According to the form of representation, the errors of

measurements and measurement instruments are classified as:

- the absolute error of measurement;

- the relative error of measurement;

- the fiducial error of a measurement instrument.

The absolute error of measurement is the difference between the measurement result and the true (in practice, the conventional true or actual) value of the measurand. The absolute error is always represented in the same units as the measurand:

Δ_X = X_result  – X_true,

where Δ_X is the absolute error of measurement; X_result is the measurement result ; X_true is the true, but in practice, the conventional true (actual) value of the measurand.

For example, by weighing one -hundred gramme weight (material measure of mass) with the dial scales we obtain a result of 95 g. So, the absolute error is:

                  Δ_m = 95 g - 100 g = minus 5 g.

It is necessary to remember that the absolute error has a sign. Therefore the first place in the formula must occupy the measurement result, from which the true (conventional true, or actual) value of the measurand is subtracted. In the opposite case, we shall get not the absolute error but the correction.

The correction is the quantity value, that is algebraically added to the measurement result to exclude the systematic error.

The correction is the absolute error with the inverse sign, it is a value to be added to the measurement result to get the true (in practice, the actual) value of the measurand. Of course, the correction as well as the absolute error is always represented in the same units as the PQ being measured.

Thus, in the previous example with the weighing of the weight we have the correction of +5 g. So, to get the true (in practice, actual) value of the weight weighed, it is necessary to add the correction (5 g) to the measurement result (95 g). We shall get the actual value of the weighed weight- 100g.

It is necessary to note, that in all countries of the world the same definitions of the absolute error and the correction are accepted. The addlement may cause very undesirable, sometimes fatal, consequence.

For example, an airplane, which takes off the sea level located airfield, has to land at a high-altitude airdrome.

To be able to use a barometric altimeter before a descent a pilot, by radio must get information on the atmosphere pressure at the landing ground. Using those data he applies the correction for the altimeter zero indication .

The addlement between the correction and the absolute error may cause that the plane will fly not over a mountain on the certain altitude, but “under the mountain” that means a smash.

It also should be noted that the absolute error alone cannot characterize the measurement quality. For instance, one weighing gave us an absolute error of 1 g, the second – of 1 kg. What is better? The answer is impossible if we have no information about the masses weighed. If, for example, in the first case it is a pencil weighing only 10 g and in the second one – an airplane, weighing 10000 kg, it becomes clear, that the quality of the second weighing is much higher. In order to compare the characteristics of the different measurements the concept of a“relative error” was introduced.

The relative error of measurement is a ratio of the absolute measurement error to the true (in practice, the conventional true or actual) value of a measurand.

In practice, calculation of the relative error is possible even by dividing the absolute error on the measurement result.

    δ = Δ_X / X_true ≈ Δ_X / X_a  ≈ Δ_X / X_result

where:

Δ_X is the absolute error of the measurement, X_true is the true value of the measurand, X_a is the actual (conventional true) value of the measurand, X_result is the measurement result .

As it is seen from the formula above, the relative error is a dimensionless quantity. Very often the relative error is presented in percentage:

  δ% = (Δ_X / X_true) ∙ 100 ≈ (Δ_X / X_a) ∙ 100 ≈ (Δ_X / X_result) ∙ 100

Unlike the absolute error, the relative error gives possibility to compare the quality of the different measurements. For example, in the above-mentioned examples of the pencil and the airplane weighing we have the respective relative errors in percentage:

δ_p% = (1 g / 10 g)·100 = 10% ;

δ_a% = (1 kg / 10000 kg)·100 =0,01% ,

where:

δ_p% - is the relative error in weighing of the pencil;

δ_a% -is the relative error in weighing of the airplane.

Having compared the obtained relative errors we may come to the conclusion that the plane weighing quality is 1000 times better. Here it would be useful to introduce a concept of the “accuracy of measurements”.

The accuracy of measurements is characterized by the proximity of their errors to zero.

Numerically, the measurement accuracy is rated by the inverse value of the dimensionless relative error modulus.

For example, for the pencil weighing we got the relative error of 1/10 and for the airplane weighing of 1/10000 (it is necessary to take them not in percentage). Then the accuracy will be as follows: A_p = 10, A_a =10000 or 10¹ and 10⁴, respectively. Thus, the relative error and the accuracy of measurements give the possibility of qualitative comparison of the different measurements.

 Is it possible to compare the accuracy of the deflectional instruments by this indexes? Let us analyze it by means of a numerical example.

 We have a deflectional voltmeter with a voltage range of 100 V. An operator has paid no regard to that the pointer of the voltmeter is not fixed on the zero scale mark before the measurement start but indicates 1 V. It means that any measurement result will be overrated by 1 V, so an absolute error of 1 V has already entered. Let us estimate the relative errors for measurements of different voltages, for example, 100, 50, 10, 1 V, with that instrument. We shall gain the following results: 1, 2, 10 and 100%. We can observe a very great disparity between these numbers for the same instrument.

Thus, it is impossible to characterize a deflectional instrument accuracy by the relative error of measurements held with it.

By the way, this example illustrates a very important for practice rule concerning with the deflectional instrument operating range selection. It is expedient to choose the upper limit of the operational range of a deflectional instrument as close as possible to the quantity being measured. When the reading of the results is taken in the last third of the scale, the relative error is not large. On the contrary, the reading of the results at the beginning of the scale of a deflectional instrument may cause very large relative errors of measurements and that is why one should avoid it and substitute the instrument with a smaller operating range.

Then, with what parameter is it possible to characterize the accuracy of the instrument, just the MI accuracy, but not the accuracy of the PQ measurement with it? For this purpose we shall introduce the following concept.

The fiducial error of a measurement instrument is a ratio of the measurement instrument absolute error to its fiducial value.

                     γ%=(Δ_X / X_f ) ∙ 100 ,

where:

γ% is the fiducial error of the instrument in percents, %;

Δ_X is the instrument absolute error ;

X_f is fiducial value of the measurement instrument .

The fiducial value is a conditionally accepted constant value, specified for the given instrument. As a rule it is given in the technical documentation of the instrument. Commonly used variants of the fiducial values are the following: