## Principal concepts of the theory of errors
We can't define the true values of a physical quantity. We can define only the interval ( a. For example: we can affirm, that students' height may be defined between 1.5 m and 2.0 m with probability of 0.9. Then we can prove, that students' height may be defined between 1.6 m and 1.8 m with smaller probability of 0.6 and so on. Value of this interval is called . On fig.2.1 interval of quantity being investigated the entrusting interval is represented.xFigure 2.1
Where is the half width of the entrusting interval of the measured quantity with probability of Dx.aTherefore we can estimate, that true value of the measured quantity may be defined as ,aor . If a quantity times and n are the results of the individual measurements then the most probable measured value or the arithmetic mean is:x_{1} , x_{2} ,..., x_{n}(2.1) The deviation is called the accidental error (deviation) of a single measurement. (2.2) is called Mean root square is defined as (2.3) where and a. The ratio ofn(2.4) is called . (2.5)
Absolute error of instrumental , (2.6) where is the true value of the quantity measured. Typically X is quantity of the instruments minimum value scale. For example: the ruler error is d = 1 mm.d
. (2.7) It is usually expressed in percent . (2.8)
, (2.9) expressed in percent. For example: electric current is measured by the instrument with interval 0 ÷ 1 A, precision class is 0.5. This means, that = 0.5 %, andg. If the instrument shows 0.3 A, then .
1. The error of table quantity is defined as , (2.10) where, is half price of category from last significance figure in table quantity. For example: quantity p may be 3.14. In this case v = 0.005 andv. If quantity p is 3.141 and v = 0.0005 then and so on. 2. Error of count may occur when we measure quantity by an instrument. Typically, the error of count is half price of minimum value of instruments scale. For example a ruler has error of count v 3. Rules of approximation: quantity = 0.01865, approximated quantity is 0.019; x = 0.896, approximated quantity is 0.9 and so on.x
Errors of direct measurements are defined as , (2.11) if there is one measurement ( , (2.12) if there are several measurements ( In these equations and column with n; a is error of an instrument; d is error of count, v =d/2.vFor example: the length of a body was measured three times:
The error of this measurement will be: The relative error is . The final result is x = (13.3 + 0.5) mm, a= 0.7 , E = 3.6 % .
Let . y = f(x_{1} ,x_{2} , ..., x_{n} ) is defined as the direct measurements.x_{1} ,x_{2} , ... , x_{n}1. Errors of indirect measurements are defined as (2.13) if the functional dependence of investigated quantity is a polynomial. 2. Errors of indirect measurements are defined as (2.14) if the functional dependence of investigated quantity, is a monomial and we can define 1. If the functional dependence is then Then 2. If functional dependence is , Then and The final result: .
Graph is built on the millimeter paper. In fig.2.2 you can see an example of graph. Figure 2.2
The experimental curve is drawn through the experimental points. This curve describes the experimental data.
1. Definition of direct and indirect measurements. Examples. 2. Definition of the most probable value of the measured quantity 3. What is called a relative error? 4. What is called an accidental deviation? 5. What is the equation of square mean of errors? 6. How do we define errors of instruments? 7. How do we define errors of table quantities and count errors? 8. Rules of approximation. 9. What is the equation of errors of direct measurements? 10. What is the equation of errors of indirect measurements?
Authors: S.P. Lushchin, the reader, candidate of physical and mathematical sciences. Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences. Approved by the chair of physics. Protocol № 3 from 01.12.2008 . |
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