Evaluation of the accuracy of the result of repeated direct measurements

Let the repeated measurement of a physical quantity of its values were obtained: Then, the arithmetic mean of all values obtained, which is the most significant as well

= = (1.1.1.)

Each individual measurement differs from the arithmetic mean by an amount equal

(1.1.2.)

The deviation is called absolute error of i - that dimension.

The absolute error of individual measurements take both positive and negative values.

From experience we know that the probability of error is smaller, the more it. Additionally, if a very large number of measurements , the errors are the same magnitude but different in sign occur equally often. Speed reducing the probability of occurrence and its increase is characterized by a dispersion of error. It is

(1.1.3)

The less , the less the likelihood of greater in magnitude of random error , less scatter of individual values.

(1.1.4)

 

Consequently,

(1.1.5)

The square root of the variance measurement is called the mean square error

(1.1.6)

If, instead of the expression (1.1.6) to substitute the value of the equation (1.1.2), then

(1.1.7)

In the expression (1.1.7) includes the value , not . The very magnitude determined some oshibkoy.Sredney square error of the arithmetic mean of a number of measurements is the value equal to the mean square of the number of measurements.

(1.1.8)

 

Or

(1.1.9)

Estimates of variance and are marginal, fair, just , ie, at large . For small values of these estimates are themselves random, at best, only determine the order of magnitude of dispersion.

Measurement processing task is to determine the range of up to + , which concluded with a probability of the true value of the measured value. The interval from to is called the confidence interval is called a confidence probability (reliability). If the number of measurements is sufficiently large, then the confidence level expresses the proportion of the total number of measurements in which the measured value is within the confidence interval. For example, if a measurement is made of 100, then the confidence level of 95 measurement values were obtained, without departing from its scope. Of course, the greater reliability is required, the more turns the corresponding confidence interval, and conversely, the larger the confidence interval is given, the more likely that the measurement results will not go beyond it.

It follows from the theory of errors that a large number of measurements (over a hundred) confidence level interval of equal to 68%, and the interval from to is 95%. When submitting any of the measured values of the main lead and the confidence interval of the probability of corresponding to this interval.

In cases where the number of measurements is small, there are no conditions for the existence of a strict statistical regularities that underlie the determination of random errors. This leads to the fact that the values ​​of the standard deviation of the average S_a, calculated from (1.1.9) is not accurate and the more inaccurate the smaller number of measurements . Therefore, to garanatirovat that the true value of the measured value with a given probability is within the confidence interval must be increased. With a limited number of measurements are not taken abroad Sa confidence interval S_a, and а:

(1.1.11)

The measurement results can not be compared values dissimilar to each other by their absolute errors. For comparison, the measurement accuracy of these values is entered relative error - the ratio of absolute error to the mean value of the measured value.

 

(1.1.12)

According to the relative error is convenient to compare the results of measurements of similar values.

Values Ctyudenta coefficients.

 

 

Table 1.1.1

 

	0,6	0,7	0,9	0,95	0,99
	1,38	2,01	6,31	12,71	63,66
	1.06	1,3	2,92	4,30	9,92
	0,98	1,2	2,35	3,18	5,84
	0,94	1,1	2,13	2,78	4,60
	0,92	1,1	2,02	2,57	4,03
	0,90	1,1	1,94	2,45	3,71
	0,90	1,1	1,90	2,36	3,50
	0,90	1,1	1,86	2,31	3,36
	0,90	1,1	1,83	2,26	3,25

Order of work

As an example, consider the processing of the results of measurements of the oscillation period of a simple pendulum (mathematical pendulum is called the body suspended on a weightless and inextensible thread, which is much greater than its size).

1. Use the stopwatch to measure the time tі 30 oscillations of a pendulum. The measurements were repeated n times (n is given by the teacher).

2. According to the formula

(1.1.13)

calculate the period of oscillation.

3. Definitions sredngo arithmetic mean value of the oscillation period

4. According to the formula (1.1.9) to find the standard deviation of T.

5. From the table to find the value of the Student's coefficient corresponding to reliability, said the teacher (Student coefficient table is available in the laboratory).

6. When the values found and the formula (1.1.10) to calculate the absolute error.

7. Calculate the relative error by the formula (1.1.12).

8. The results of measurement and computation recorded in the table

Table (1.1.2)

N	п	ti с	Тi с	Тi с

 

 

Control questions

1. What kinds of measurements are divided and associated errors.

2. How is the absolute error.

3. What is the Student factor is introduced.

4. What do the confidence interval and confidence level.

5. Why is it necessary to calculate the relative error.

6. What is called a mathematical pendulum.

7. What is called the period of oscillation.

 

Literature

1. Kassandrova ON, VV Lebedev Processing of observations. M.: Nauka, 1970.

2. Agekyan TA Fundamentals of the theory of errors for astronomers and physicists. M.: Nauka, 1972.