RELATIVE ERROR: FORMULAS, HOW IT IS CALCULATED, EXERCISES - PHYSICAL - 2025

The relative error of a measurement, denoted as ε, is defined as the quotient between the absolute error Δ X and the measured quantity X. In mathematical terms it remains as ε _r = ΔX / X.

It is a dimensionless quantity, since the absolute error shares the same dimensions with the quantity X. It is often presented in terms of percentage, in this case we speak of the percentage relative error: ε _r% = (ΔX / X). 100%

Figure 1. Every measurement always has a degree of uncertainty. Source: Pixabay.

The word "error" in the context of physics, does not necessarily have to do with mistakes, although of course it is possible that they occur, but rather with the lack of certainty in the result of a measurement.

In science, measurements represent the support of any experimental process, and therefore must be reliable. Experimental error quantifies how reliable a measure is or not.

Its value depends on various factors, such as the type of instrument used and the state it is in, whether a suitable method has been used to carry out the measurement, the definition of the object to be measured (the measurand), whether there are faults in the calibration of the instruments, the skill of the operator, the interaction between the measurand and the measurement process, and certain external factors.

These factors result in the measured value differing from the actual value by a certain amount. This difference is known as uncertainty, uncertainty or error. Every measure that is carried out, no matter how simple, has an associated uncertainty that naturally always seeks to reduce.

Formulas

To obtain the relative error of a measure, it is necessary to know the measure in question and its absolute error. The absolute error is defined as the modulus of the difference between the real value of a quantity and the measured value:

ΔX = -X _real - X _measured -

In this way, although the real value is not known, there is an interval of values ​​where it is known to be: X _measured - Δx ≤ X real ≤ X _measured + Δx

ΔX takes into account all possible sources of error, each of which must in turn have an assessment that the experimenter assigns, considering the influence they may have.

Possible sources of error include the appreciation of the instrument, the error from the measurement method, and the like.

Of all these factors, there are usually some that the experimenter does not take into account, assuming that the uncertainty introduced by them is very small.

Appreciation of a measuring instrument

Since the vast majority of experimental determinations require the reading of a graduated or digital scale, the instrument's error of appreciation is one of the factors that must be taken into account when expressing the absolute error of the measurement.

The appreciation of the instrument is the smallest division of its scale; for example, the rating of a millimeter ruler is 1 mm. If the instrument is digital, the appreciation is the smallest change that has the last digit to the right shown on the screen.

The higher the appreciation, the lower the precision of the instrument. On the contrary, the lower the appreciation, the more accurate it is.

Figure 2. The rating of this voltmeter is 0.5 Volts. Source: Pixabay.

How is the relative error calculated?

Once the X measurement has been made and the absolute error ΔX is known, the relative error takes the form indicated at the beginning: ε _r = ΔX / X or ε _r% = (ΔX / X). 100%.

For example, if a length measurement has been made, which yielded the value of (25 ± 4) cm, the relative percentage error was ε _r% = (4/25) x 100% = 16%

The good thing about relative error is that it allows you to compare measurements of both the same and different magnitudes and determine their quality. In this way it is known whether the measure is acceptable or not. Let's compare the following direct measures:

- An electrical resistance of (20 ± 2) ohms.

- Another (95 ± 5) ohm.

We might be tempted to say that the first measure is better, since the absolute error was smaller, but before deciding, let's compare the relative errors.

In the first case, the percentage relative error is ε _r% = (2/20) x 100% = 10% and in the second it was ε _r% = (5/95) x 100% ≈ 5%, in which case we will consider this measure of higher quality, despite having a higher absolute error.

These were two illustrative examples. In a research laboratory the maximum acceptable percentage error is considered to be between 1% and 5%.

Solved exercises

-Exercise 1

In the packaging of a piece of wood, the nominal value of its length is specified in 130.0 cm, but we want to make sure of the true length and when measuring it with a tape measure we obtain 130.5 cm. What is the absolute error and what is the percentage relative error of this single measure?

Solution

Let's assume that the factory-specified value is the true value of the length. You can never really know this, since the factory measurement also has its own uncertainty. Under this assumption, the absolute error is:

Note that Δ X is always positive. Our measure is then:

And its percentage relative error is: e _r% = (0.5 / 130.5) x 100% ≈ 0.4%. Nothing bad.

-Exercise 2

The machine that cuts the bars in a company is not perfect and its parts are not all identical. We need to know the tolerance, for which we measure 10 of your bars with a tape measure and forget about the factory value. After taking the measurements, the following figures are obtained in centimeters:

- 130.1.

- 129.9.

- 129.8.

- 130.4.

- 130.5.

- 129.7.

- 129.9.

- 129.6.

- 130.0.

- 130.3.

What is the length of a bar from this factory and its respective tolerance?

Solution

The length of the bar is properly estimated as the average of all the readings:

And now the absolute error: since we have used a tape measure whose appreciation is 1 mm and assuming that our eyesight is good enough to distinguish half of 1 mm, the error of appreciation is set at 0.5 mm = 0.05 cm.

If you want to take into account other possible sources of error, of those mentioned in previous sections, a good way to assess them is by means of the standard deviation of the measurements made, which can be found quickly with the statistical functions of a scientific calculator:

σ _n-1 = 0.3 cm

Calculation of absolute error and relative error

The absolute error Δ L is the instrument's error of appreciation + the standard deviation of the data:

The length of the bar is finally:

The relative error is: ε _r% = (0.4 / 130.0) x 100% ≈ 0.3%.

References

1. Jasen, P. Introduction to the theory of measurement errors. Recovered from: fisica.uns.edu.ar
2. Laredo, E. Laboratory of Physics I. Simón Bolívar University. Recovered from: fimac.labd.usb.ve
3. Prevosto, L. On physical measurements. Recovered from: frvt.utn.edu.ar
4. Technological University of Peru. General Physics Laboratory Manual. 47-64.
5. Wikipedia. Experimental error. Recovered from: es.wikipedia.org