the error of the result obtained is usually carried out on the basis of information about the limit of the main error of the measuring instrument (according to the regulatory and technical documentation for the measuring instruments used) and the known values of additional errors from the influence of influencing quantities. The maximum value of the total error of the measurement result (without taking into account the sign) can be found by summing the components in absolute value:
A more realistic estimate of the error can be obtained by statistical addition of the components of the error:
where is the boundary of the i-th non-excluded component of the systematic error; k- coefficient determined by the accepted confidence probability (at P = 0,95, coefficient k=1.11); m is the number of non-excluded components.
The measurement result is recorded according to the first form of recording results:
where is the result of a single measurement; - total error of the measurement result; Р - confidence probability (at Р = 0,95 may not be specified).
When measuring under normal conditions, we can assume
2. Direct multiple measurements.
It is possible to accurately assess the actual value of the measured quantity only by its multiple measurements and appropriate processing of their results. Correctly processing the obtained results of observations means obtaining the most accurate estimate of the actual value of the measured quantity and the confidence interval in which its true value is located.
In the process of processing the results of observations, it is necessary to consistently solve the following main tasks:
Determine point and integral estimates of the law of distribution of measurement results by the formulas:
where D(x) is a point estimate of the variance;
Eliminate "misses" (according to one of the criteria);
Eliminate systematic measurement errors;
Determine the confidence limits of the non-excluded balance of the systematic component, the random component and the total error of the measurement result;
Record the measurement result.
Estimation of the error of indirect measurements. Basic principles and stages of calculations. GOSTs for processing results.
Errors of indirect measurements
The estimation of errors arising from indirect measurements is based on the following assumptions:
1. The relative errors of the values obtained by direct measurements and involved in the calculation of the desired value must be small compared to unity (in practice, they should not exceed 10%).
2. For the errors of all quantities involved in the calculation, the same confidence probability is accepted. The error of the desired value will also have the same confidence probability.
3. The most probable value of the desired value is obtained if the most probable values of the initial values are used for its calculation, i.e. their arithmetic averages.
Error in the case of one initial value.
Absolute error. Let the desired value y, measured indirectly, depends on only one quantity a obtained by direct measurement. The boundaries of the interval in which the value lies with a given probability a, are determined by the arithmetic mean and the total absolute error a quantities a. This means that the value a may lie inside an interval with bounds ± a.
With indirect measurement for the quantity y(a) such boundaries will be determined by its most probable value =y() and error y, i.e. values y lie inside the interval with boundaries ± y. Upper bound for y(with monotonic increase) there will be a value corresponding to the upper bound a, i.e. value + y= y( + a) . Thus, the absolute error y quantities y has the form of a function increment y(a) caused by incrementing its argument a by the amount a its absolute error. Therefore, we can use the rules of differential calculus, according to which, for small values a increment y can be approximately expressed as
Here is the derivative with respect to a functions y(a) at a = .
Thus, the absolute error of the final result can be calculated using formula (1), and the confidence probability corresponds to the confidence probability that a.
Relative error. To find the relative error of a value y, divide (1) by y and take into account that
is the derivative with respect to a natural logarithm y. The result will be
If we substitute into this expression a= and y= , then its value will be the relative error of the quantity y.
To process the results of measurements, it is used GOST 8.207-76 “GSI. Direct measurements with multiple observations. Methods for processing the results of observations.
8.3. Measurement result and estimation of its standard deviation:
1. Methods for detecting gross errors should be specified in the measurement procedure. If the results of observations can be considered to belong to a normal distribution, gross errors are excluded.
2. The measurement result is taken as the arithmetic mean of the observation results, in which corrections have been previously introduced to eliminate systematic errors.
3. Standard deviation S the result of observation is evaluated according to the NTD.
4. The standard deviation of the measurement result is estimated by the formula
,
where x i - i-th observation result;
Measurement result (arithmetic mean of the corrected observation results);
n- number of observation results;
Estimation of the standard deviation of the measurement result.
8.4. Confidence limits of the random error of the measurement result:
1. Confidence limits of the random error of the measurement result in accordance with this standard are established for the results of observations belonging to the normal distribution. If this condition is not met, methods for calculating the confidence limits of a random error should be specified in the procedure for performing specific measurements.
1.1. With the number of observation results n>50 to check whether they belong to the normal distribution according to the NTD, one of the criteria is preferable: χ 2 Pearson or ω 2 Mises - Smirnov.
When processing the results of indirect measurements of a physical quantity that is functionally related to the physical quantities A, B and C, which are measured in a direct way, first determine the relative error of indirect measurement e = DX / X pr using the formulas given in the table (without evidence).
The absolute error is determined by the formula DX \u003d X pr * e,
where e is expressed as a decimal, not as 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=A*B*C | |
| X=A n | |
| X=A/B | |
| X= |
(+ http://fiz.1september.ru/2001/16/no16_01.htm useful) How to take measurements http://www.fizika.ru/fakultat/index.php?theme=01&id=1220
Example: Let us calculate the error in measuring the friction coefficient using a dynamometer. The experience is that the bar is uniformly pulled along a horizontal surface and the applied force is measured: it is equal to the force of sliding friction.
Using a dynamometer, we weigh a bar with loads: 1.8 N. F tr \u003d 0.6 N
μ=0.33. The instrumental error of the dynamometer (find from the table) is Δ and \u003d 0.05N, Reading error (half of the scale division)
Δ o \u003d 0.05N. 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 of μ is 0.22*0.33=0.074
Answer:
To measure a physical quantity means to compare it with another homogeneous quantity taken as a unit of measurement. The measurement can be made using:
1. measures, which are samples of a unit of measurement (meter, weight, liter vessel, etc.),
2. measuring instruments (ammeter, pressure gauge, etc.),
3. measuring installations, which are understood as a set of measures, measuring instruments and auxiliary elements.
Measurements are either direct or indirect. In direct measurements the physical quantity is measured directly. Direct measurements are, for example, measuring the length with a ruler, time with a stopwatch, current strength with an ammeter.
In indirect measurements they directly measure not the quantity whose value needs to be known, but other quantities with which the desired quantity is associated with a certain mathematical dependence. For example, the density of a body is determined by measuring its mass and volume, and the resistance is determined by measuring current and voltage.
Due to the imperfection of measures and measuring instruments, as well as our sense organs, measurements cannot be performed accurately, i.e. any measurement gives only an approximate result. In addition, the nature of the measurand itself is often the reason for the deviation of measurement results. For example, the temperature measured by a thermometer or thermocouple at a certain point in the furnace fluctuates due to convection and thermal conductivity within certain limits. The measure for assessing the accuracy of the measurement result is measurement error (measurement error).
To assess the accuracy, either the absolute error or the relative measurement error is indicated. Absolute error expressed in units of the measured quantity. For example, the segment of the path traveled by the body, , is measured with an absolute error . Relative measurement error is the ratio of the absolute error to the value of the measured quantity. In the given example, the relative error is . The smaller the measurement error, the higher its accuracy.
According to the sources of their origin, measurement errors are divided into systematic, random and gross (misses).
1. Systematic errors- measurement errors, the value of which remains constant during repeated measurements carried out by the same method, using the same measuring instruments. The reasons for systematic errors are:
malfunctions, inaccuracies of measuring instruments
illegality, inaccuracy of the measurement technique used
An example of systematic errors can be the measurement of temperature with a thermometer with a shifted zero point, the measurement of current with an incorrectly calibrated ammeter, the weighing of a body on a balance using weights without taking into account the buoyancy force of Archimedes.
To eliminate or reduce systematic errors, it is necessary to carefully check the measuring instruments, measure the same quantities by different methods, and introduce corrections when the errors are known (corrections for buoyancy force, corrections for thermometer readings).
2. Gross mistakes (misses)- a significant excess of the error expected under the given measurement conditions. Misses appear as a result of incorrect recording of instrument readings, incorrect readings on the instrument, due to errors in calculations during indirect measurements. The source of misses is the inattention of the experimenter. The way to eliminate these errors is the accuracy of the experimenter, the exclusion of rewriting measurement protocols.
3. Random errors- errors, the value of which changes randomly during repeated measurements of the same value by the same method using the same instruments. The source of random errors is the uncontrolled reproducibility of the measurement conditions. For example, during the measurement, temperature, humidity, atmospheric pressure, voltage in the electrical network, and the state of the experimenter's senses can change in an uncontrolled way. It is impossible to rule out random errors. With multiple measurements, random errors obey statistical laws, and their influence can be taken into account.
