This section looks at how to deal with errors arising from several sources in an experimental procedure.
Propagation of errors
Whenever experiments are carried out involving more than one step, the uncertainties of each step must accumulate. It is not usually appropriate to add uncertainties in measurements of different types as absolute errors. Instead, the percentage error accrued at each measurement should be added together.
Once all of the percentage errors have been added together, this percentage can be applied to any final answer to determine the absolute uncertainty in the final value.
The best way to understand the process is to see how the uncertainties are propagated through a simple procedure, such as the preparation of a standard solution of potassium hydrogen phthalate (relative mass 204.23).
Step 1 The mass of solid needed for preparation of the solution is weighed out and then recorded.
- Mass of potassium hydrogen phthalate = 1.66 ± 0.005 g
Step 2 The solid is dissolved in approximately 100 ml water and transferred to a volumetric flask, which is then made up to the 250 ml mark with distilled water. (the volumetric flask has a tolerance of ± 0.46 ml written on the side)
- Volume of solution = 250 ± 0.46 ml
Calculation: The solution concentration = mass/(relative mass x volume in litres)
= 1.66/(204.23 x 0.25) = 0.052 mol dm-3
Absolute error = final concentration x percentage error/100 = 0.052 x 0.372/100 = 0.0002 mol dm-3
In reality this value is too small to be included, as it is smaller than the number of decimal places in the final concentration. This is to be expected, as preparation of a standard solution should be accurate.
Approximation of errors
In situations where the errors arising from one measurement are much larger than from another, it is a safe approximation to ignore the smaller value.
In the example given above both of the error margins were fairly similar and carry equal importance. However, if the expermiment is continued by extracting a 25 ml aliquot using a pipette and timing a reaction, such as the time taken for a sulfur precipitate to appear and obscure a mark placed below the reaction vessel when 1.0 ml of 2 mol dm-3 hydrochloric acid is added.
- The uncertainty in the pipette = 25 ± 0.04 ml
- The uncertainty in HCl addition = 1.0 ± 0.1 ml
- The uncertainty in time taken = 32 ± 2 s
- The percentage uncertainty in the pipette = 0.04/25 x 100 = 0.16%
- The percentage uncertainty in the HCl volume = 0.1/2 x 100 = 5%
- The percentage uncertainty in the time = 2/32 x 100 = 6.25%
We can see that the uncertainly in the pipette measurement is far less than that of either the HCl volume or the time. It would be a reasonable approximation to ignore to pipette uncertainty when calculating the overall uncertainty in the final value.
It is important to realise that errors arise from all sources, but that when one error is much larger than the others they become relatively insignificant.