absolute error and relative error
As a consequence of the existence of different sources of error, the scientist systematically considers to what extent or to what degree the results obtained are reliable, that is, trustworthy. For this reason, the result of a measurement is associated with a complementary value that indicates the quality of the measurement or its degree of precision. The errors or inaccuracies in the results are expressed mathematically in two forms that are called absolute error and relative error. The absolute error ΔE is defined as the difference between the result of the measurement M and the true value m 0 of the magnitude to be measured.
ΔE = M – m 0
The relative error E r is the ratio between the absolute error ΔE and the true value. When expressed as a percentage, its expression is
|E r (%) =||ΔE 100|
Strictly speaking, such definitions are only applicable when they refer not to actual physical measurements, but to mathematical operations, since the exact value of a magnitude is not accessible. For this reason, it is often preferred to speak of uncertainties rather than errors. In this case, the value that is closest to the true value is taken as m, that is, the average value obtained by repeating the same measurement several times.
Scientists try to ensure that their experimental data does not say more than what they can say according to the measurement conditions in which they were obtained. For this reason, they take care in the number of figures with which to express the result of a measurement with the purpose of including only those that have some experimental meaning. Such figures are called significant figures. A figure is significant when it is known with acceptable precision. Thus, when measuring with a thermometer that shows up to 0.1 °C, it makes no sense to write results of the type 36.25 °C or 22.175 °C, for example.
All figures in a result must be significant. This same general criterion must be respected when operating with experimental data; it is a matter of common sense that by simply operating with numbers it is not possible to improve the precision of the results if they have an experimental basis. When a result is written so that all of its figures are significant, it itself provides information about the precision of the measurement.
If the sources of error are only random, that is, if they sometimes influence the measurement result by excess and sometimes by default, it can be shown that the value that is closest to the true value is precisely the mean value. This is due to the fact that when averaging all the results, the excess errors will tend to be compensated by the default errors and this will be all the more true the greater the number of times the measurement is repeated. For this reason, the usual procedure to establish a reliable value of a quantity M and its corresponding uncertainty is as follows:
1) Repeat the measurement operation of M n times and record the results M 1 , M 2 … M n
2) Calculate the arithmetic mean M of all of them:
|▫ M =||M1 + M2 + … + Mn|
3) Calculate the mean deviation ΔM, that is, the arithmetic mean of the absolute values of the deviations of the different measurement results with respect to their mean M:
|ΔM =|||M 1 – M | + |M 2 – M | + … + |M n – M ||
▫ Taking the absolute values and not their sign is equivalent to deliberately placing oneself in the most disadvantageous situation in which the errors do not cancel each other
5) Consider ΔM as a bound or error limit, so that the true value M of the measured magnitude will be between the extreme values M – ΔM and M + ΔM:
▫ M – ΔM < M < M + ΔM
6) Express the result in the form:
▫ M ± ΔM
Sometimes, if you work with a large number n of measurements, it is useful to arrange the results and their errors in an orderly manner in the form of a table. In the following example, the measurement of the time of fall of a ball made by a chronometer that appreciates up to two tenths of a second is collected.
Use of significant figures
To correctly handle the results expressed by means of significant figures, it is necessary to follow the following rules:
When zeros appear as the first figures of a result, they are not considered as significant figures, therefore the number of significant figures of a result is the same, whatever the unit in which it is expressed. Thus, for example, if you want to express in meters the result of measuring a length l of 3.2 cm with a ruler that measures up to the millimeter, you will have:
I = 3.2cm = 0.032m
And the result will still have two significant figures. For this reason it is customary to write it using the powers of 10:
I = 3.2 10 -2m
When zeros appear as the last figures of integers, this does not imply that they must necessarily be considered as significant figures. Thus, for example, when the above quantity is expressed in microns, the result is I = 32,000 µ (1 µ = 1 thousandth of a mm = 10 -3 mm); this does not mean that the result has five significant figures, but only two in this case. To avoid this type of confusion, the most appropriate thing is to write the data resorting, again, to the powers of 10:
I = 3.2 10 -5
You may wonder how to carry over significant figures in operations such as multiplication or division. When you have an electronic calculator it seems as if you are tempted to write the results with as many decimal places as they appear on the screen, but this is mostly meaningless. Take the following case as an example:
It is desired to find the surface area of a strip of paper. Its length and width are measured using a ruler that measures up to millimeters, obtaining 53.2 and 4.1 cm, respectively. Multiplying both results gives:
S = 53.2 4.1 = 218.12 cm²
But how many of these figures are truly significant? The rule that follows is the following: the number of significant figures of a product (or a quotient) between data corresponding to measurement results cannot be greater than that of any of the factors. In the present case 4.1 has two significant figures, so the result would strictly be written as:
S = 220 cm² = 22.10 cm²
When, as in this example, it is necessary to round off a figure because it is not significant, it is disregarded if it is equal to or less than half the value of the unit of the last significant figure and if it is higher, it is considered increased by one unit. Since in the present example 8 is above the half unit of the tens (10/2) the result has been written as 220 cm² and not as 210 cm²