Relative uncertainty formula and calculation

Relative uncertainty , often represented by the symbol δ (the lowercase Greek letter delta) , is the ratio between the absolute uncertainty of an experimental measurement and the value accepted as true , or the best estimate of that measurement. It is a quantity that gives us an idea about how large or small the uncertainty of a measurement is compared to its magnitude.

Remember that the uncertainty of a measurement refers to the width of the range of possible values ​​within which we assume that the true value of a measurement lies. This comes from the fact that it is impossible to carry out perfect experimental measurements, completely free of error, so the best we can do is estimate its value. We do this by reporting the value of a measurement together with its uncertainty:

where x is the value of the measurement and ∆x is its absolute uncertainty. This expression is interpreted by saying that the value of the measure lies between x – ∆x and x + ∆x with a certain level of confidence.

Interpretation of relative uncertainty

In the case of relative uncertainty, the value is usually represented as a percentage, and is interpreted as saying that the actual value of the measurement is within a range of a few percent around the value of the experimental measurement.

For example, if the speed of a car traveling at 150 km/h is measured with a relative uncertainty of 5%, this is interpreted as the car’s true speed being within a 5% range of around 150 km/h. .

Importance of relative uncertainty

Relative uncertainty, sometimes also called relative error (although this term is not strictly correct), allows you to put the uncertainty of a measurement into perspective. For example, having an absolute uncertainty of 0.5 cm when measuring the length of a 400 m long running track is not a serious problem. One could say that the uncertainty of the measurement is relatively small, since the magnitude of the measurement is large compared to the uncertainty.

On the other hand, if we have the same uncertainty of 0.5 cm when measuring the size of a mobile phone that measures 10 cm, then it is easy to see that this uncertainty is much higher, despite the fact that both absolute uncertainties are the same.

On the other hand, if instead of comparing the absolute uncertainties of two measurements we compare their relative uncertainties, then we will have a direct idea of ​​which of the two measurements has a lower uncertainty.

Formula for calculating relative uncertainty

In general terms, the relative uncertainty is calculated as the ratio between the absolute uncertainty and the magnitude of the measurement. That is to say:

Relative uncertainty units

Contrary to absolute uncertainty, which is reported in the same units as the measurement to which it refers, relative uncertainty has no units; It is therefore a dimensionless quantity. This is one of the reasons that makes it possible to compare the relative uncertainty of different measurements of different physical magnitudes, which are obviously expressed in different units.

On the other hand, in some cases it is customary to express the relative uncertainty as a percentage, in which case it is accompanied by the symbol %.

How to calculate the relative uncertainty?

The formula for calculating relative uncertainty is very simple. However, its application depends on the context in which it is used, since absolute uncertainty can be defined in different ways.

Relative uncertainty of reported values

In those cases in which you want to calculate the relative uncertainty of a measurement reported in the literature, you usually already have everything you need to calculate the relative uncertainty, since these values ​​are always reported together with their absolute uncertainty.

Example

The density of water is 997 ± 1kg/m ³ , so x = 997 1kg/m ³ (the magnitude) and ∆x = 1 1kg/m ³ (the absolute uncertainty), so the relative uncertainty in this case is:

Relative uncertainty of individual experimental measurements

What to do when we want to determine the relative uncertainty of a single experimental measurement? In these cases, we take the error of appreciation of the measuring instrument with which we are working as relative uncertainty. For example, if we are measuring the length of a table with a tape measure that has an appreciation of 0.1cm (that is, 1mm), then the appreciation error will be 0.05cm.

Example

We weigh a sample of an unknown liquid on an analytical balance whose appreciation is 0.001g. The weight of the sample is 0.489g. If we want to determine the relative uncertainty, we take half the estimate as uncertainty, so we report the mass as 0.489 ± 0.0005g and the relative uncertainty of the measurement will be:

Relative uncertainty for a set of experimental measurements

To obtain a better estimate of the true value of a measurement and to counteract the effect of random errors , the measurement of the same quantity is often carried out several times. In these cases, statistical tools are used to estimate the best value of the measure.

In this sense, the mean of the experimental data is taken as the accepted value of the measurement, and the standard deviation of the measurements with respect to the mean is usually taken as the uncertainty.

This is given by the equation:

This equation may seem complex, but we don’t really need to carry out the calculations, since any scientific calculator comes equipped with statistical functions that allow you to enter individual data and produce the value of the standard or standard deviation with the press of a button. pair of keys.

Example

Suppose a biology lab professor asks his students to measure the pH of a bacterial culture broth that has been incubating for the past 48 hours. There are 15 groups of students who carried out the experiment independently and whose results are summarized in the following table:

Using a scientific calculator or a spreadsheet such as Excel, the mean and standard deviation of the measurements are determined. The result is 4.42 ± 0.15. So, the relative uncertainty will be, in this case:

References

Bohacek P, and Schmidt I G. (nd). Integrating Measurement and Uncertainty into Science Instruction. Retrieved from https://serc.carleton.edu/sp/library/uncertainty/index.html

The mathematical treatment of the measurement results. (n.d.). Retrieved from https://espanol.libretexts.org/@go/page/1798

The measures. (2020, October 30). Retrieved from https://espanol.libretexts.org/@go/page/1796

National Institute of Standards and Technology (2009). NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. Retrieved from https://www.nist.gov/pml/nist-technical-note-1297

Stanbrough, J,L, (2008), Uncertainty Dictionary, Retrieved from http://www,batesville,k12,in,us/physics/apphynet/measurement/UncertaintyDictionary,html