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Error & uncertainty

The importance of error and uncertainty

As part of the assessment for the Individual Scientific Investigation students do need to take errors and uncertainties into account. They should know (or be able to estimate) the uncertainty associated with each piece of apparatus they use and have an awareness of which might significantly affect the result of their experiment. They should also have an awareness of uncertainties due to assumptions made. For example: Have all the chemicals reacted? How pure are the chemicals? Are there other products formed or side-reactions taking place, etc.? Students are expected to be able to work out the percentage error by comparing their result with the ‘literature’ or data book value. Students are also expected to be able to work out the total uncertainty by summing all the individual uncertainties.

A good guide, The Beginner's Guide to Uncertainty of Measurement by Stephanie Bell can be downloaded free from the National Physical Laboratory.

Direct student access

Note that students have direct access to a version of this page written specifically for students (as opposed to teachers) at Error & uncertainty in Complete course for students.

Error

The error is the difference between the result obtained and the generally accepted 'correct' result found in the data book or other literature. If the 'correct' result is available it should be recorded and the percentage error calculated and commented upon in the conclusion. Without the 'correct ' value no useful comment on the error can be made.

The percentage error is equal to:

(the difference between the value obtained and the literature value ÷ the literature value)  x 100

Uncertainty

Uncertainty occurs due to the limitations of the apparatus itself and the taking of readings from scientific apparatus. For example, during a titration there may be four separate pieces of apparatus, each of which contributes to the uncertainty.

e.g. when using a balance that weighs to ± 0.001 g the uncertainty in weighing 1.500 g will equal

(0.001 ÷ 1.500) x 100 = 0.0667%

Similarly a pipette measures 25.00 cm3 ± 0.04 cm3.

The uncertainty due to the pipette is thus (0.04 ÷ 25.00) x 100 = 0.16%

Assuming the uncertainties due to the burette and the volumetric flask are 0.50% and 0.10% respectively the overall uncertainty is obtained by summing all the individual uncertainties:

Overall uncertainty = 0.067 + 0.16 + 0.50 + 0.10 = 0.83% ≈ 1.0%

Hence if the answer is 0.750 mol dm-3 then the uncertainty is 1.0% of 0.750 mol dm-3

The answer should be given as 0.750 ± 0.008 mol dm-3 or, even better, as 7.50 ± 0.08 x 10-1 mol dm-3

Sometimes it is not possible to give precise percentage uncertainties. For example, in a titration the end-point taken could vary according to the person carrying out the titration. In such cases you should train your students to state the colour change taken (e.g. until a faint permanent pink colour was obtained). Any assumptions made which can add to the uncertainty (e.g. the specific heat capacity of the solution was taken to be the same as that for pure water) should also be stated.

You will need to train students to be able to recognize when the uncertainty of one of the measurements is much greater than that of the others. This will then have the major effect on the uncertainty of the final result, and the approximate uncertainty can be taken as being due to that quantity alone. This is often the case whenever a thermometer is used. Compared with an analytical balance, or accurate and precise volumetric apparatus, the percentage uncertainty in the temperature readings will be much greater, particularly when the temperature difference is relatively small.

Consider an experiment to measure the heat required to raise the temperature of 500 g of water by 6.0 oC.

If the uncertainty in the mass of water is ± 1 g then the percentage uncertainty in the mass is only ± 0.2%. However if the thermometer reads to ± 0.5 °C, and two readings were taken to measure the 6.0 oC temperature rise, then the absolute uncertainty is ± 1.0 °C. This gives a percentage uncertainty for the temperature rise of 16.7%, which is much higher than any of the other uncertainties inherent in the experiment. Students need to bear this in mind when they make realistic suggestions to improve their experimental method.

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