Calculation of relative measurement error. Calculation of measurement errors

1. Introduction

The work of chemists, physicists and representatives of other natural science professions often involves performing quantitative measurements of various quantities. In this case, the question arises of analyzing the reliability of the obtained values, processing the results of direct measurements and assessing the errors of calculations that use the values ​​of directly measured characteristics (the latter process is also called processing of results indirect measurements). For a number of objective reasons, the knowledge of graduates of the Faculty of Chemistry of Moscow State University about calculating errors is not always sufficient for correct processing of the received data. One of these reasons is the absence in the faculty curriculum of a course on statistical processing of measurement results.

By now, the issue of calculating errors has, of course, been thoroughly studied. There are a large number of methodological developments, textbooks, etc., in which you can find information about calculating errors. Unfortunately, most of these works are overloaded with additional and not always necessary information. In particular, most of the work of student workshops does not require such actions as comparing samples, assessing convergence, etc. Therefore, it seems appropriate to create a brief development that outlines the algorithms for the most frequently used calculations, which is what this development is devoted to.

2. Notation adopted in this work

The measured value, - the average value of the measured value, - the absolute error of the average value of the measured value, - the relative error of the average value of the measured value.

3. Calculation of errors of direct measurements

So, let's assume that they were carried out n measurements of the same quantity under the same conditions. In this case, you can calculate the average value of this value in the measurements taken:

(1)

How to calculate the error? According to the following formula:

(2)

This formula uses the Student coefficient. Its values ​​at different confidence probabilities and values ​​are given in.

3.1. An example of calculating the errors of direct measurements:

Task.

The length of the metal bar was measured. 10 measurements were made and the following values ​​were obtained: 10 mm, 11 mm, 12 mm, 13 mm, 10 mm, 10 mm, 11 mm, 10 mm, 10 mm, 11 mm. It is required to find the average value of the measured value (length of the bar) and its error.

Solution.

Using formula (1) we find:

mm

Now, using formula (2), we find the absolute error of the average value with confidence probability and the number of degrees of freedom (we use the value = 2.262, taken from):

Let's write down the result:

10.8±0.7 0.95 mm

4. Calculation of errors of indirect measurements

Let us assume that during the experiment the quantities are measured , and then c Using the obtained values, the value is calculated using the formula .

In this case, the errors of directly measured quantities are calculated as described in paragraph 3.

The calculation of the average value of a quantity is carried out according to the dependence using the average values ​​of the arguments.

,(3)

The error value is calculated using the following formula:

where is the number of arguments, is the partial derivative of the function with respect to the arguments, is the absolute error of the average value of the argument.

The absolute error, as in the case of direct measurements, is calculated using the formula.

Task.

4.1. An example of calculating the errors of direct measurements:

5 direct measurements of and were carried out. The following values ​​were obtained for the value: 50, 51, 52, 50, 47; the following values ​​were obtained for the quantity: 500, 510, 476, 354, 520. It is required to calculate the value of the quantity determined by the formula and find the error of the obtained value.

Physics is an experimental science, which means that physical laws are established and verified by accumulating and comparing experimental data. The purpose of the physics workshop is for students to study basic physical phenomena through experience, learn 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 device. So, for example, length is measured with a ruler, time is measured by a clock, etc. If the desired physical quantity cannot be measured directly by the device, but is expressed through the measured quantities using a formula, then such measurements are called.

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

Errors are usually divided into systematic straight random.

Systematic called an error that remains constant throughout the entire series of measurements. Such errors are caused by the imperfection of the measuring instrument (for example, the 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 is called an error that varies in different experiments and can be both positive and negative. Random errors are caused by reasons that depend both on the measuring device (friction, gaps, etc.) and on external conditions (vibration, voltage fluctuations in the network, etc.).

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

Calculation of error in direct measurements - average value and average absolute error.

Let us assume that we carry out a series of measurements of the value X. Due to the presence of random errors, we obtain n different meanings:

X 1, X 2, X 3… X n

The average value is usually taken as the measurement result

Difference between average and result i – of the th measurement we will call the absolute error of this measurement

As a measure of the error of the average value, we can take the average value of the absolute error of an individual measurement

(2)

Magnitude
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)

Let the systematic errors in measurements be negligible. Let us consider the case when the measurement is carried out a large number of times (n→∞).

As experience shows, the deviation of measurement results from their average value up or down is the same. Measurement results with small deviations from the average value are observed much more often than with large deviations.

Let us arrange all the numerical values ​​of the measurement results in a series in ascending order and divide this series into equal intervals
. Let – number of measurements with results falling within the interval [
]. Magnitude
there is a probability ΔP i (x) of obtaining a result with a value in the interval [
].

Let's present it graphically
, corresponding to each interval [
] (Fig. 1). The stepped curve shown in Fig. 1 is called a histogram. Let us assume that the measuring device has extremely high sensitivity. Then the width of the interval can be made infinitesimal dx. The stepped curve in this case is replaced by a curve represented by the function φ(x) (Fig. 2). The function φ(x) is usually called the distribution density function. Its meaning is that the product φ(x)dx is the probability dP(x) of obtaining results with a value in the range from x to x+dx. Graphically, the probability value is represented as the area of ​​a shaded rectangle. Analytically, the distribution density function is written as follows:

. (5)

The function φ(x) presented in the form (5) is called the Gaussian function, and the corresponding distribution of measurement results is Gaussian or normal.

Options
and σ have the following meaning (Fig. 2).

–average value of measurement results. At
=
the Gaussian function reaches its maximum value. If the number of dimensions is infinitely large, then
equal to the true value of the measured quantity.

σ – characterizes the degree of scatter of measurement results from their average value. The parameter σ is calculated using the formula:

. (6)

This parameter represents the root mean square error. The quantity σ 2 in probability theory is called the dispersion of the function φ(x).

The higher the measurement accuracy, the closer the measurement results are to the true value of the measured quantity, and, therefore, the smaller σ.

The form of the function φ(x) obviously does not depend on the number of dimensions.

Probability theory shows that 68% of all measurements will give a result that is in the interval, 95% in the interval and 99.7% in the interval.

Thus, with a probability (reliability) of 68%, the deviation of the measurement result from the average value lies in the interval [
], with a probability (reliability) of 95% – in the interval [
] and with a probability (reliability) of 99.7% – in the interval [
].

The interval corresponding to a particular probability of deviation from the average value is called confidence.

In real experiments, the number of dimensions obviously cannot be infinitely large, so it is unlikely that
coincided with the true value of the measured value
. In this regard, it is important to estimate, based on probability theory, the magnitude of the possible deviation
from
.

Calculations show that when the number of measurements is more than 20, with a probability of 68%
falls within the confidence interval [
], with a probability of 95% – in the interval [
], with a probability of 99.7% – in the interval [
].

Magnitude , which defines the boundaries of the confidence interval, is called the standard deviation or simply the standard.

Standard calculated by the formula:

. (7)

Taking into account formula (6), expression (7) takes the following form:

. (8)

The greater the number of dimensions n, the closer X is to
. If the number of measurements is not large, less than 15, then instead of the Gaussian distribution, the Student distribution is used, which leads to an increase in the width of the confidence interval of the possible deviation of X from
int n, p times.

The factor t n, p is called the Student coefficient. The indices P and n indicate with what reliability and to what number of measurements the Student coefficient corresponds. The value of the Student coefficient for a given number of measurements and a given reliability is determined according to Table 1.

Table 1

Student's coefficient.

For example, with a given reliability of 95% and the number of measurements n = 20, Student’s coefficient t 20.95 = 2.1 (confidence interval
) with the number of measurementsn=4, t 4.95 =3.2 (confidence interval
). That is, with an increase in the number of measurements from 4 to 20, a possible deviation
fromX decreases by 1.524 times.

Below is an example of calculating the absolute random error

Using formula (2) we find the average value of the measured value
(without indicating the dimension of the physical quantity)

.

Using formula (8) we calculate the standard deviation

.

Student's coefficient determined for n=6, and P=95%, t 6.95 =2.6 final result:

X=20.1±2.6·0.121=20.1±0.315 (with P=95%).

We calculate the relative error:

.

When recording the final measurement result, it must be kept in mind that the error must contain only one significant figure (other than zero). Two significant figures in the error are recorded only if the penultimate figure is 1. It is useless to record a larger number of significant figures, since they will not be reliable. In the recording of the average value of the measured value, the last digit must belong to the same digit as the last digit in the recording of the error.

X=(243±5)·10 2;

X=232.567±0.003.

Taking several measurements may yield the same result. This is possible if the sensitivity of the measuring device is low. When the measurement is made with a device with low sensitivity, a single measurement is sufficient. It makes no sense, for example, to repeatedly measure the length of the table with a tape measure with centimeter divisions. The measurement result in this case will be the same. The error during a single measurement is determined by the value of the smallest division of the device. It is called instrument error. Its meaning
calculated using the following formula:

, (10)

where γ is the division price of the device;

t ∞, p – Student coefficient corresponding to an infinitely large number of measurements.

Taking into account the instrument error, the absolute error with a given reliability is determined by the formula:

, (11)

Where
.

Taking into account formulas (8) and (10), (11) is written as follows:

. (12)

In the literature, to shorten the record, the magnitude of the error is sometimes not indicated. The error is assumed to be one half of one last significant digit. For example, the radius of the Earth is written in the form
m. This means that the error should be taken as a value equal to ±
m.