What is the relative and what is the absolute error in this question?

Answer

Dear Lucy,

Before I give an answer based on misunderstanding(s), let’s check the conventions used:

1. May I assume that the numbers in brackets represent the absolute measurement error on both measuring instruments (resp. a 250 ml measuring cup, and presumably a 50 ml pipette)?

2. Can I also assume that the measuring cup is first completely filled to the 250 ml mark, and that therefore the (relatively large, hence my question) measurement error on the measuring cup itself must also be taken into account?

3. The fact that the error on each of the measurements is only shown by means of 1 number in brackets is actually based on the assumption that the measurement operations
(resp. measuring the volume in the volumetric flask, and each of the 4 pipette operations of 50 ml) Samples from normally distributed distributions are with the central value (Mu) the nominal value, and the standard deviation (Sigma) the error limit in parentheses.

A first remark about this: some use 3 sigma values ​​as error limits instead of the sigma values ​​themselves!
That is already a convention that must be pointed out again and again.

If these assumptions are all correct then the considered operator is simply a sum of 5 normally distributed quantities,
and then the standard deviation of the obtained statistical distribution satisfies the relevant theorem:
This is therefore normally distributed, with the Sigma:

Numerically this becomes:

This gives the absolute error limit according to the notation used. This is even independent of the chosen convention with regard to the distribution on the Normal distributions
(ie 3 sigma error limits also give rise to 3 sigma error limits).

As already indicated, in this particular issue, the error on the volumetric flask drowns out the (relatively much smaller) errors on the pipette operations!
As a result, the absolute error on the remaining 50 ml in the volumetric flask remains practically the same: about 12 ml.

So the relative error on this becomes 12 ml/50 ml = 24%.

Philippe J. Roussel
Senior Reliability Research Engineer
imec