Principal concepts of the theory of errors


We can't define the true values of a physical quantity. We can define only the interval (xmin, xmax) of the investigated quantity with some probability 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 the entrusting interval. On fig.2.1 interval of quantity being investigated x is represented.

Figure 2.1


Where x is the most probable value of quantity being measured; Dx is the half width of the entrusting interval of the measured quantity with probability of a.

Therefore we can estimate, that true value of the measured quantity may be defined as x = x D x, with probability a,

or .

If a quantity x has been measured n times and x1 , x2 ,..., xn are the results of the individual measurements then the most probable measured value or the arithmetic mean is:


The deviation is called the accidental error (deviation) of a single measurement.


is called the mean accidental deviation of the measurements.

Mean root square is defined as


where t Students constant for definite a and n. The ratio of


is called the relative error of measurement and is usually expressed in percents:

. (2.5)


Errors of instruments


Absolute error of instrumental d is a deviation

, (2.6)

where a is an index of an instrument; X is the true value of the quantity measured. Typically d is quantity of the instruments minimum value scale. For example: the ruler error is d = 1 mm.

Relative error of the measurement is the ratio of

. (2.7)

It is usually expressed in percent

. (2.8)

Brought error of the measurement or precision class is the ratio

, (2.9)

expressed in percent. D is maximum value on the instrument scale.

For example: electric current is measured by the instrument with interval 0 ÷ 1 A, precision class is 0.5. This means, that D = 1 A, g = 0.5 %, and


If the instrument shows 0.3 A, then



Error of table quantities, count and rules of approximations


1. The error of table quantity is defined as

, (2.10)

where, α is probability; v 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 and


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 vl=0.5 mm.

3. Rules of approximation: quantity x may be approximated only to two significant figures, if the first significant figure is 1 or 2. In other cases, the quantity is approximated only to one significant figure. For example: x = 0.01865, approximated quantity is 0.019; x = 0.896, approximated quantity is 0.9 and so on.


Errors of direct measurement


Errors of direct measurements are defined as

, (2.11)

if there is one measurement (n = 1). And

, (2.12)

if there are several measurements (n > 1).

In these equations t¥ is Student's constant, it may be defined from the table on the crossing of line with n and column with a; d is error of an instrument; v is error of count, v =d/2.

For example: the length of a body was measured three times:


xi, mm Dxi, mm (Dxi)2, mm2
12.8 -0.466 0.217
13.6 0.334 0.111
13.4 0.134 0.018
=13.266, n=3, t=1.4, t¥=1, d=1 mm.


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 % .


Errors of indirect measurements


Let y be indirectly measured quantity, it is defined as y = f(x1 ,x2 , ..., xn ). x1 ,x2 , ... , xn is defined as the direct measurements.

1. Errors of indirect measurements are defined as


if the functional dependence of investigated quantity is a polynomial.

2. Errors of indirect measurements are defined as


if the functional dependence of investigated quantity, is a monomial

and we can define Dy as: . For example:

1. If the functional dependence is then


2. If functional dependence is ,



The final result: .



Graph presentation of the experimental results


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.


Control questions


1. Definition of direct and indirect measurements. Examples.

2. Definition of the most probable value of the measured quantity x.

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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