Principal concepts of the theory of errors

We can't define the true values of a physical quantity. We can define only the interval (x _min, x _max) 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,..., x_n are the results of the individual measurements then the most probable measured value or the arithmetic mean is:

(2.1)

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

(2.2)

is called the mean accidental deviation of the measurements.

Mean root square is defined as

(2.3)

where t – Student’s constant for definite a and n. The ratio of

(2.4)

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

. (2.5)