Introduction. Processing of measurement results in physical practice Measurements and measurement errors Analysis of direct measurement results

Random errors have the following properties.

With a large number of measurements, errors of the same magnitude but opposite in sign occur equally often.

Large errors are less likely to occur than small ones. From relations (1), rewriting them in the form

X \u003d x 1 + x 1

X = x 2 + x 2

X = x n + x n

and adding up in a column, you can determine the true value of the measured value as follows:

or
.

(2)

those. the true value of the measured quantity is equal to the arithmetic mean of the measurement results, if there are an infinite number of them. With a limited, and even more so with a small number of measurements, which we usually deal with in practice, equality (2) is approximate.

Let the following values of the measured quantity X be obtained as a result of several measurements: 13.4; 13.2; 13.3; 13.4; 13.3; 13.2; 13.1; 13.3; 13.3; 13.2; 13.3; 13.1. Let's build a diagram of the distribution of these results, plotting the instrument readings along the abscissa axis in ascending order. The distances between adjacent points along the abscissa axis are equal to twice the maximum reading error on the instrument. In our case, the countdown is made up to 0.1. This is equal to one division of the scale marked on the x-axis. On the ordinate axis, we plot values proportional to the relative number of results corresponding to a particular reading of the device. The relative number, or the relative frequency of results equal to x k, will be denoted by W(x k). In our case

We assign each x to

(3)

where A is the coefficient of proportionality.

The diagram, which is called a histogram, differs from the usual graph in that the points are not connected by a smooth curved line, but steps are drawn through them. It is obvious that the area of the step over some value of x k is proportional to the relative frequency of occurrence of this result. By choosing the proportionality coefficient in expression (3) in an appropriate way, this area can be made equal to the relative frequency of the result x k. Then the sum of the areas of all steps, as the sum of the relative frequencies of all results, should be equal to one

From here we find A=10. Condition (4) is called the normalization condition for function (3).

If you make a series of measurements with n measurements in each series, then with a small n the relative frequencies of the same value x k found from different series can differ significantly from each other. As the number of measurements in the series increases, the fluctuations in the values of W(x k) decrease and these values approach a certain constant number, which is called the probability of the result x k and is denoted by P (x k).

Let us assume that, while making an experiment, we do not count the result to whole divisions of the scale or their shares, but we can fix the point where the arrow stopped. Then, for an infinitely large number of measurements, the arrow will visit each point on the scale. The distribution of measurement results in this case acquires a continuous character and is described by a continuous curve y=f(x) instead of a stepped histogram. Based on the properties of random errors, it can be concluded that the curve must be symmetrical and, therefore, its maximum falls on the arithmetic mean of the measurement results, which is equal to the true value of the measured quantity. In the case of a continuous distribution of measurement results, there is no

it makes sense to talk about the probability of any of their values, because there are values arbitrarily close to the one under consideration. Now we should already raise the question of the probability of meeting during measurements the result in a certain interval around the value of x k, equal to
,
. Just as on the histogram the relative frequency of the result x to equaled the area of the step built over this result, on the graph for a continuous distribution the probability of finding the result in the interval (
,
) is equal to the area of the curvilinear trapezoid constructed over this interval and bounded by the curve f(x). The mathematical notation of this result is

if
little, i.e. the area of the hatched curvilinear trapezoid is replaced by the approximate area of a rectangle with the same base and a height equal to f(xk). The function f(x) is called the probability density of the distribution of measurement results. The probability of finding x in some interval is equal to the probability density for the given interval multiplied by its length.

The distribution curve of the measurement results obtained experimentally for a certain section of the instrument scale, if it is continued, asymptotically approximating the abscissa axis from the left and right, is analytically well described by a function of the form

(5)

Just as the total area of all the steps on the histogram was equal to one, the entire area between the f (x) curve and the abscissa axis, which has the meaning of the probability of meeting at least some value of x during measurements, is also equal to one. The distribution described by this function is called the normal distribution. The main parameter of the normal distribution is the variance  2 . The approximate value of the dispersion can be found from the measurement results using the formula

(6)

This formula gives a dispersion close to the real value only for a large number of measurements. For example, σ 2 found from the results of 100 measurements may have a deviation from the actual value of 15%, found from 10 measurements already 40%. The variance determines the shape of the normal distribution curve. When the random errors are small, the dispersion, as follows from (6), is small. The curve f(x) in this case is narrower and sharper near the true value of X and tends to zero faster when moving away from it than with large errors. The following figure will show how the form of the curve f(x) for a normal distribution changes depending on σ.

In probability theory, it is proved that if we consider not the distribution of measurement results, but the distribution of arithmetic mean values found from a series of n measurements in each series, then it also obeys the normal law, but with a dispersion that is n times smaller.

The probability of finding the measurement result in a certain interval (
) near the true value of the measured value is equal to the area of the curvilinear trapezoid built over this interval and bounded from above by the curve f(x). Interval value
usually measured in units proportional to the square root of the variance
Depending on the value of k per interval
there is a curvilinear trapezoid of a larger or smaller area, i.e.

where F(k) is some function of k. Calculations show that for

k=1,

k=2,

k=3,

This shows that in the interval
accounts for approximately 95% of the area under the curve f(x). This fact is in full agreement with the second property of random errors, which states that large errors are unlikely. Errors greater than
, occurs with a probability of less than 5%. The expression (7) rewritten for the distribution of the arithmetic mean of n measurements takes the form

(8)

Value in (7) and (8) can be determined on the basis of measurement results only approximately by formula (6)

Substituting this value into expression (8), we will get on the right not F (k), but some new function, depending not only on the size of the considered interval of values X, but also on the number of measurements made
And

because only for a very large number of measurements does formula (6) become sufficiently accurate.

Having solved the system of two inequalities in brackets on the left side of this expression with respect to the true value of X, we can rewrite it in the form

Expression (9) determines the probability with which the true value of X is in a certain interval of length about value . This probability in the theory of errors is called reliability, and the interval corresponding to it for the true value is called the confidence interval. Function
calculated depending on t n and n and a detailed table has been compiled for it. The table has 2 inputs: pt n and n. With its help, for a given number of measurements n, it is possible to find, given a certain value of reliability Р, the value of t n , called the Student's coefficient.

An analysis of the table shows that for a certain number of measurements with the requirement of increasing reliability, we obtain growing values of t n , i.e. an increase in the confidence interval. A reliability equal to one would correspond to a confidence interval equal to infinity. Given a certain reliability, we can make the confidence interval for the true value narrower by increasing the number of measurements, since S n does not change much, and decreases both by decreasing the numerator and by increasing the denominator. Having made a sufficient number of experiments, it is possible to make a confidence interval of any small value. But for large n, a further increase in the number of experiments very slowly reduces the confidence interval, and the amount of computational work increases much. Sometimes in practical work it is convenient to use an approximate rule: in order to reduce the confidence interval found from a small number of measurements by several times, it is necessary to increase the number of measurements by the same factor.

EXAMPLE OF DIRECT MEASUREMENT RESULTS PROCESSING

Let's take as experimental data the first three results out of 12, according to which the histogram X was built: 13.4; 13.2; 13.3.

Let's ask ourselves the reliability, which is usually accepted in the educational laboratory, P = 95%. From the table for P = 0.95 and n = 3 we find t n = 4.3.

with 95% reliability. The last result is usually written as an equality

If the confidence interval of such a value does not suit (for example, in the case when the instrumental error is 0.1), and we want to halve it, we should double the number of measurements.

If we take, for example, the last 6 values of the same 12 results (for the first six, it is proposed to do the calculation yourself)

X: 13.1; 13.3; 13.3; 13.2; 13.3; 13.1,

then

The value of the coefficient t n is found from the table for Р = 0.95 and n = 6; tn = 2.6.

In this case
Let's plot the confidence interval for the true value in the first and second cases on the numerical axis.

The interval calculated from 6 measurements is, as expected, within the interval found from three measurements.

The instrumental error introduces a systematic error into the results, which expands the confidence intervals depicted on the axis by 0.1. Therefore, the results written taking into account the instrumental error have the form

1)
2)

In the general case, the procedure for processing the results of direct measurements is as follows (it is assumed that there are no systematic errors).

Case 1 The number of measurements is less than five.

1) According to formula (6), the average result is found x, defined as the arithmetic mean of the results of all measurements, i.e.

2) According to the formula (12), the absolute errors of individual measurements are calculated

3) According to the formula (14), the average absolute error is determined

4) According to formula (15), the average relative error of the measurement result is calculated

5) Record the final result in the following form:

, at
.

Case 2. The number of measurements is over five.

1) According to formula (6), the average result is found

2) According to the formula (12), the absolute errors of individual measurements are determined

3) According to the formula (7), the mean square error of a single measurement is calculated

4) Calculate the standard deviation for the average value of the measured value by the formula (9).

5) The final result is recorded in the following form

Sometimes random measurement errors may turn out to be less than the value that the measuring device (instrument) is able to register. In this case, for any number of measurements, the same result is obtained. In such cases, as the average absolute error
take half the scale division of the instrument (tool). This value is sometimes called the limiting or instrumental error and denoted
(for vernier instruments and stopwatch
equal to the accuracy of the instrument).

Assessment of the reliability of measurement results

In any experiment, the number of measurements of a physical quantity is always limited for one reason or another. Due with this may be the task of assessing the reliability of the result. In other words, determine with what probability it can be argued that the error made in this case does not exceed the predetermined value ε. This probability is called the confidence probability. Let's denote it with a letter.

An inverse problem can also be posed: to determine the boundaries of the interval
so that with a given probability it could be argued that the true value of the measurements of the quantity will not go beyond the specified, so-called confidence interval.

The confidence interval characterizes the accuracy of the result obtained, and the confidence interval characterizes its reliability. Methods for solving these two groups of problems are available and have been developed in particular detail for the case when the measurement errors are distributed according to the normal law. Probability theory also provides methods for determining the number of experiments (repeated measurements) that provide a given accuracy and reliability of the expected result. In this work, these methods are not considered (we restrict ourselves to mentioning them), since such tasks are usually not posed when performing laboratory work.

Of particular interest, however, is the case of assessing the reliability of the result of measurements of physical quantities with a very small number of repeated measurements. For example,
. This is exactly the case with which we often meet in the performance of laboratory work in physics. When solving this kind of problems, it is recommended to use the method based on Student's distribution (law).

For the convenience of practical application of the method under consideration, there are tables with which you can determine the confidence interval
corresponding to a given confidence level or solve the inverse problem.

Below are those parts of the mentioned tables that may be required when evaluating the results of measurements in laboratory classes.

Let, for example, produced equal (under the same conditions) measurements of some physical quantity and calculated its average value . It is required to find the confidence interval corresponding to the given confidence level . The problem is generally solved in the following way.

According to the formula, taking into account (7), calculate

Then for given values n and find according to the table (Table 2) the value . The value you are looking for is calculated based on the formula

(16)

When solving the inverse problem, the parameter is first calculated using formula (16). The desired value of the confidence probability is taken from the table (Table 3) for a given number and calculated parameter .

Table 2. Parameter value for a given number of experiments

and confidence level

Table 3 The value of the confidence probability for a given number of experiments n and parameter ε

The main provisions of the methods for processing the results of direct measurements with multiple observations are defined in GOST 8.207-76.

Take as the measurement result average data n observations, from which systematic errors are excluded. It is assumed that the results of observations after the exclusion of systematic errors from them belong to the normal distribution. To calculate the result of the measurement, it is necessary to exclude the systematic error from each observation and, as a result, obtain the corrected result i-th observation. The arithmetic mean of these corrected results is then calculated and taken as the measurement result. The arithmetic mean is a consistent, unbiased, and efficient estimate of a measurand under a normal distribution of observational data.

It should be noted that sometimes in the literature, instead of the term observation result the term is sometimes used single measurement result, from which systematic errors are excluded. At the same time, the arithmetic mean value is understood as the measurement result in this series of several measurements. This does not change the essence of the results processing procedures presented below.

When statistically processing groups of observation results, the following should be performed: operations :

1. Eliminate the known systematic error from each observation and obtain the corrected result of the individual observation x.

2. Calculate the arithmetic mean of the corrected observation results, taken as the measurement result:

3. Calculate the estimate of the standard deviation

observation groups:

Check Availability gross errors – are there any values that go beyond ±3 S. With a normal distribution law with a probability practically equal to 1 (0.997), none of the values of this difference should go beyond the specified limits. If they are, then the corresponding values should be excluded from consideration and the calculations and evaluation should be repeated again. S.

4. Calculate the RMS estimate of the measurement result (average

arithmetic)

5. Test the hypothesis about the normal distribution of the results of observations.

There are various approximate methods for checking the normality of the distribution of observational results. Some of them are given in GOST 8.207-76. If the number of observations is less than 15, in accordance with this GOST, their belonging to the normal distribution is not checked. The confidence limits of the random error are determined only if it is known in advance that the results of the observations belong to this distribution. Approximately, the nature of the distribution can be judged by constructing a histogram of the results of observations. Mathematical methods for checking the normality of a distribution are discussed in the specialized literature.

6. Calculate the confidence limits e of the random error (random component of the error) of the measurement result

where tq- Student's coefficient, depending on the number of observations and the confidence level. For example, when n= 14, P= 0,95 tq= 2.16. The values of this coefficient are given in the appendix to the specified standard.

7. Calculate the limits of the total non-excluded systematic error (TSE) of the measurement result Q (according to the formulas in Section 4.6).

8. Analyze the ratio of Q and :

If , then the NSP is neglected in comparison with random errors, and the error limit of the result D=e.. If > 8, then the random error can be neglected and the error limit of the result D=Θ . If both inequalities are not met, then the margin of error of the result is found by constructing a composition of distributions of random errors and NSP according to the formula: , where To– coefficient depending on the ratio of random error and NSP; S e- assessment of the total standard deviation of the measurement result. The estimate of the total standard deviation is calculated by the formula:

The coefficient K is calculated by the empirical formula:

The confidence level for calculating and must be the same.

The error from applying the last formula for the composition of uniform (for NSP) and normal (for random error) distributions reaches 12% at a confidence level of 0.99.

9. Record the measurement result. There are two options for writing the measurement result, since it is necessary to distinguish between measurements, when obtaining the value of the measured quantity is the ultimate goal, and measurements, the results of which will be used for further calculations or analysis.

In the first case, it is enough to know the total error of the measurement result, and with a symmetrical confidence error, the measurement results are presented in the form: , where

where is the measurement result.

In the second case, the characteristics of the components of the measurement error should be known - the estimate of the standard deviation of the measurement result , the boundaries of the NSP , the number of observations made. In the absence of data on the form of distribution functions of the error components of the result and the need for further processing of the results or analysis of errors, the measurement results are presented in the form:

If the boundaries of the NSP are calculated in accordance with clause 4.6, then the confidence probability P is additionally indicated.

Estimates and derivatives of their value can be expressed both in absolute form, that is, in units of the measured quantity, and relative, that is, as the ratio of the absolute value of a given quantity to the measurement result. In this case, calculations according to the formulas of this section should be carried out using quantities expressed only in absolute or relative form.

Physics is an experimental science, which means that physical laws are established and tested by accumulating and comparing experimental data. The goal of the physical workshop is for students to experience the basic physical phenomena, learn how to correctly measure the numerical values of physical quantities and compare them with theoretical formulas.

All measurements can be divided into two types - straight and indirect.

At direct In measurements, the value of the desired quantity is directly obtained from the readings of the measuring instrument. So, for example, length is measured with a ruler, time by the clock, etc.

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.

Measurement of any quantity does not give an absolutely accurate value of this quantity. Each measurement always contains some error (error). The error is the difference between the measured value and the true value.

Errors are divided into systematic and random.

Systematic is called the error that remains constant throughout the entire series of measurements. Such errors are due to the imperfection of the measuring tool (for example, zero offset of the device) or the measurement method and can, in principle, be excluded from the final result by introducing an appropriate correction.

Systematic errors also include the error of measuring instruments. The accuracy of any device is limited and is characterized by its accuracy class, which is usually indicated on the measuring scale.

Random called error, which varies in different experiments and can be both positive and negative. Random errors are due to causes that depend both on the measuring device (friction, gaps, etc.) and on external conditions (vibrations, voltage fluctuations in the network, etc.).

Random errors cannot be ruled out empirically, but their influence on the result can be reduced by repeated measurements.

Calculation of the error in direct measurements, the average value and the average absolute error.

Assume that we are making a series of measurements of X. Due to the presence of random errors, we obtain n different meanings:

X 1, X 2, X 3 ... X n

As a measurement result, the average value is usually taken

Difference between mean and result i- th measurement is called the absolute error of this measurement

As a measure of the error of the mean value, one can take the mean value of the absolute error of a single measurement

(2)

Value
is called the arithmetic mean (or mean absolute) error.

Then the measurement result should be written in the form

(3)

To characterize the accuracy of measurements, the relative error is used, which is usually expressed as a percentage

(4)

In the general case, the procedure for processing the results of direct measurements is as follows (it is assumed that there are no systematic errors).

Case 1 The number of measurements is less than five.

x, defined as the arithmetic mean of the results of all measurements, i.e.

2) According to the formula (12), the absolute errors of individual measurements are calculated

3) According to the formula (14), the average absolute error is determined

4) According to formula (15), the average relative error of the measurement result is calculated

5) Record the final result in the following form:

Case 2. The number of measurements is over five.

1) According to formula (6), the average result is found

2) According to the formula (12), the absolute errors of individual measurements are determined

3) According to the formula (7), the mean square error of a single measurement is calculated

4) Calculate the standard deviation for the average value of the measured value by the formula (9).

5) The final result is recorded in the following form

Sometimes random measurement errors may turn out to be less than the value that the measuring device (instrument) is able to register. In this case, for any number of measurements, the same result is obtained. In such cases, half the division value of the scale of the device (tool) is taken as the average absolute error. This value is sometimes called the limiting or instrumental error and denoted (for vernier instruments and a stopwatch, it is equal to the accuracy of the instrument).

Assessment of the reliability of measurement results

In any experiment, the number of measurements of a physical quantity is always limited for one reason or another. In this regard, the task can be set to assess the reliability of the result. In other words, determine with what probability it can be argued that the error made in this case does not exceed the predetermined value ε. This probability is called the confidence probability. Let's denote it with a letter.

An inverse problem can also be set: to determine the boundaries of the interval , so that with a given probability it can be argued that the true value of the measurements of the quantity will not go beyond the specified, so-called confidence interval.

Of particular interest, however, is the case of assessing the reliability of the result of measurements of physical quantities with a very small number of repeated measurements. For example, . This is exactly the case with which we often meet in the performance of laboratory work in physics. When solving this kind of problems, it is recommended to use the method based on Student's distribution (law).

For the convenience of the practical application of the method under consideration, there are tables with which you can determine the confidence interval corresponding to a given confidence probability or solve the inverse problem.

Below are those parts of the mentioned tables that may be required when evaluating the results of measurements in laboratory classes.

Let, for example, equal-accurate (under the same conditions) measurements of a certain physical quantity be made and its average value calculated. It is required to find a confidence interval corresponding to a given confidence level. The problem is generally solved in the following way.

According to the formula, taking into account (7), calculate

Then for given values n and find the value according to the table (Table 2). The value you are looking for is calculated based on the formula

When solving the inverse problem, the parameter is first calculated by formula (16). The desired value of the confidence probability is taken from the table (Table 3) for a given number and a calculated parameter .

Table 2. Parameter value for a given number of experiments

and confidence level

n	0,5	0,6	0,7	0,8	0,9	0,95	0.98	0,99	0.995	0,999
	1,000	1,376	1,963	3,08	6,31	12,71	31,8	63,7	127,3	637,2
	0,816	1,061	1,336	1,886	2,91	4,30	6,96	9,92	14,1	31,6
	0,765	0,978	1,250	1,638	2,35	3,18	4,54	5,84	7,5	12,94
	0,741	0,941	1,190	1,533	2,13	2,77	3,75	4,60	5,6	8,61
	0,727	0,920	1,156	1,476	2,02	2,57	3,36	4,03	4,77	6,86
	0.718	0,906	1,134	1,440	1,943	2,45	3,14	3,71	4,32	5,96
	0,711	0,896	1,119	1,415	1,895	2,36	3,00	3,50	4,03	5,40
	0,706	0,889	1,108	1,397	1,860	2,31	2,90	3,36	3,83	5,04
	0,703	0,883	1,110	1,383	1,833	2,26	2,82	3,25	3,69	4,78

Table 3 The value of the confidence probability for a given number of experiments n and parameter ε

n		2,5		3,5
	0,705	0,758	0,795	0,823
	0,816	0,870	0,905	0,928
	0,861	0,912	0,942	0,961
	0,884	0,933	0,960	0,975
b	0,898	0,946	0,970	0,983
	0,908	0,953	0,976	0,987
	0,914	0,959	0,980	0,990
	0,919	0.963	0,983	0,992
	0,923	0,969	0,985	0,993

Processing the results of indirect measurements

Very rarely, the content of a laboratory work or a scientific experiment is reduced to obtaining the result of a direct measurement. For the most part, the desired quantity is a function of several other quantities.

The task of processing experiments with indirect measurements is to calculate the most probable value of the desired value and estimate the error of indirect measurements based on the results of direct measurements of certain quantities (arguments) associated with the desired value by a certain functional dependence.

There are several ways to handle indirect measurements. Consider the following two methods.

Let some physical quantity be determined by the method of indirect measurements.

The results of direct measurements of its arguments x, y, z are given in Table. 4.

Table 4

Experience number	x	y	z	…
				…
				…
…	…	…	…	…
…	…	…	…	…
…	…	…	…	…
n				…

The first way to process the results is as follows. Using the calculated (17) formula, the desired value is calculated based on the results of each experiment

(17)

The described method of processing the results is applicable, in principle, in all cases of indirect measurements without exception. However, it is most expedient to use it when the number of repeated measurements of the arguments is small, and the calculation formula for the indirectly measured value is relatively simple.

In the second method of processing the results of experiments, first, using the results of direct measurements (Table 4), the arithmetic mean values of each of the arguments, as well as the errors of their measurement, are first calculated. Substituting , , ,... into the calculation formula (17), determine the most probable value of the measured quantity

(17*)