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 will limit 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
|
| ||||
To reduce the influence of random errors, it is necessary to measure this value several times. Suppose we are measuring some value x. As a result of the measurements, we obtained the following values:
x1, x2, x3, ... xn. (2)
This series of x values is called a sample. Having such a sample, we can evaluate the measurement result. We will denote the value that will be such an estimate. But since this evaluation value of the measurement results will not represent the true value of the measured quantity, it is necessary to estimate its error. Let us assume that we can determine the estimate of the error Δx. In this case, we can write the measurement result in the form
Since the estimated values of the measurement result and the error Dx are not accurate, the record (3) of the measurement result must be accompanied by an indication of its reliability P. Reliability or confidence probability is understood as the probability that the true value of the measured quantity is contained in the interval indicated by record (3). This interval itself is called the confidence interval.
For example, when measuring the length of a certain segment, we wrote the final result as
l = (8.34 ± 0.02) mm, (P = 0.95)
This means that out of 100 chances - 95 that the true value of the length of the segment lies in the range from 8.32 to 8.36 mm.
Thus, the task is to, having a sample (2), find an estimate of the measurement result, its error Dx and reliability P.
This problem can be solved with the help of probability theory and mathematical statistics.
In most cases, random errors follow the normal distribution law established by Gauss. The normal distribution of errors is expressed by the formula
where Dx - deviation from the value of the true value;
y is the true mean square error;
2 - variance, the value of which characterizes the spread of random variables.
As can be seen from (4), the function has a maximum value at x = 0, in addition, it is even.
Figure 16 shows a graph of this function. The meaning of function (4) is that the area of the figure enclosed between the curve, the Dx axis and two ordinates from the points Dx1 and Dx2 (shaded area in Fig. 16) is numerically equal to the probability with which any sample falls into the interval (Dx1, Dx2 ) .
Since the curve is distributed symmetrically about the y-axis, it can be argued that errors of equal magnitude but opposite in sign are equally likely. And this makes it possible to take the average value of all elements of the sample as an estimate of the measurement results (2)
where - n is the number of measurements.
So, if n measurements are made under the same conditions, then the most probable value of the measured quantity will be its average value (arithmetic). The value tends to the true value m of the measured value at n > ?.
The mean square error of a single measurement result is the value (6)
It characterizes the error of each individual measurement. When n > ? S tends to a constant limit y
With an increase in y, the scatter of readings increases, i.e. measurement accuracy becomes lower.
The root-mean-square error of the arithmetic mean is the value (8)
This is the fundamental law of increasing accuracy as the number of measurements increases.
The error characterizes the accuracy with which the average value of the measured value is obtained. The result is written as:
This error calculation technique gives good results (with a reliability of 0.68) only when the same value is measured at least 30 - 50 times.
In 1908, Student showed that the statistical approach is also valid for a small number of measurements. Student's distribution for the number of measurements n > ? goes into the Gaussian distribution, and at a small number it differs from it.
To calculate the absolute error for a small number of measurements, a special coefficient is introduced that depends on the reliability P and the number of measurements n, called the coefficient
Student t.
Omitting the theoretical justifications for its introduction, we note that
Dx = t. (ten)
where Dx is the absolute error for a given confidence level;
mean square error of the arithmetic mean.
Student's coefficients are given in the table.
It follows from what has been said:
The value of the root-mean-square error allows you to calculate the probability that the true value of the measured value will fall into any interval near the arithmetic mean.
When n > ? > 0, i.e. the interval in which the true value of m is found with a given probability tends to zero with an increase in the number of measurements. It would seem that by increasing n, one can obtain a result with any degree of accuracy. However, the accuracy increases significantly only until the random error becomes comparable with the systematic one. Further increase in the number of measurements is inexpedient, because the final accuracy of the result will depend only on the systematic error. Knowing the value of the systematic error, it is easy to set the admissible value of the random error, taking it, for example, equal to 10% of the systematic error. By setting a certain value P for the confidence interval chosen in this way (for example, P = 0.95), it is easy to find the required number of measurements, which guarantees a small effect of a random error on the accuracy of the result.
To do this, it is more convenient to use the Student's coefficient table, in which the intervals are given in fractions of the value of y, which is a measure of the accuracy of this experiment with respect to random errors.
When processing the results of direct measurements, the following order of operations is proposed:
Record the result of each measurement in a table.
Calculate mean of n measurements
Find the error of an individual measurement
Calculate Squared Errors of Individual Measurements
(Dx 1)2, (Dx 2)2, ... , (Dx n)2.
Determine the standard error of the arithmetic mean
Specify the reliability value (usually take P = 0.95).
Determine the Student's coefficient t for a given reliability P and the number of measurements made n.
Find the confidence interval (measurement error)
If the value of the error of the measurement result Δx turns out to be comparable with the value of the error of the instrument d, then take as the boundary of the confidence interval
If one of the errors is less than three or more times the other, then discard the smaller one.
Write the final result as
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 the 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.
Measurement results
Basic concepts, terms and definitions
Measurement - determination of the value of a physical quantity empirically. Measurements are divided into two groups: direct and indirect. Direct measurement - finding the value of a physical quantity directly with the help of instruments. Indirect measurement – finding the desired value based on the known relationship between this value and the values found in the process of direct measurements. For example, to determine the acceleration of a uniformly accelerated motion of a body, you can use the formula , where S - distance traveled, t- travel time. The path and time of movement are found directly in the course of the experiment, that is, in the process of direct measurements, and the acceleration can be calculated using the above formula and, therefore, its value will be determined as a result of indirect measurement.
The deviation of the result of a direct or indirect measurement from the true value of the desired quantity is called measurement error . The errors of direct measurements are due to the capabilities of the measuring instruments, the measurement technique, and the conditions of the experiment. The errors of indirect measurements are due to the “transfer” to the desired value of the errors of direct measurements of those quantities on the basis of which it is calculated. According to the method of numerical expression, absolute errors are distinguished (Δ BUT), expressed in units of the measured quantity ( BUT), and relative errors δ A=(Δ A/A) 100%, expressed as a percentage.
There are three types of errors: systematic, random, and misses.
Under systematic errors understand those, the cause of which remains constant or regularly changes during the entire measurement process. Sources of systematic errors are usually incorrect adjustment of instruments, regularly changing external factors, and an incorrectly chosen measurement technique. To identify and eliminate systematic errors, it is necessary to first analyze the measurement conditions, conduct control checks of measuring instruments and compare the results obtained with data from more accurate measurements. The non-excludable systematic errors that must be taken into account when processing the results include the errors of the instruments and instruments used (instrumental errors).
instrument room ness equal to half of the scale division of the device Δ A pr \u003d CD / 2 (for instruments such as a ruler, caliper, micrometer) or is determined by the accuracy class of the instrument (for pointer electrical measuring instruments).
Under instrument accuracy class γ understand the value equal to:
where ∆ A etc instrumental error (the maximum permissible absolute error, the same for all points of the scale); A max measurement limit (maximum value of instrument readings).
For electronic devices, the formulas for calculating the instrumental error are given in the instrument's passport.
Random errors arise as a result of the action of various random factors. This type of error is detected when repeatedly measuring the same quantity under the same conditions using the same instruments: the results of a series of measurements differ somewhat from each other randomly. The contribution of random errors to the measurement result is taken into account in the process of processing the results.
Under misses understand large errors that sharply distort the measurement result. They arise as a result of gross violations of the measurement process: instrument malfunctions, experimenter errors, power surges in the electrical circuit, etc. Measurement results containing misses should be discarded during the preliminary analysis.
In order to identify misses and subsequently take into account the contribution of random and instrumental errors, direct measurements of the desired value are carried out several times under the same conditions, that is, a series of equally accurate direct measurements is carried out. The purpose of subsequent processing of the results of a series of equally accurate measurements is:
The result of a direct or indirect measurement shall be presented as follows:
A=(
where < BUT> is the average value of the measurement result, Δ BUT is the half-width of the confidence interval, α is the confidence probability. In this case, it should be taken into account that the numerical value of Δ BUT must contain no more than two significant digits, and the value ‹ BUT> must end with a digit of the same digit as Δ BUT.
Example: The result of measuring the time of movement of the body is:
t= (18.5 ± 1.2) s; α = 0.95.
From this record it follows that with a probability of 95% the true value of the movement time lies in the interval from 17.3 s to 19.7 s.
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 formula through the measured quantities, 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)
